Cooling Dynamics of a Brownian Particle and the Markovian Mpemba Eﬀect

Cooling dynamics of a Brownian particle and the Markovian Mpemba eﬀect

by Lisa Zhang

B.A.Sc., The University of British Columbia, 2013

Thesis Submitted in Partial Fulﬁllment of the Requirements for the Degree of Master of Science

in the Department of Physics Faculty of Science

Copyright in this work rests with the author. Please ensure that any reproduction or re-use is done in accordance with the relevant national copyright legislation. Approval

Name: Lisa Zhang Degree: Master of Science (Physics) Title: Cooling dynamics of a Brownian particle and the Markovian Mpemba eﬀect Examining Committee: Chair: Malcolm Kennett Associate Professor

John Bechhoefer Senior Supervisor Professor

David Sivak Supervisor Assistant Professor

Barbara Frisken Internal Examiner Professor

Date Defended/Approved: September 26, 2018

ii Abstract

I experimentally and numerically investigate a Mpemba-like behaviour in a colloidal particle diffusing in a bath under the influence of an externally applied potential. Multiple particle trajectories were recorded and used to obtain the spatial probability distribution of the particle at different times. As a temperature quench is applied, the probability distribution shifts from one equilibrium distribution to another that correspond to the initial and final temperatures in the process, respectively. I experimentally and numerically study the change in value of a measure for the degree of cooling calculated from the measured probability distributions that is compatible with the characteristics of temperature when the system is at equilibrium, and can equally be applied to a system that is out-of-equilibrium. I demonstrate that probability distributions can be estimated using a limited amount of data at sufficiently high accuracy to permit experimental observation of the Markovian Mpemba effect.

Keywords: Markov Process; Mpemba eﬀect; stochastic thermodynamics

iii Contents

Approval ii

Abstract iii

Table of Contents iv

List of Tables vi

List of Figures vii

List of Variables ix

1 Introduction 1 1.1 Possible explanations for the Mpemba effect ...... 2 1.2 Extrinsic factors affecting the cooling of water ...... 5 1.3 Observation of Mpemba-like effects in other systems ...... 6 1.4 Mpemba-like effect for a colloidal particle in an external potential ...... 8 1.5 Thesis Overview ...... 10

2 Theory 12 2.1 Markov process ...... 12 2.1.1 Markovian dynamics ...... 13 2.1.2 Long-time limit ...... 14 2.2 The Markovian Mpemba effect ...... 16 2.2.1 The distance-from-equilibrium function ...... 16 2.2.2 A Three-State Model ...... 21 2.2.3 Markovian Mpemba effect in a Continuous-state System ...... 25 2.2.4 How the condition for the Markovian Mpemba effect could occur . . 26 2.3 Using the Langevin Equation for simulations ...... 27 2.3.1 Solving the deterministic part of the Langevin equation ...... 28 2.3.2 Solving the full Langevin equation ...... 29 2.4 Physical Model ...... 30 2.5 Experimental Model ...... 32

iv 2.5.1 Minimum width of the potential ...... 33 2.5.2 Power spectrum for free diﬀusion ...... 36 2.5.3 Implementing the Physical Model ...... 37

3 Experiment 43 3.1 Experimental Setup ...... 43 3.2 Experimental Protocol ...... 47 3.2.1 Estimating the time-dependent distribution ...... 48 3.2.2 Data resampling ...... 49 3.3 The effect of height fluctuations on the particle diffusion coefficient . . . . . 50 3.4 Benchmarking the Feedback Trap ...... 54 3.4.1 Free diffusion ...... 54 3.4.2 Harmonic potential ...... 58 3.4.3 Harmonic potential at different positions ...... 60

4 Results 63 4.1 Sampling from a distribution ...... 64 4.2 Estimating the equilibrium distribution and potential ...... 67 4.2.1 Optimum temperature ...... 68 4.2.2 Why there is a minimum allowable barrier height ...... 71 4.2.3 Trajectory length ...... 73 4.3 Observation of the Markovian Mpemba Eﬀect ...... 74 4.3.1 Experimental Boltzmann distribution and potential ...... 74 4.3.2 Noise ﬂoor ...... 77 4.3.3 Measured distance from equilibrium ...... 78

5 Conclusion 83

Bibliography 87

Appendix A Derivation of Master equation discretization 91

Appendix B Testing the distance-from-equilibrium function 93

v List of Tables

Table 1.1 Possible mechanisms for the Mpemba eﬀect ...... 3

vi List of Figures

Figure 2.1 “Potential” of absolute energy levels and kinetic barriers ...... 15 Figure 2.2 Mapping temperature to distance from equilibrium ...... 20 Figure 2.3 Illustration of a convex function ...... 20 Figure 2.4 Schematic of energy landscape: Three-state model ...... 22 Figure 2.5 Conﬁguration space: Three-state model ...... 23 Figure 2.6 Cooling pathways: Three-state model ...... 24 Figure 2.7 Predictor of occurrence of the Markovian Mpemba eﬀect ...... 24 Figure 2.8 Distance from equilibrium versus time: three-state model ...... 25 Figure 2.9 Deterministic Euler’s Method ...... 29 Figure 2.10 Physical model of temperature quench ...... 31 Figure 2.11 Minimum width of the potential well ...... 34 Figure 2.12 Constraint on curvature ...... 39 Figure 2.13 Potential and corresponding Boltzmann distributions for three temperatures ...... 40 Figure 2.14 Convergence of Fokker-Planck Solution ...... 41 Figure 2.15 Evolution of probability distributions ...... 42

Figure 3.1 Schematic of experimental setup ...... 44 Figure 3.2 Image of the particle ...... 46 Figure 3.3 Colourmap image ...... 47 Figure 3.4 Schematic of data structure ...... 48 Figure 3.5 Estimating error bars using data resampling ...... 51 Figure 3.6 Diffusion coefficient vs height ...... 52 Figure 3.7 Vertical probability distribution ...... 53 Figure 3.8 Effect of varying diffusion coefficient on the Markovian Mpemba effect 55 Figure 3.9 Trajectory of a Browniam Particle ...... 57 Figure 3.10 Power spectrum of free diffusion: no windowing ...... 58 Figure 3.11 Power spectrum of free diffusion: with windowing ...... 59 Figure 3.12 Power spectrum of motion in a harmonic potential: α = 0.1 . . . . 60 Figure 3.13 Harmonic potential at different locations ...... 62

Figure 4.1 Sampling the initial position ...... 65

vii Figure 4.2 Distribution of sampled positions ...... 66 Figure 4.3 Boltzmann distributions on log-scale ...... 68 Figure 4.4 Simulation to ﬁnd optimal temperature ...... 70 Figure 4.5 Estimating probability distributions ...... 72 Figure 4.6 Simulation to choose trajectory length ...... 75 Figure 4.7 Measured potential and equilibrium distribution ...... 76 Figure 4.8 Experimental observation of the Markovian Mpemba Eﬀect . . . . 79 Figure 4.9 Convergence to the Fokker-Planck solution ...... 82

Figure B.1 Discrete energies and energy barriers for three-state model...... 94

Figure B.2 Plot of a2 versus T...... 95

Figure B.3 Plot of Distance versus time for Tb = 0.1, and Th = 1 and Tc = 0.5. 95

Figure B.4 Plot of Distance versus time for Tb = 0.1, and Th = 3 and Tc = 0.5. 96

Figure B.5 Plot of Distance versus time for Tb = 0.1, and Th = 5 and Tc = 3. . 97

Figure B.6 Plot of Distance versus time for Tb = 0.1, and Th = 10 and Tc = 1.4. 97

viii List of Variables

α Dimensionless parameter, α = ts/tr

χ Standard deviation of observation noise

χn Observation noise

∆¯xn Measured one-step displacement √ ` Fundamental length scale, ` = Dts √ `0 Fundamental length scale, `0 = 2D∆t

η Dynamic viscosity of water

γ Drag constant

D Entropic distance function

π Boltzmann distribution

ρ Probability density

τ Fundamental time scale, τ = ts

~p Probability distribution of states

ξ(F )(t) Thermal force

ξn Thermal noise

ζ Drag constant a2 Coeﬃcient of the term with the second-largest eigenvalue in the eigenfunction expansion of ~p(t)

Bij Energy barrier between states i and j

D Diﬀusion coeﬃcient

ix d Particle diameter

F (x) The negative slope of the potential at position x h Time step size

∂2U k Maximum curvature, maxx ∂x2 kB Boltzmann’s constant m Mass of particle

P (x1; x2; x3; ...; xn) Joint probability density pi ith element in ~p

Tb Bath temperature tc Duration of camera exposure tr Relaxation time, tr = γ/k ts Duration of a feedback cycle

U Energy landscape

Ui Discrete energy level

Wij Transition probabilities per unit time

x Chapter 1

Introduction

The ability to transfer heat to an from its environment is a fundamental property of any physical system. Every physical object can transfer heat to another object that is at a lower temperature; the net flow of heat ceases when both objects reach the same temperature [1]. In the absence of external inputs, a hot object in a cool environment will experience a decrease in temperature. A key aspect of heat transfer is the rate at which it occurs. The rate of heat flow is affected by whether heat is transferred by conduction, through direct contact, without motion of particles; radiation, through empty space; and convection, through motion of particles [2]. In physical systems, it can be impossible to determine exactly what proportion of heat is transferred by conduction, radiation, and convection, which sometimes leads to unintuitive consequences. In recent years, there has been an increasing interest in studying deviations from quasi-equilibrium behaviour occurring when the rate of heat transfer is increased. The first serious discussions and analyses of anomalous heat transfer emerged during the 1960s with the work done by a student named Erasto Mpemba. In 1963, Erasto Mpemba put boiled milk with sugar into the refrigerator without cooling it at the same time that another student put unboiled milk with sugar into the refrigerator, and discovered that his turned into ice cream first [3]. Later, he carried out an experiment using glasses of water at 100°C and 35°C, and found by qualitatively observing the amount of ice formed, as well as determine when ice was first formed, that the initially hotter water froze first [3]. There is evidence that suggests this phenomenon had a long history prior to Mpemba’s rediscovery of it. This anomaly has been used to the benefit of ice-cream makers [3] and ice fishermen [4, 5], and Burridge and Linden [5] and Jeng [4] point to excerpts of Aristo- tle’s Meteorologica, Bacon’s Novum Organum, and Descartes’ Discourse on Method, Optics, Geometry, and Meteorology that indicate this effect was known to philosophers well before the scientific revolution. This effect was recently reintroduced into the scientific literature with the publication of Mpemba et al.’s experimental paper [3]. In a graph in that paper showing the time to freezing plotted versus the initial temperature of a sample of water, it

1 can be seen that that the time ice first forms is not monotonic with initial temperature, but rather, starts to decrease at temperatures above about 30°C. However, a strict increase in time that ice first forms with increasing initial temperature is expected using classical thermodynamics, and questions have been raised about whether identical systems and processes are being compared. What is intriguing about Mpemba et al.’s [3] experiments is that extensive measures are taken to mitigate the effects of mechanisms, such as evaporation, dissolved gases, and conduction, that are known to play a role in expediting hot water to cool faster, yet the effect still occurred. Mpemba et al. explain there is an insufficient amount of mass loss due to evaporation to account for the amount of cooling necessary for initially hotter water to freeze before initially cooler water. They boiled all the samples of water regardless of initial temperature in the experiment to remove dissolved gases in both the initially hot and initially cold samples, to make the samples the same. They also insulated the bottom of the sample container using a block of polystyrene foam, in order to make the thermal contact with the refrigerator floor the same for all samples, to make conduction the same. Mpemba et al. draws attention to a paradox in applying classical thermodynamics in this type of situation. Most studies in the field of classical thermodynamics have only focused on gradual changes in thermodynamic equilibrium. Such an assumption is unsatisfactory here; Mpemba et al. show that a single temperature cannot be assigned to the system. Graphs in their paper of the temperature near the top and bottom of the sample plotted versus time show that temperature gradient exists during cooling. Mpemba et al.’s hypothesis for why hot water sometimes freezes faster than cold water is based on their finding that cooling occurs mainly through the top surface of the water, whose temperature they found can be up to 14° higher than the temperature at the bottom of the sample chamber, due to a temperature gradient created by convective forces. The cooling rate, calculated using Newton’s law of cooling for the the top-surface temperature and the temperature of the surroundings, would be higher than if a middle-of-the-range temperature was used instead of the top-surface temperature.

1.1 Possible explanations for the Mpemba eﬀect

Since Mpemba et al.’s [3] 1969 paper was published, a considerable amount of literature has been published on the Mpemba effect. Table 1.1 lists and describes several of the mechanisms that have been identified in those studies as contributing factors to the Mpemba effect, including evaporation, cooling history, supercooling, convection, and the environment. Con- tradictory theoretical and experimental evidence make it a challenge to determine whether any of the listed attributes have a sufficiently large effect to result in the Mpemba effect. Evaporation, the first attribute, was the explanation that was given in another paper which was independently published at around the same time as the Mpemba et al. paper

2 Attribute Hypothesis Reference

Evaporation Smaller volume freezes faster [3, 4, 6–9] Latent heat of vaporization Cooling history Gases expelled when heated [4, 7, 8, 10, 11] Water cooled from the same temperature cools faster when boiled prior Supercooling The end of supercooling is when ice is formed [4, 8, 11–14] Convection Hotter top loses heat faster [3, 4, 15] Environment Convection currents in air [3, 4, 8, 16] Thermal contact with refrigerator Ambient temperature

Table 1.1: Possible mechanisms for the Mpemba eﬀect

[17]. In this other paper, to determine the effects of evaporation, Kell [6] calculated the amount of cooling that can be attributed to mass reduction due to evaporation, as well as the latent heat of vaporization. Providing both theoretical and experimental evidence, he found that, in a system that cools primarily by evaporation, a sample of initially hotter water can freeze faster than a sample of initially cooler water. This view is supported by Vynnycky et al. [9] who presented a more detailed model of evaporation building on Kell’s model. However, there are numerous experimental conditions for which evaporation is not necessarily the main cooling method. As Freeman [7] confirmed, Kell’s theory did not apply to his experimental conditions. Similarly, Mpemba et al. notes that only a small amount of water evaporated from their samples, despite the Mpemba effect being observed, and suggests that it is likely that other mechanisms are involved [3]. Despite the limitations of Kell’s theory, Jeng believes that evaporation is one of the better hypotheses for why the Mpemba effect occurs [4]. The second attribute in Table 1.1 is cooling history. The cooling history could be important because the solubility of impurities in water changes with temperature [4, 8, 10, 11], and solutes can lower the freezing temperature of water such that it takes longer to freeze [18]. The importance of the cooling history to the Mpemba effect remains unclear due to contradicting claims by different experimental groups. Freeman [7] concluded that the cooling history is what causes the Mpemba effect, based on his finding that the Mpemba effect only occurred in samples containing carbon dioxide. On the other hand, Brownridge

[11] reported that dissolved CO2 and minerals in water did not aﬀect the freezing time of the water. Mpemba et al. suggested that they had removed most of the dissolved air by using recently boiled water [3]; but, a calculation done by Auerbach [8] suggests that gas reenters the water sample soon after heating. Katz’s [10] in-depth analysis of the eﬀect of solutes

3 on cooling rate appears to provide the only quantitatively-sound explanation. He points out that the amount of solute that precipitates out during heating leads to minute changes in solute concentration. He proposes the relevance of a process called zone refining that makes it possible for a small change in solute concentration to have a disproportionately large effect leading to the Mpemba effect. In zone refining, solutes can be incorporated more slowly than water into ice, such that solute concentration is higher ahead of the freezing front than in surrounding regions [10, 19]. Katz calculates the concentration of solutes at the freezing front, and uses it to predict the temperature at which water will freeze, with which predictions of the Mpemba effect can be made. However, he did not do experiments to verify this theory. The third attribute in Table 1.1 is supercooling. The amount of supercooling of two seemingly identical samples of water can vary, which can affect when freezing occurs for a particular sample [4, 8, 11, 12, 14]. This property could lead to identical samples of water freezing at different times randomly. In an experimental investigation of supercooling, Auer- bach [8] found that initially cooler water actually supercools more, resulting in its taking longer to freeze than a sample of initially hotter water under some circumstances. Unlike Auerbach, Dorsey [14], had previously argued that boiling water causes it to supercool more. A broader perspective has been adopted by Jeng [4], who suggests that Auerbach’s result is unrelated to the Mpemba effect, but rather, is likely an artifact of the random freezing times of different samples. In another major study, Brownridge [11] observed that freezing can occur at the same supercooled temperature after reheating. Using this prior knowledge about his samples, Brownridge claims to have been able to select samples that exhibit the Mpemba effect with certainty. In view of the clear relationship between the time it takes for water to freeze and supercooling, one may suppose the supercooling is a possible explanation for the Mpemba effect; but, with the high sensitivity of the supercooling temperature to impurities [12], this observation will be difficult to reproduce. The fourth attribute in Table 1.1 is convection. As previously mentioned, Mpemba et al.’s [3] primary hypothesis for why the Mpemba effect occurs is convection. In contrast, a detailed theoretical model that included convection as the only mechanism for cooling by Vynnycky et al. [15] did not find that the Mpemba effect would result. However, their experimental results had some deviations from their theoretical prediction that could allow for the Mpemba effect to be observed, which they suggest could be attributed to supercooling. In the same vein, Jeng [4] questions whether systems for which convection is not the main mechanism of cooling might exhibit the Mpemba effect through other mechanisms, such as evaporation. The fifth attribute in Table 1.1 is the environment. Jeng [4] offers examples of environmental factors that affect the cooling rate of water between trials including convection currents in the surrounding air, and the melting of frost that could affect the rate of freezing of water. Some authors discussed the strategies they used to maintain control of the envi-

4 ronmental factors. For example, Mpemba et al. [3] addressed the problem of the melting of frost by adding a layer of insulation between the sample and the refrigerator and found the effect still occurred. Another example, Firth [16] hypothesized that air currents had a strong effect on the time of cooling, and took steps observe the sample without disturbing it. Similarly, differences in environmental factors between experiments were also problematic. For example, not all experiments have been done in the refrigerator making results more difficult to compare; for example, a cryostat, which could produce a larger range of ambient temperatures, was used by Auerbach [8]. Overall, there seems to be some evidence to indicate that the Mpemba effect could occur, but whether it is due to some combination of the above factors, or a not yet identified factor remains to be determined. Together, these studies provide important insights into how complex the cooling of water really is, highlighting the need for explicitly stating the experimental parameters, which is the topic of the following section.

1.2 Extrinsic factors aﬀecting the cooling of water

In Section 1.1, we discussed that there are many explanations given for why the Mpemba effect occurs, and that experimentalists sometimes disagree whether a particular attribute is positively linked to the occurrence of the Mpemba effect, or has no effect. This is likely to be the case because there are many external variables that affect the rate of change of temperature in a sample of water. In different experimental setups, the effect of these external variables could enhance or minimize a particular mechanism of cooling, leading to different outcomes and different conclusions between researchers. A few of them are listed as follows:

• Mass. A sample of water can be cooled faster by taking a smaller sample [4].

• Temperature of surroundings. Cooler surroundings cool the sample faster. Auer- bach [8] does a series of experiments that change the ambient temperature using a cryostat.

• Mode of cooling. The sample can be placed on a cold metal block and cooled by conduction [15]. Brownridge [11] suspended his sample in a vacuum chamber and cooled or heated the sample by radiation. Auerbach [8] put his sample in a bath of ethanol, and the sample was heated by conduction, while the temperature of the ethanol bath was maintained with stirring by convection. Mpemba et al. [3] placed their sample on an insulating surface and air cooled the sample by conduction, maintaining the ambient temperature with air ﬂow by convection.

5 • Container. The insulating properties can aﬀect whether the main mode of cooling is by evaporation or through heat loss following Newton’s law of cooling1 [6]. Kell [6] notes that the use of wooden pails that are more insulating than metal pails would have increased the amount of evaporation and contributed to more people observing the Mpemba eﬀect when they were being commonly used.

These factors do not need to be specifically controlled in the design of the experiment in order for the Mpemba effect to occur. The main reason for wanting to control these factors is to standardize them, in order to be able to compare results with other experimental groups more easily. When different groups use setups that have variations in these factors, their results cannot be compared directly. Burridge et al. [5] have non-dimensionalized the results from different groups in an effort to compare them. However, as the data they analyzed have limited accuracy, more experiments should be done. By way of illustration, Brownridge [11] reported that, in his experiments, changing the position of the temperature probe by as little as a few millimeters significantly changes the measured values. Even within the same experimental setup, the results show great variability. This is certainly true in the case of Auerbach’s [8] experiments in which the Mpemba effect occurred in only a little over half the experimental runs. Therefore, better experimental design is needed that yields more significant results, before others begin to attempt to replicate the experiment.

1.3 Observation of Mpemba-like eﬀects in other systems

Sections 1.1 and 1.2 have focused on studies of the Mpemba effect in water. However, what we know about the Mpemba effect is still very limited. Observations of Mpemba-like effects in systems other than water open the possibility for studying the phenomenon in an easier- to-manage system, offering reprieve from the challenging experimental conditions that are associated with studying the “classic” Mpemba effect. In this section, Mpemba-like effects that have been observed in systems including magnetoresistant manganites [21], clathrate hydrates [22], granular fluids [23] and carbon nanotube resonators [24] are discussed. A Mpemba-like effect has been experimentally measured in magnetoresistant manganites [21]. Chaddah et al. [21] measure the time for a system that is initially far from equilibrium to reach equilibrium and compare it with that of a system that is initially closer to equilibrium. To determine how close the system is from equilibrium, they measure the magnetization, which they define to be the relative sizes of the equilibrium phase to the non-equilibrium phase regions in the system. For example, a larger magnetization means that the system is closer to equilibrium [21]. They observe that equilibrium regions start to form at nucleation sites, nucleation sites expand during relaxation, and the number and size of nucleation sites depend on the initial temperature, with more nucleation sites in

1Newton’s law of cooling states that the temperature of an object decreases at a rate proportional to the diﬀerence in temperature between the object and its surroundings [20].

6 systems at a higher initial temperature [21]. Then, under certain conditions, it is possible that a system that is initially farther from equilibrium can sometimes reach equilibrium faster than a system that is initially closer to equilibrium, which is a Mpemba-like effect. A Mpemba-like effect has been experimentally measured in clathrate hydrates [22]. Ahn et al. [22] find that, for a certain temperature range, the formation time of the clathrate hydrate decreases with an increase in the initial temperature. They measure Raman spectra of the clathrate hydrate tetrahydrofuran hydrate at various temperatures and observed that at lower temperatures, the Raman peak of the O:H bond exhibited a blue shift, which indicates a shortening of the bond, and the Raman peak of the O—H bond exhibited a red shift, which indicates a lengthening of the bond [22]. This effect has been studied in water [25]. Huang et al. [25] explain that the weak O:H bond thermally expands when the temperature is increased, leading to a shortening of the strong O—H bond, increasing the amount of energy that is stored in the O—H bond, and subsequently the rate of energy release during cooling. The fact that the O—H bond changes in length in response to the change in length of the O:H bond, and the difference in energy between the strength of the O:H bond and the O—H bond, results in a Mpemba-like effect. In granular fluids modelled as hard spheres in a fluid that has slightly inelastic collisions, a Mpemba-like effect has also been found using analytical predictions and is confirmed by molecular dynamics and Monte Carlo simulations [23]. The kurtosis of the velocity distribution, with mean zero is [26]

v4 kurt(v) = , (1.1) hv2i2 where the angle brackets h·i is the mean. For a Gaussian distribution, the kurtosis is three, and excess kurtosis is generally deﬁned as the amount that the kurtosis is more than three [26], but Lasanta et al. use the deﬁnition

3 v4 excess kurt(v) = 5 − 1. (1.2) hv2i2

Lasanta et al. [23] find that the time evolution of temperature depends on the excess kurtosis. They find that a Mpemba-like effect is observed over a certain range of values of the difference in excess kurtosis divided by the difference in temperature between two systems that differ only in their initial temperature. Furthermore, they propose that the same method of analyzing the excess kurtosis can be used to predict a Mpemba-like effect in other non-Gaussian systems. A Mpemba-like effect has been found using computer simulations of flexural modes in single-walled carbon nanotube resonators [24]. Greaney et al. [24] investigate the decay in the amplitude of vibrations over time through intrinsic damping and found that vibrations with a larger initial amplitude decayed to zero faster, which is a Mpemba-like effect. They

7 find that when exciting just one flexural mode, other modes become excited too by the energy that is dissipated from the initially excited mode, and these modes all interact to increase the overall rate of dissipation. As little as three additionally excited modes is sufficient for increasing the rate of dissipation [24]. However, they acknowledge that their model is not actually of a physical system because they use classical dynamical equations at a temperature below the Debye temperature where quantum dynamics should also play a role. While the study of the Mpemba effect in clathrate hydrates, an ice-like structure, was done because they are water-like, the others do not have the same cooling mechanisms as water. This suggests that a more general theory is needed. It also suggests that there a likely to be other systems in which a Mpemba-like effect could occur, that are completely different from any of these systems. The general theory is likely to aid in the search for identifying such systems [27]. While hypothesis that are specific to water are no longer hold the key to explain the Mpemba effect, the discovery of Mpemba-like systems also point a new direction for studying the Mpemba effect. Commonalities in systems that show Mpemba-like effects can inform new hypotheses.

1.4 Mpemba-like eﬀect for a colloidal particle in an external potential

In light of the discoveries of Mpemba-like effects in diverse systems, a more generally applicable theory seems appropriate. So far, however, there has been little discussion about a general mechanism explaining how the Mpemba effect and Mpemba-like effect can be produced. This thesis will examine a recent theory [27] which applies to a general class of systems. This theory, which the authors call the “Markovian Mpemba effect”, predicts a Mpemba-like effect in Markovian systems which are systems involving thermal processes that can be modelled using Markovian dynamics [27]. Such thermal processes, known as Markovian processes, are stochastic processes in which the measured value at a future time depends on the present value but not on past values in the process [28]. The Markovian Mpemba effect predicts that a Markovian system that starts at a higher initial temperature can reach a lower final temperature before an identical system which starts at a lower initial temperature, following a temperature quench of both systems to the same low temperature. It is necessary here to clarify exactly what is meant by the Markovian Mpemba effect. Throughout this thesis, the term the Markovian Mpemba effect will be used to refer to the theory, as well as, in some cases, the observation of a Mpemba-like effect in a Markovian system. The one which is being used can be told depending on the context. There are a number of attractive features of the Markovian Mpemba effect. These are:

8 • Measurements are straightforward to make, and the quantity being measured is un- ambiguously deﬁned.

• The statement of the eﬀect is precise.

• The theory can be applied to any system with Markovian dynamics.

• Fully nondimensionalized equations are used, so results are comparable even with variability in the experimental setup.

• Large amounts of data can be taken since the system can have arbitrary design, and it is possible to design a system with short cooling time.

This theory can be used to predict a Mpemba-like effect in systems not previously known to exhibit this phenomenon as long as they fall into the class of Markovian systems described above. One such system is a colloidal particle diffusing in a bath with an externally applied potential. This system is accessible in Dr. Bechhoefer’s laboratory in which this thesis seeks to experimentally observe a Mpemba-like effect. The objectives of this research are to implement the specific details of the experiment using Labview, to simulate the experiment before it is carried out to determine the required amount of data, and to assess whether the experimental results agree with the Markovian Mpemba effect theoretical prediction The key research question of this thesis is whether or not a Mpemba-like effect could occur in the experimental system consisting of a colloidal particle diffusing in a water bath in a potential landscape. This study provides an exciting opportunity to experimentally observe a Mpemba-like effect in a colloidal particle system for the very first time. Since the trajectory of a small system during a process encodes information about changes in quantities such as heat, work, and entropy [29–31], a small system could be studied by making repeated experimental measurements of its trajectories while it is undergoing the same process. Data for this study are collected using a device that approximately imposes the desired process [32–34]. This study offers some important insights into our ability to make meaningful comparisons between two out-of-equilibrium processes. Given two out-of-equilibrium distributions, it can be difficult to conclude whether they have been estimated sufficiently accurately to make a worthwhile comparison, much less quantitatively assess which one is closer to equilibrium. The fact that this study could be done as well as it has indicates there exist robust metrics which can be used to extract useful information from seemingly poor quality measurements. This can inform future experiments to probe a different quantity when one is not working well. It is beyond the scope of this thesis to establish the equilibrium behaviour of the system and the Boltzmann equation is used to relate the potential to the measured distribution for equilibrium distributions. Due to practical constraints, initial and final equilibrium states cannot be established directly during the quenching process. The quenching process can

9 be divided into three distinct phases: the initialization, the temperature quench, and the cooling. Each phase is deemed to have been implemented by achieving a particle distribution that is known to be characteristic of each phase through any viable means. Firstly, the particle is initialized as though it were at equilibrium with a bath of a prescribed temperature, diffusing within a designed potential landscape. To achieve this, a random number generator was used to obtain a number of sampled positions from the initial equilibrium distribution, and, using a narrow harmonic well, the particle was localized at one of those positions for the initial position of every measured trajectory. Next is the temperature quench, in which the particle is released into a bath at a lower temperature, within the same potential landscape; this is achieved by instantaneously replacing the harmonic potential by the desired potential landscape. The cooling phase is the subsequent relaxation of the particle towards the new equilibrium. After a sufficient amount of time has passed, the position of the particle should be close to being distributed as the equilibrium distribution of the designed potential at the lower temperature. The particle trajectory is recorded beginning with the last measured position while the particle was localized in the harmonic well, and continued to be measured while the particle explored the designed potential, until a time that was predetermined using computer simulation such that the Markovian Mpemba effect had occurred. The reader should bear in mind that the traditional notion of temperature is based on equilibrium dynamics, while the quench being applied to the system is a far-from-equilibrium process. Hence, a metric that is analogous to temperature from the measured distributions is derived through finding the distribution of the particle over an ensemble in an analysis identical to what is used in equilibrium dynamics, thus extending the laws of classical thermodynamics to non-equilibrium processes. From calculating this metric, we can quantify the cooling of the system and determine whether a Mpemba-like effect has occurred.

1.5 Thesis Overview

My thesis is composed of five chapters, including this introductory chapter. Chapter 2 begins by laying out the theoretical foundation on which this experiment-centered thesis is based, and looks at how the Markovian Mpemba effect is defined using a Markovian description. The Markovian model including the dynamical and stationary behaviour is introduced, followed by an exact statement of the Markovian Mpemba effect. A description of what constitutes an observation is given for a finite-state model, illustrated by an example with a the three-state, continuous-time Markov process. Then, the generalization of the Markovian Mpemba effect from this three-state model to a continuous-state, continuous-time Markov process is discussed. Chapter 3 is concerned with the methodology used for this research. Measurements are taken and analyzed for a basic model to show that the experimental apparatus is working

10 to specification, and the experimental protocol for how data was collected and analyzed is given. I further identify large fluctuations in the diffusion coefficient, which are not included in the basic model, and explore its effect on measurements of the Markovian Mpemba effect using simulations. Chapter 4 is the core of my work, focusing on how the process is implemented, what measurements were made and the analyses that were done. A 1.49 ± 0.1 µm particle is trapped using a narrow harmonic well at a position that is randomly chosen from the initial equilibrium distribution. This choice of the initial position is discussed. The equilibrium distribution of the particle inside the designed potential is measured, and the corresponding potential is calculated and compared with the designed potential. Optimization of this measurement by adjusting temperature is discussed. The particle is observed in repeated trials of quenching it from a high temperature to a low temperature, and the particle trajectory is recorded for each trial. The data is analyzed to determine whether the Markovian Mpemba effect has occurred. The final chapter summarizes key findings and provides recommenda- tions for future work.

11 Chapter 2

Theory

As described in the Introduction, the study of the Mpemba effect in water is complicated by the many experimental variables that could contribute to the effect. It is possible that classical thermodynamics does not apply to this situation because the dynamics are far from equilibrium. Reformulating the Mpemba effect under a generalized theory in terms of stochastic thermodynamics, namely, the Markovian Mpemba effect, opens the possibility for the phenomenon to be studied in alternate systems that are unsusceptible to the same challenges, such as having too many experimental variables. Model systems commonly used in experiments for testing stochastic thermodynamic theory are colloidal particles, DNA, RNA, proteins, enzymes, and molecular motors in aqueous solution, which are collectively referred to as small systems [30]. In these systems, the aqueous solution is much larger in extent than the small system itself and is often referred to as the heat bath because transfers of energy to and from the small system do not affect the temperature of the solution. Hence, only the small system needs to be observed, and not the countless variables that affect the cooling of water. This chapter begins with some background information about Markovian dynamics, which describes the behaviour of systems in which the Markovian Mpemba effect occurs. Then the Markovian Mpemba effect is defined. Next, an example of a system in which the Markovian Mpemba effect occurs is given using a three-state model, followed by some general comments about the Markovian Mpemba effect in continuous systems. This chapter ends with a description of the physical process that is being studied, and the experimental model used to implement it.

2.1 Markov process

This introductory section provides a brief overview of the mathematical terminology for describing Markovian processes. It then goes on to providing the dynamical equations and long-time properties that characterize Markovian processes.

12 A stochastic process that is Markovian has no memory of the past, and only the present state of the system matters in determining what the future state of the system will be [35]. In general, the Markovian model of a particle undergoing Brownian motion must include both position and velocity [28], but in the case of overdamped motion, only knowledge of the positions is required. In general, the conditional probability density

P (x1; x2; ...|y1; y2; ...) = P (x1; x2; ...|y1) (2.1) can be written in terms of xm ≡ x(t = m∆t) and yn ≡ y(t = n∆t) where x1, x2, ... are predicted future positions of the particle, where m are positive integers, and y1, y2, ... are past positions, where n are zero or negative integers. Then the joint probability density for a Markovian process can be speciﬁed using only the initial position and the one-step transition probabilities between states [36]

P (x1; x2; x3; ...; xn) = P (x1)P (x2|x1)...P (xn−1|xn−2)P (xn|xn−1), (2.2) where P (xn|xn−1) are the one-step transition probabilities. As can be seen from Eq. (2.2), only the current position x1, and none of the past positions yn is needed to predict the probability that the particle takes a particular trajectory deﬁned by the sequence of points xm. This property of Markovian processes allows the dynamics of the process to be described by a single equation, the master equation, which is the topic of the next section.

2.1.1 Markovian dynamics

Consider the dynamics of a discrete-state, continuous-time Markov process with N states, given by [28] N dpi(t) X = {W p (t) − W p (t)}, (2.3) dt ij j ji i j=1 where Wij is the transition probability per unit time from state j to i, and pi are elements representing state i in the probability distribution of states, ~p. The two terms on the right hand side of Eq. (2.3) represent the flow of probability into and out of state i [28]. For a finite state, continuous-time Markov process, we can define a finite transition-rate matrix, or W-matrix, which satisfies two properties: 1) every off-diagonal element is zero or positive, Wij ≥ 0 for i =6 j, by considering positive probability flow, and 2) the rows sum to P zero, i Wij = 0 for all j, to satisfy the normalization of probability [28]. The W-matrix [28] can be compactly described as

! X Wij = Wij − δij Wki . (2.4) k

13 Equation 2.3 can be rewritten in the form of the master equation

d~p(t) = ~p(t) ⇒ ~p(t) = etW~p(0), (2.5) dt W where ~p(0) is the initial distribution. The transition-rate matrix, W, has eigenvalues λ1, λ2, ..., λn with right eigenvectors V1, V2, ..., Vn which satisfy WVi = λiVi, where i = 1, 2, ..., n. Then, the time-dependent probability distribution can be written in terms of the eigenfunction expansion, ~p(t) = P λit P i ciVie , and the initial probability distribution is ~p(0) = i ciVi.

2.1.2 Long-time limit

The eigenfunction expansion of p(t) diverges as t → ∞, if λi > 0, but, for λi ≤ 0 it converges. Properties of the eigenvalues for nonnegative square matrices are given by the Perron-Frobenius theorem [37], which guarantees the existence of a stationary state in ﬁnite Markov chains [28]. For a general physical system with continuous states and continuous time, if the system is known to have a stationary state, the continuous model can be approximated using a discrete model to which the Perron-Frobenius theorem is applied [28]. For discrete-state, continuous-time systems, an alternate proof of the existence of a stationary state in the long-time limit uses the property of detailed balance, which guarantees −1 that the W-matrix is symmetric in the sense that a matrix S exists where S WS is a diagonal matrix, and that the W-matrix is negative semideﬁnite: that is, all eigenvalues are zero or negative [28]. Detailed balance is the statement that, at thermodynamic equilibrium, for every pair of states i, j, the number of transitions from state i to j must equal the number of transitions from state j to i, or equality of currents [28] given by

Wijπj = Wjiπi, (2.6) where ~π is the thermodynamic equilibrium probability distribution.

For a system with a unique equilibrium state, the eigenvalues can be ordered as 0 = λ1 > λ t λnt λ2 ≥ λ3 ≥ ... ≥ λn, giving the eigenfunction expansion ~p(t) = V1+a2V2e 2 +...+anVne . In the long-time limit t → ∞, solutions to the master equation will tend to a stationary distribution, ~p = ~p s that does not change with time, d~p s(t)/dt = 0 [28]. The stationary distribution is the Boltzmann distribution, or canonical distribution, π, and is given by [27, 38] e−Ui/kBTb πi(Tb) = , (2.7) P −Ui/k T i e B b where Ui is a discrete energy level, and Tb is the bath temperature.

14 6,-

5,- 1&',23,4 !# *

$%&'() #!"

+- +, #!# ./0/&*, ./0/&*-

Figure 2.1: “Potential” of energy levels and kinetic barriers. Unscaled energies are shown. Black dotted lines intersected with the solid red lines indicates allowed energy levels for states, and a threshold energy level for barriers.

The transition probabilities per unit time Wij are chosen to be in Arrhenius form [27], given by Bij −Uj − k T Wij(Tb) = Γe B b i =6 j, (2.8) where Γ is a rate constant, Tb is the bath temperature, Bij = Bji is the energy level of the barrier between states i and j, and Ui is the energy of state i. The relationship between the variables in Eq. (2.8) is illustrated in Fig. 2.1. In Fig. 2.1, the dotted lines intersected with the red lines indicate the allowed energy levels. A system with three states is shown. The system can be thought of as being periodic because transitions are made between the leftmost and rightmost states. To illustrate the elements which are in Eq. (2.8), the discrete states, i and j are labelled, along with their energy levels, Ui and Uj, the energy barrier between those two states Bij, and the transition probabilities per unit time Wij. Equation 2.8 has the same form as the ratio of two Boltzmann distributions with energies corresponding to the bottom of the well and the top of the barrier. This equivalence demonstrates how energy barriers dictate the shape of the probability distribution. For transition rate Wij from state j to i, deﬁned in Eq. (2.8), the state i is at energy Ui, and the state j is at energy Uj. Between them is a barrier of height Bij. Despite having similar features, this is not a potential landscape, because the independent axis is not to be viewed as having any spatial relevance. At any given time, a single system is in one of the states i = 1, ..., N with the energy associated with that state, assuming that transitions are instantaneous and the system does not spend any time at the top of the barrier.

15 2.2 The Markovian Mpemba eﬀect

Having deﬁned what is a Markov process, I will now move on to discuss the mathematical condition necessary for the Markovian Mpemba eﬀect to occur.

Let the eigenvalues of the transition-rate matrix be ordered such that 0 = λ1 > λ2 ≥ λ t λnt λ3 ≥ ... ≥ λn. In the eigenfunction expansion ~p(t) = V1 + a2V2e 2 + ... + anVne , at small times, all the terms must be taken into account because they are on the same order of magnitude. At long times, and assuming there are no degenerate eigenvalues, since the decay is exponential for negative λk, all terms beyond the second term are much smaller than the first two terms, and can be ignored [27]. Then the comparison between two non-equilibrium c c −λ2t states amounts to a comparison between the truncated functions ~p (t) ≈ ~π(Tb) + a2~v2e h h −λ2t and ~p (t) ≈ ~π(Tb) + a2~v2e , where ~π is the eigenvector corresponding to the eigenvalue equal to zero, the equilibrium distribution, which does not decay with time [27]. These two c h functions differ only in the coefficients a2 and a2 . The larger in magnitude these coefficients are, the longer it will take for those terms to decay away. For a discrete Markov process, the definition of the Markovian Mpemba effect is: “When c h λ2 is strictly larger than λ3 and |a2| > |a2 |, the Markovian Mpemba effect occurs” [27]. The Markovian Mpemba effect for a continuous Markov process is defined using a metric called the distance from equilibrium, which will be introduced next.

2.2.1 The distance-from-equilibrium function

Temperature is an easy-to-measure metric for cooling, in the sense that a thermometer or thermocouple is all that is required. However, care needs to be exercised in the in- terpretation of the measurement. In the Mpemba effect that occurs in water where an overall temperature is undefined. Instead, when temperature given, it implicitly refers to the temperature in the small region surrounding and adjacent to the temperature probe. When observing the Markovian Mpemba effect, the amount of cooling of non-equilibrium processes in a small system is measured. A small system has stochastic variations and is described by its probability distribution. When the probability distribution is the equilibrium distribution, temperature can be defined via the Boltzmann distribution and is consistent with the macroscopic definition. When the probability distribution ~p is not the Boltzmann distribution, a proxy for temperature needs to be defined that is consistent with temperature when the system is at equilibrium and extends the notion of temperature for when the system is out-of-equilibrium. While the distribution ~p holds all the relevant information about the system, a single number would be more convenient to use to make quantitative comparisons. This is also necessary for compatibility with the existing notion of temperature. Therefore, we can try to reduce the elements of ~p to a single number through a mathematical transformation.

16 One possible candidate is known as the entropic distance function, D, alternatively called the distance-from-equilibrium function, or distance from equilibrium, whose discrete and continuous versions are [27]

b ! X Ui(pi − π ) D[~p; ~π(T )] = i + p ln p − πb(x) ln πb (2.9a) b T i i i i i b Z U(x)(p(x, t) − πb(x)) D[p(x, t); π(x, Tb)] = + p(x, t) ln p(x, t) − πb(x) ln πb(x) dx Tb (2.9b) where x is position, t is time elapsed, Tb is bath temperature, Ui is the discrete energy of b a state, U(x)the continuous energy at position x, and πi and πb(x) are chosen to be the Boltzmann distribution for the discrete and continuous systems, respectively.

Interpreting what the entropic distance physically means

The name entropic distance seems to be a misnomer because it contains both terms for energy, as well as terms for entropy. Moreover, the deﬁnition of D diﬀers from the usual notion of distance. Equation 2.9a has the property D[~p; ~π(Tb)] =6 D[~π(Tb); ~p]. However, one would expect that the magnitude of the distance between two points is independent of which point is used as the reference. To clarify what this quantity represents, the entropic distance is actually the change in stochastic entropy.

In a stochastic process that takes place from time 0 to τ, a single trajectory x0:τ is measured for each individual system. Each individual system has a different trajectory, and can have a different amount of entropy change. However, a distribution can be measured for an ensemble of trajectories p(x0:τ , τ) [30] Thus, the entropy change of the stochastic process should be used to describe the entropy change of an ensemble of systems. The ensemble average is denoted using <>. The change in stochastic entropy in a process can be separated into two parts, ∆stotal ≡ ∆sb + ∆s, where ∆sb is the change in entropy of the bath, and ∆s is the change in entropy of the system. The change in entropy of the b bath is defined as ∆s ≡ hq[x0:τ ]/Tbi, with q the heat transferred, and Tb the equilibrium temperature. The change in entropy of the system is defined as ∆s ≡ h− ln p(x0:τ , τ)i = P − p(x0:τ , τ) ln p(x0:τ , τ) [30]. In Eq. (2.9a), since there is no change in work in the process, U (p −πb) the first term, P i i i is the average change in heat per temperature, which is ∆sb, i Tb P b b the entropy change of the bath. Then the latter two terms together, i(pi ln pi − πi ln πi ) is the entropy change of the system [27]. The entropy is alternatively separated into a relative entropy that is “additional entropy” of an out-of-equilibrium system which is in excess of the entropy of an equilibrium state, and e the equilibrium entropy, S = −kDKL + S , where k is a constant that depends on the units P pn used, DKL = pn log e is the excess entropy, known as the Kullback-Leibler divergence n pn e e or the relative entropy between the states pn and pn, and S is the equilibrium entropy

17 [28]. The equilibrium entropy does not change during cooling; therefore, cooling that is represented by the entropic distance is entirely encoded in the Kullback-Leibler divergence. Hence, the Kullback-Leibler divergence satisﬁes the three conditions needed to accurately measure the cooling rate of a Markovian process [27]. Other functions also exist that satisfy the three conditions [27].

Statement of the Markovian Mpemba eﬀect

Using the above definition of the distance from equilibrium, the Markovian Mpemba effect for a continuous Markov process is defined as follows:

If there exist three temperatures, Tb < Tc < Th such that two systems which h are initially prepared at the hot and a cold equilibrium (~p (0) = π(Th) and c ~p (0) = π(Tc)) relax toward the same equilibrium π(Tb) according to Eq. (2.5), h c and if there exist some ﬁnite time tm such that D[~p (t); Tb] < D[~p (t); Tb] for

all t > tm, namely the distance from equilibrium of the initially hot system becomes smaller than that of the cold system, then the Mpemba eﬀect exists in the system. [27]

Three conditions the entropic distance must satisfy

A distance-from-equilibrium function must be aligned with the conventional deﬁnition of temperature and is a proxy for temperature when the system is out of equilibrium; a system that is farther from equilibrium always has a larger distance from equilibrium than that calculated for a system that is closer to equilibrium. Lu and Raz [27] identify three conditions a distance-from-equilibrium function must satisfy. Any function satisfying these three conditions can be used to measure cooling to an equilibrium temperature Tb, and every function that does will give the same conclusion as to whether or not the Mpemba eﬀect occurs. The three conditions are [27]:

1. D[~p(t); ~π(Tb)] is a monotonically non-increasing function of t for a relaxation process. In a spontaneous process, the system can only maintain or decrease its level of free energy, whereas an increase in free energy would be needed to push a system further away from equilibrium. Therefore, the distance from equilibrium can only stay the same or decrease over time.

2. D[~π(T ); ~π(Tb)] is a monotonically increasing function of T for T > Tb (the result is

true even when T = Tb, so that T ≥ Tb is a stronger valid statement). The distance- from-equilibrium function must follow the direction of changes in temperature such that the rank order of a set of systems by temperature is the same as the rank order by distance from equilibrium.

18 3. D[~p(t); ~π(Tb)] is a continuous, convex function of ~p. That is, D[E~p(t); E~π(Tb)] ≤

ED[~p(t); ~π(Tb)], where E is the expected value [39]. This condition ensures that any function valid for observing the Markovian Mpemba effect will concur whether the Markovian Mpemba effect has occurred. In particular, this convexity requirement ensures that the sign of the difference of the distance from equilibrium at large times

for the two systems is the same as the sign of the difference of the a2 coefficients, so that Markovian Mpemba effect result for discrete Markov processes and continuous Markov processes is the same.

All three conditions are satisfied by both the discrete and continuous entropic distance defined in Eq. (2.9); but, in practice, only the discrete form Eq. (2.9a) is used to observe the Markovian Mpemba effect [27]. The first condition is satisfied, which can be seen in the D[~p(t); ~π(Tb)] versus t curves in Chapter 4. The individual distance-from-equilibrium curves are monotonically non-increasing (decrease or stay the same) with time, as expected for a relaxation to equilibrium. Figure 2.2 is consistent with the second condition. Here, the distribution ~p(t) is strictly the equilibrium distribution, ~p = ~π(T ) for T > Tb. The curve is monotonically increasing with temperature. Therefore, the distance from equilibrium can be used as a proxy for temperature: a higher temperature corresponds to a larger distance from equilibrium. The two initial temperatures and the bath temperature that are used for the experiment are especially marked: hot temperature is 5295 K, the green square; warm temperature is 1995 K, the maroon pentagon; cold temperature is 295 K, the pink triangle. The blue line is the distance from equilibrium calculated at temperatures evenly spaced at every 100 K starting from 295 K. d2f(x) A twice-differentiable scalar function f(x) is convex if dx2 ≥ 0 [28]. Then for x > 0, d2f(x) f(x) = x ln x is convex since dx2 = 1/x is positive for x > 0. Figure 2.3 is a plot of this function for 0 ≤ x ≤ 1. The dashed line is an arbitrary line joining two points on the curve. Every point on the dashed line lies above the curve, which holds when the function is convex. The convexity condition for the distance-from-equilibrium function is used to obtain a lower bound in the proof that establishes that the distance-from-equilibrium function D[~p(t); ~π(Tb)] is smaller for ~p(t) that is closer to equilibrium, ~π(Tb). The third condition allows comparison between distributions from non-overlapping cooling processes. If two processes partially or completely overlap, it would be trivial to tell which one is closer to equilibrium because they would both follow the same overlapped pathway once the overlap begins. Consider two D[~p(t); ~π(Tb)] versus t curves, starting from different initial conditions and cooling to the same equilibrium temperature. Even though the two curves cross at time tMME when the Markovian Mpemba effect is observed, the h c distributions ~p (tMME) and ~p (tMME) are actually distinct. Indeed, they remain distinct at all finite times. This condition highlights the fact that the value of the distance from equilibrium is not a statement about the amount of time it will take ~p(t) to relax to ~π(Tb).

19 Hot 20 Warm

10 Cold

Distance from equilibrium, 0

0 5000 10000

Temperature, T (K)

Figure 2.2: Distance from the equilibrium state ~π(295 K), is calculated as a function of ~π(T ) using Eq. (2.7). Hot temperature is 5295 K (green square). Warm temperature is 1995 K (maroon pentagon). Cold temperature is 295 K (pink triangle).

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!"$ %&'()'*+&'(

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Figure 2.3: Illustration of a convex function. Solid red curve is the function, f(x) = x ln x. Intermediate points in dashed line that joins two arbitrary points on the curve are always above the curve.

20 This section has analysed the features of a Markovian system that are used to predict and observe a Markovian Mpemba eﬀect. The next section of this thesis gives an example of the simplest system in which a Markovian Mpemba eﬀect can occur.

2.2.2 A Three-State Model

The three-state model is a finite-state, continuous-time Markov process for the case where there are only three states, i = 1, 2, 3, which will be called state 1, state 2, and state 3. The potential is periodic, and the particle transitions are from state three to either state two or state one, from state one into either state two or state three, and from state two to either state one or state three. At any given time, a single system can be in only one of the three states. The three-state model is the simplest discrete system in which the Markovian Mpemba effect occurs. A one-state system can only be at equilibrium, and in a two-state system, the temperature only affects the ratio of probability to be in the higher-energy versus the lower-energy states, and there is a single pathway for cooling. A continuous system could have two wells and exhibit the Markovian Mpemba effect, but it could not then be coarse grained into a discrete two-state system while preserving that property.

Consider the three-state model shown in Fig. 2.4 with U1 = 0, U2 = 0.1, and U3 = 0.7, and energy barriers between states: B12 = 1.5, B23 = 1.2, and B13 = 0.8 [27]. This example is the one published in Lu and Raz’s [27] paper. Additional examples of the three- state model, with parameters I chose, are in Chapter B of the Appendix. The dynamics of the system is given by Eq. (2.5) because this is a Markov process, which means it obeys Markovian dynamics. Transition rates are calculated using Eq. (2.8) for i, j = 1, 2, 3. They are defined up to a scaling factor, Γ in Eq. (2.8), which sets the time scale. Here, Γ is defined to be 1. This choice of time scaling does not affect whether or not the Markovian Mpemba effect occurs since the entropic distance is a dimensionless quantity, but it does change the time dependence. A larger Γ corresponds to faster transitions, and the cooling would occur more quickly, through the same sequence of states, according to the master equation, Eq. (2.5), for i = 1, 2, 3.

The configuration space of the three-state model is the set of states that satisfy 0 ≤ pi ≤ P 1, and i pi = 1 for i = 1, 2, 3. The three corners correspond to the three states. Figure 2.5 shows the configuration space of the three-state model in Fig. 2.4. As shown, state 1 has the lowest energy, then state two, and state 3 has the highest energy. Under the conditions of the process, the probability that the particle is found in each of the three states varies. This information is encoded as a point in the configuration space. For example, when temperature is at absolute zero, the point is on the corner corresponding to state 1, and the particle is found in state 1 with probability 1. For a quasi-equilibrium cooling process, the points that represent the process lies along a curve in configuration space, shown as the black equilibrium curve in Fig. 2.5. It is calculated using Eq. (2.7) for temperatures

21 . , !" .,- 2&'34536 . !# - +- *

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Figure 2.4: Schematic of energy landscape for the three-state model. Black dotted lines intersected with the solid lines indicates allowed energy levels for states, and a threshold energy level for barriers. The energies of each state are U1 = 0, U2 = 0.1, and U3 = 0.7, and the energy barriers between states are B12 = 1.5, B23 = 1.2, and B13 = 0.8 from 0.1 ≤ T ≤ 1.5, with an added equilibrium state, state 1, the lowest-energy state, which is the state of the system when the temperature is absolute zero. Two systems are prepared at a hotter and a cooler initial temperature and are rapidly cooled to a third equilibrium temperature. The two initial temperatures are Th = 1.3, Tc = 0.42, and the

ﬁnal equilibrium temperature is Tb = 0.1. Figure 2.5 shows in blue circle, orange square and red triangle the equilibrium states for these three temperatures [27]. Figure 2.6 shows the curve that corresponds to a quasi-equilibrium cooling process, and two examples of curves that correspond to non-equilibrium cooling processes, from initial states, ~π(Th), and ~π(Tc), marked by a red circle and orange disk on the equilibrium curve, to the ﬁnal state ~π(Tb), marked by a blue asterix. The dashed red line is the non-equilibrium cooling curve from the initial state, ~π(Th), calculated from the Fokker-Planck equation for 20 data points. The solid orange line is the non-equilibrium cooling curve from the initial state, ~π(Tc), calculated from the Fokker-Planck equation for 20 data points. The green arrow with tail that is a dotted line and blue dashed arrow with tail that is pairs of dots are eigenvectors, ~v2 and ~v3, of the transition-rate matrix, Wij, associated with the second- and third-largest eigenvalues. ~v2 is parallel to the line from the lowest-energy state, ~p1, to the middle-energy state, ~p2. ~v3 is parallel to the line from the lowest-energy state, ~p1, to the highest-energy state, ~p3. As temperature is decreased, redistribution of probability into state 1 are made. Eigen- vectors ~v2 and ~v3 that are associated with the second- and third-largest eigenvalues of the transition-rate matrix, Wij correspond to the redistribution of probability from state 2 to state 1, and state 3 to state 1, respectively [27]. The barrier between state 2 to state 1 is

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Figure 2.5: Conﬁguration space of the three-state model shown in Fig. 2.4, where the state energies are U1 = 0, U2 = 0.1, and U3 = 0.7, and the energy barriers between states are B12 = 1.5, B23 = 1.2, and B13 = 0.8. Gray triangle is the set of all allowed states. Black curve is ~π(T ) calculated from Eq. (2.7). Blue circle, orange square and red triangle denote equilibrium states for the three temperatures used, which are 0.1, 0.42, and 1.3, respectively. higher than the barrier between state 3 and state 1. Transitions which occur at long times correspond mainly to transitions from state 2 to state 1, and transitions which occur at short times correspond mainly to transitions from state 3 to state 1, while at intermediate times, transitions are from both state 2 to state 1 and state 3 to state 1.

We denote a2(T ) by the coefficient of the term in the eigenfunction expansion of a state containing the second-largest eigenvalue. From the definition of the Markovian Mpemba effect for a discrete Markov process, the Markovian Mpemba effect occurs when the magnitude of the a2 coefficient at the higher temperature, |a2(Th)|, is smaller than the a2 coefficient at the lower temperature, |a2(Tc)|. In other words, for two temperatures, Th and Tc where d|a2| Th > Tc, the Markovian Mpemba effect will occur if dT < 0 [27]. Figure 2.7 shows a plot of |a2(T )| against T . The a2 coefficient is maximized at temperature Tm = 0.4. For T > Tm, the magnitude of a2 decreases with further increases in temperature. Then, for two points on the equilibrium curve, ~π(Th) and ~π(Tc), where h Th > Tc ≥ Tm, in the eigenfunction expansion of the equilibrium states, the coefficient a2 c is smaller in magnitude than a2. According to the definition of the Markovian Mpemba effect for a discrete Markov process, the system with the initial state ~π(Th) will reach the state ~π(Tb), where Tb < Tm, sooner than the system with the initial state ~π(Tc), under spontaneous cooling. In Fig. 2.6, since the fast mode of cooling dominates over the slow mode of cooling, the out-of-equilibrium cooling curves are initially parallel to ~v3 until all the probability of being in state 3 is transferred to state 1, and then the cooling curve is parallel to ~v2 until the

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Figure 2.6: Cooling pathways of the three-state model shown in Fig. 2.4, where the energies of each state U1 = 0, U2 = 0.1, and U3 = 0.7, and the energy barriers between states are B12 = 1.5, B23 = 1.2, and B13 = 0.8. Black curve denotes the locus of equilibrium states. Green arrow with tail that is a dotted line and blue arrow with alternating dot-dash tail are the eigenvectors of the transition-rate matrix, Wij, deﬁned in Eq. (2.8), that are associated with the second- and third-largest eigenvalues. Dashed red line starting from the red triangle marker, ~π(Th), and solid orange line starting from the orange square marker, ~π(Tc), and ending at the blue circle marker, the ﬁnal equilibrium state, ~π(Tb) is the rapid cooling process from Th = 1.3 and Tc = 0.42 to Tb = 0.1.

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Figure 2.7: Plot of the absolute value of the coeﬃcient of the term in the eigenfunction expansion of the Boltzmann distribution containing the second-largest eigenvalue against initial temperature for the three-state model shown in Fig. 2.4, where the energies of each state U1 = 0, U2 = 0.1, and U3 = 0.7, and the energy barriers between states are B12 = 1.5, B23 = 1.2, and B13 = 0.8. The bath temperature is 0.1. The vertical dashed line is Tm = 0.4, close to the temperature at which the slope of the curve is zero.

24 Hot 1

MME 0.1 Warm

Distance from equilibrium, 0.01

0 10 20

Time

Figure 2.8: Distance from equilibrium versus time plots for the three-state model [27] from solving the discrete-state Fokker-Planck equation. The crossover between the two curves indicates the Markovian Mpemba eﬀect (MME). excess probability in state 2 has been transferred into state 1 and equilibrium is reached.

Both the cooling curves for systems with initial temperatures Th and Tc can be seen to follow this pattern. Each state can be written as an eigenfunction expansion in terms of the vectors ~v2 and ~v3. This is done by projecting the tangent to the curve of equilibrium ~v ~v da2 | a states at the state being expanded onto 2 and 3. Since dT Tm = 0, the 2 coeﬃcient of the eigenfunction expansion of the Boltzmann distribution at a slightly higher or lower temperature is approximately constant at Tm. This happens when the tangent to the curve of equilibrium states is parallel to ~v3. Since Tc ≈ Tm, we see that the cooling curve for Tc is approximately tangent to the curve of equilibrium states. Figure 2.8 shows the distance from equilibrium versus time plot for the three-state model with values as given above from solving the discrete-state Fokker-Planck equation. Red squares are the initially hot system and orange triangles are the initially warm system. h c The Markovian Mpemba eﬀect is observed since D[~p (t); Tb] 9. This is seen in the red squares falling below the orange triangles.

2.2.3 Markovian Mpemba eﬀect in a Continuous-state System

Since an approximation to a continuous potential is applied by the feedback trap, the theory in the previous section needs to be extended to continuous-state, continuous-time Markov processes, which will be the topic of this section. For a colloidal particle diﬀusing in an arbitrary potential U(x), coupled to a heat bath at temperature T [28], the master equation

25 is approximated by the Fokker-Planck equation [28],

1 h i ∂ ρ = ∂ [(∂ U)ρ] + k T ∂2ρ (2.10) t γ x x B x where ρ = ρ(x, t) is probability density, γ is the drag constant, U = U(x) is the potential landscape, kB is Boltzmann’s constant, and T is the temperature of the bath. Einstein’s relation for the diﬀusion coeﬃcient D,

kBT D = (2.11) γ and the deﬁnition, h(∆X)2i D = , (2.12) 2∆t for displacements ∆X in one dimension, give rise to the ﬂuctuation-dissipation relation, 2 h(∆X) i 2kBT ∆t = γ [28]. This is the analogous relation to detailed balance that was true for discrete Markov Processes which injects properties of thermodynamic systems into the Fokker-Planck equation. While a continuous Markov process does not necessarily have a stationary state [28], we will only consider thermodynamic systems that have a stable equilibrium, with a stationary distribution given by Boltzmann’s equation,

e−U(x)/kBT π(x) = R . (2.13) e−U(x)/kBT dx

Lu and Raz [27] did a calculation for the three-state model for how likely we would succeed in observing the Markovian Mpemba effect using a trial and error method for 1 choosing system parameters. They determined that there was a 8 or higher chance that randomly chosen parameters would result in the Markovian Mpemba effect being observed. Based on this result, they suggested that the odds are even better for a continuous-state model because there are an infinite number of ways to modify a continuous model while retaining its basic shape.

2.2.4 How the condition for the Markovian Mpemba eﬀect could occur

The previous section suggested that it would be easy to find a continuous model in which the Markovian Mpemba effect, and the authors of the Markovian Mpemba effect believed that all it would take would be a few iterations of trial and error to design a system. As it turned out, designing the potential U(x) to have the Markovian Mpemba effect is tricky. To slow down the cooling of the system initially at a lower temperature, one might try increasing the probability density in the metastable state. However, this adjustment leaves less probability density in the high-energy states for both the lower and higher temperatures,

26 thereby diminishing the ability of the system initially at a higher temperature to rapidly cool. To increase the amount of rapid cooling in the initially hotter system, one might increase the temperature, but that puts the system farther from equilibrium, which means it takes longer for the system to cool. Then, one could try to increase the speed of rapid cooling, but that causes fewer high-energy states to be accessible. The tradeoffs that exist when one tries to either slow down the rate of cooling in the system initially at a lower temperature, or to increase the rate of cooling in the system initially at a higher temperature make it so that it is difficult to manipulate a system to have the Markovian Mpemba effect. This analysis emphasizes the fact that the phenomenon we are studying is an anomaly and thus, rarely occurs. It also highlights an interesting aspect of the Markovian Mpemba effect, which purports to provide a formula for bypassing the roadblocks that result in the more typical behaviour of the system initially at a lower temperature cooling more quickly than a system initially at a higher temperature. For designing the potential, Lu and Raz identify qualitative features that contribute to increasing the likelihood of having the Markovian Mpemba effect occur. A local minimum in the energy landscape can delay the rapid cooling of systems to equilibrium [27]. When a particle is trapped within the local energy minimum, it must overcome a high-energy barrier before reaching the global minimum energy. At high temperature, the probability density is spread out to more high-energy states. If there is a low-energy barrier blocking the high-energy states from reaching the global minimum energy, it is possible that the Markovian Mpemba effect occurs. Whether it actually occurs depends on the amount of probability that is trapped, how long it takes to escape, and the amount of probability that is in the high-energy states. A time-scale difference for the time it takes for probability to flow from the different regions is needed so that the probability from high-energy states reach the global minimum energy before the probability that is stuck in the local energy minima can escape the barrier. To have this occur quickly so that repeated experiments can be done, one tries to increase the number of high-energy states in the basin of attraction containing the probability that is destined for the global minimum energy when temperature is lowered.

2.3 Using the Langevin Equation for simulations

Before preceding to examine the experimental system in which the Markovian Mpemba effect will be observed, it will be necessary to introduce the Langevin equation, that will be used to simulate the experiment to determine what is a sufficient amount of data to enable observation of the Markovian Mpemba effect. This discussion is divided into two parts. The first examines the Langevin equation when the thermal noise is zero. The second part examines the Langevin equation with the thermal noise. The equations used in the simulations of this thesis are provided at the end of this section.

27 The Langevin equation can be derived using Newton’s second law. From Newton’s second law, the equation for Brownian motion with no external potential is [40]

d2x dx m` d2x0 ` dx0 m = −ζ + ξ(F )(t) ⇒ = − + other terms, (2.14) dx2 dt ζτ 2 dt02 τ dt0 where ξ(F )(t) is the thermal force, ζ is the drag constant, and m is the mass of the particle. The non-dimensionalized form is obtained using the non-dimensional substitutions, t0 = t/τ, x0 = x/`, where the other terms correspond to the thermal force, and other forces, if applicable. We can see from Eq. (2.14) that the inertial term is negligible in comparison m m with the drift term if ζτ → 0. This is satisfied when τ ζ , which Einstein, Smoluchowski and Langevin refer to as the overdamped limit [40]. The mass of a 1.49 µm silica particle with density 2 g/cm3 in water is m = 3.46 × 10−15 kg. At T=295 K, the dynamic viscosity of water, η = 1 × 10−3 Ns/m2, so ζ = 1.34 × 10−8 Js/m2. Thus, the overdamped timescale τ = 2.6 × 10−7 s is on the order of 100 ns. When the dynamics of the system being studied are at time scales larger than this time scale, the inertial term in Eq. (2.14) can be neglected. To simulate the trajectory of the particle in a potential, we employ Euler’s Method [36, 41, 42], a numerical method typically used for solving differential equations. First, I will show how Euler’s method works on the deterministic part of the Langevin equation, an ordinary differential equation, which is the more commonly encountered usage of Euler’s Method. Then, I will use Euler’s method on the Langevin equation inclusive the noise term, a stochastic differential equation.

2.3.1 Solving the deterministic part of the Langevin equation

The deterministic part of the Langevin equation is given by

dx dx k − kt 0 = −kx − γ ⇒ = − x ⇒ x = e γ . (2.15) dt dt γ

We consider the motion of a particle diﬀusing in a harmonic well U = kx, which is α ts t γ parametrized by the dimensionless parameter = tr , where r = k . The experimental conditions are matched by setting α = 0.1, and ts = 0.005 s. This equation can be solved using the general form of Euler’s method [41]

dx x = x + h (t , x ), (2.16) n n−1 dt n−1 n−1 where x is position and t is time, with time index n and n − 1. dx Given initial values, (t0, x0), and the slope, dt (t0, x0), we can estimate x1 which corresponds to t1 = t0 + h, where h is the time step size. This process can be repeated after

28 !"

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Figure 2.9: Convergence with step size of Euler’s Method for deterministic differential equation. Magenta to cyan lines are dimensionless time steps h/tr = 0.02, 0.1, 0.2, 0.4, and 0.8; black dashed line is exact solution. each step for as long as desired. The series of values (tk, xk), where tk = hk, for k ∈ Z is an approximate solution to the ordinary differential equation. We choose the initial condition that at time t = 0, x(t = 0) = 1 µm, to place the particle away from the equilibrium, but sufficiently close such that we can simulate the process of the particle returning to equilibrium in just a few iterations of Euler’s Method. We use Euler’s

Method with dimensionless time steps h/tr = 0.02, 0.1, 0.2, 0.4, and 0.8. The results are shown in Fig. 2.9. We see that as h decreases, the numerical solution converges towards the exact solution. This example shows how the particle would behave in a harmonic well, parameterized by α = 0.1, if there was no thermal force.

2.3.2 Solving the full Langevin equation

Simulations are used to determine the length of time needed to run the experiment, and the amount of data to be collected that would result in observation of the Markovian Mpemba effect. Perturbations in physical parameters can alter the process we think we are observing. Thus, we should try to simulate how the motion of the particle is affected when the parameters are perturbed. One of the parameters that fluctuates in the experiment is the diffusion coefficient. The diffusion coefficient of a particle is higher when the temperature of the water bath is higher because the water molecules interact with the particle more energetically. In the experiment, the temperature of the room and consequently the temperature of the water bath can fluctuate, thus, the diffusion coefficient of the particle has some variation. The diffusion coefficient of a particle also depends on the proximity of a particle to a surface if its motion is parallel to the surface, as discussed in Section 3.3. The stationary surface causes the diffusion coefficient of the particle to be reduced due to friction through contact with fluid molecules. Since the particle is not held at a fixed height, but

29 has a vertical distribution, the diffusion coefficient of the particle again varies. These perturbations affect the experimental results. To account for their effects in the simulation, we can estimate the magnitude of those variations and apply them to the simulation. Although we cannot remove these effects from the experiment, we can try to account for discrepancies in experimental results by showing that they agree with simulation results. The Langevin equation with units is [43]

ts x = x + F + ξ n+1 n γ n−1 n

hξni = 0 2 hξn i = 2Dts, (2.17) where Fn−1 is the force applied to be particle by the potential based on its last observed position, ξn is motion due to thermal forces, and ts is the time between positions. The non-dimensionalized form of the Langevin equation is used in the experiment in order to be able to compare the results from diﬀerent experimental runs. We assume that ﬂuctuations in parameters are small within a single trial, and treat them as being constants. The parameters in the Langevin equation can be non-dimensionalized as follows [43]

0 0 0 0 0 x = x/` t = t/τ U = U/kBTF = `F/kBT ξn = ξn/`, (2.18) √ where ` = Dts and time τ = ts.

2.4 Physical Model

What follows is a high-level description of the experimental process describing the physical changes that are being made to the system. Figure 2.10 is the physical model of the quenching process of a small system. A small system is coupled with baths of different temperatures. States A, B, and C are small systems coupled with baths of temperatures Thot = 5295 K, Twarm = 1995 K and Tcold = 295 K, respectively, each of them in thermal equilibrium. State A and B are initial states. Both systems undergo a temperature quench by somehow removing the system from its original bath and putting it into a bath at temperature Tcold = 295 K, without adding or removing energy. Both systems eventually equilibrate with the new bath and reach the final state, C, some time later. As it is experimentally difficult to prepare the system at equilibrium and to transfer it from one bath to another without disturbing it, we only mimic the behaviour of the process being described. In the process that is measured, the initial states of the system, states A and B, are prepared arbitrarily close to equilibrium, and quenched in a bath of temperature

Tcold = 295 K, but the process is terminated before state C is reached. The process starts

30 = ;

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Figure 2.10: Physical model of temperature quench in a small system. The small system (white circle) is immersed in a bath of different temperatures. We have two systems that are initially in states A and B. Both systems are quenched and re-equilibrated to the same state C. the instant of the quench and ends a short time after the Markovian Mpemba effect is observed. For most potentials, the quenching from state A to state C occurs more slowly than the quenching from state B to state C. In the case we are interested in, the quenching from state A to state C occurs more quickly than the quenching from state B to state C. This anomalous effect known as the Markovian Mpemba effect. The difference in temperature means that the initially hot system will lose more energy than the initially warm system when cooled to a final temperature that is lower than both initial temperatures. We study the impact of energy barriers on the rate of cooling. When there is a single pathway for cooling, the initially hot system will always cool more slowly than the initially warm system. However, given multiple pathways, the ability of the system to utilize more efficient pathways could impact its cooling rate. The energy landscape, U, that was used to study the Markovian Mpemba effect, which was formulated by O. Raz is

U(x)/kBT = U0{a[arctan(x/xm − b) − arctan(x/xm − c)]

+ d[arctan(x/xm − e) − arctan(x/xm − f)]

+ g[arctan(x/xm − h) − arctan(x/xm − i) + jx/xm]} (2.19)

31 where a = 1.7, b = 10, c = 5, d = 2.1, e = 16, f = 12, g = 0.4, h = 100, i = 10.5, and j =

0.35, determine the shape of the potential, and U0 and xm are vertical and horizontal scaling factors, initial temperatures Thot = 5295 K and Twarm = 1995 K, and ﬁnal temperature

Tcold = 295 K. Before proceeding to give more details about the physical model, it will be necessary to introduce some experimental considerations.

2.5 Experimental Model

So far this chapter has focused on the Markovian Mpemba effect. The following section will discuss how the previous construction of the Markovian Mpemba effect has been adapted so that it satisfies the requirements of the experimental apparatus. We consider an overdamped small system undergoing diffusion in an external field for which the Markovian Mpemba effect is predicted. For a continuous, differentiable potential,

U(x) which varies spatially in x, a particle located at a point x = x0 in the coordinate system of the potential will experience a force given by the negative slope of the potential at that ∂U(x) point, F (x0) = − ∂x |x=x0 . There is no direct method to generate a potential of the shape given by Eq. (2.19). An indirect method, is to use a feedback trap. Instead of generating a potential, the feedback trap generates arbitrary forces timed on a feedback loop with a 200 Hz cycle. The distance that the particle has diﬀused in between feedback cycles must be small such that the change in force that corresponds to the particle position in the potential is small. The forces generated by the feedback trap have been shown to closely approximate the forces that are imposed by a harmonic potential when the potential has α < 0.1, where α was deﬁned in Section 2.3.1 [32, 33]. A feedback trap iteratively measures the particle position and then applies a force which is calculated based on the measured position. The particle is observed at discrete times t0 = 0, t1 = ts, ..., tn = nts, where ts is the sampling time, and n is an integer. Associated with each time is a measured position, x0, x1, ..., xn, which are averaged over the length of the camera exposure, tc. The minimum sampling time is ts = tc, but we set tc ts by choosing the sampling time to be ts = 10tc. With this choice, the fact that the particle is in motion during the camera exposure introduces a negligible error on the measured position.

Forces F (x0),F (x1), ..., F (xn) are calculated for each measured position. This applied force is uniform in space, but is varied in time. A constant force equal to the calculated force,

F (xn), is applied from nts to (n+1)ts. The force cannot be applied before the measurement of the particle position is complete so there is a delay of td from the time that is associated with the position of the particle and the time that the force that is calculated from that position is applied. In the experiment, we set td = ts.

32 2.5.1 Minimum width of the potential

Due to time constraints, it is infeasible to have an experiment that must run for a long time. In this section, the relationship between the amount of time needed to do the experiment and the size of the potential is discussed. Although biases due to image pixellation have been observed by other students working on the same setup when they used a narrower potential, and they have found that the measured probability distribution of the particle is affected as a result [44]. This was not an issue in my work as presented here, and I have not explored the conditions that would result in pixellation, so the following discussion assumes no pixellation. Under free diffusion, a particle can explore a narrower potential in less time than it can explore a wider potential. Thus, one method of shortening the time needed to do the experiment is to design a narrower potential. The potential that was used for this experiment, Fig. 2.13, is not the narrowest potential in which the Markovian Mpemba effect can be observed. I tried unsuccessfully to find a narrower potential using polynomial functions that could have the Markovian Mpemba effect. This size constraint seems to greatly reduce the changes of finding a potential function capable of producing the Markovian Mpemba effect, as fewer potentials worked than expected, so searching for a function for a narrow potential that works will be left to future projects. The following is some crude bounds about how narrow the potential can be, and how long the experiment can take, that I followed while trying to design the potential. The potential that was published in the theoretical paper [27], exceeded both these guidelines, while the revised potential Eq. (2.19) sufficiently improved on the second guideline, how long the experiment takes, that it was feasible for experiment, but was the same size as the the original potential when both are scaled by the parameters of the experiment to meet experimental constraints. The potential that is needed to do an experiment to observe the Markovian Mpemba effect is required to have two wells separated by a barrier and a plateau, and the size of the potential must be sufficiently large to accommodate these features. Figure 2.11 is a plot of the minimum width of a potential that has two wells separated by a barrier. The length of the plateau can be extended by making the potential wider, but the wells and barrier do have a minimum size which is taken to be half the width of the harmonic well with

α = 0.1 when the energy of the landscape is 1 kBT . Dr. Bechhoefer has argued that the potential that follows the shape of the circle whose radius is given by the minimum length scale satisfies the constraint on curvature, and the potential can extend vertically from the sides of the circle, such that the barrier can be made as high as desired, with a correction in the slope from the maximum force that can be generated by the feedback trap. I find that the maximum force is sufficiently large that, for the desired barrier heights, the increase to the width of the potential due to this effect is small, so a vertical slope is used in Fig. 2.11 because the exact maximum force can vary with experimental conditions.

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Figure 2.11: Minimum width of the potential well. Dashed red line is plot of harmonic potential for α = 0.1. Shaded red region is the hypothetical minimum width of the well. The circles have a radius of the minimum length of the harmonic potential with α = 0.1.

34 To determine the allowable size of potential I should use, Dr. Bechhoefer calculated the minimum size is the characteristic length scale of the well as follows: From the relaxation time of the particle in a harmonic well, tr, we have that γ t = (2.20) r κ γ kBT = (2.21) k T ∂2U B ∂x2 !−1 1 ∂2U 0 = (2.22) D ∂x2 !−1 `2 ∂2U 0 = 0 (2.23) D ∂x02

0 0 2 where U = U/kBT and x = x/`0. Since `0/D has units of time, the nondimensionalized 0 2 ∂2U 0 −1 0 relaxation time is tr = tr/(`0/D) = ∂x02 . Taking the inverse, and using tr = tr/ts = 10 = 1/α, we have ∂2U 0 = (t0 )−1 = 0.1 = α. (2.24) ∂x02 r We can rewrite Eq. (2.24) as

∂2U 0 ∂2U 0 `2 = α ⇒ = α/`2 (2.25) 0 ∂x2 ∂x2 0

The length scale given by the radius of curvature is

!−1/2 ∂2U 0 √ ` = = ` / α . (2.26) ∂x2 0 √ 2 For `0 = 2D∆t , D = 0.24 µm /s, ∆t = 0.005 s, and α = 0.1, we have ` ≈ 0.15 µm. As shown in Fig. 2.11, this is the radius of the circle that can be inscribed in a harmonic well with α = 0.1 at an energy of 1 kBT . Therefore, the smallest length scale is 0.15 µm, which we see from Fig. 2.11 is half the width of the harmonic potential with α = 0.1, indicated by the red dashed line, at 1 kBT , indicated by the black dashed line. As an order-of-magnitude estimate, the size of the potential is taken to be ≈ 1 µm. Then, based on the estimate for the fundamental time scale for free diffusion, τ = `2/2D = (1 µm)2/0.24 µm2/s ≈ 4 s. The guidelines I followed to design my potential was to make a potential with a width of ≈1 µm, and to run the simulation for 3–4 s to check if the Markovian Mpemba effect is observed within that time frame. A reasonable expectation would be that the Markovian Mpemba effect might occur within some small multiplicative factor of this range, perhaps 3, which would suggest a range of 9–12 s. In attempting to design a potential to observe the Markovian Mpemba effect, I ended up testing much longer times because nothing worked. For the potential chosen for the experiment, the time to

35 observe the Markovian Mpemba eﬀect was about 40 s, and the total time of the experiment was 100 s.

2.5.2 Power spectrum for free diﬀusion

Plotting the power spectra of a freely diffusing particle is one method of ensuring that the apparatus is functioning properly. This following section derives the theoretical formula for a freely diffusing particle to be compared with the experimental result in Section 3.4.1. In Jun et al.’s paper [32], the power spectrum of observed positions x¯ of a freely diffusing particle, sampled at a rate of 1/ts, is calculated with a finite camera exposure time, but it does not include observation noise, which varies depending on the illumination, optics, etc. [32]. In our experiment, the camera exposure time is chosen to be a factor of 10 smaller than the sampling time such that its contribution is negligible. Since I am doing an experimental measurement, I need to take into account the observation noise. The new equation for the power spectrum for free diffusion with observation noise is straightforward to derive from scratch following the steps in the Jun et al.’s paper [32] which derives the equation for the power spectrum for diffusion in a harmonic well with observation noise. We have two coupled equations,

xn+1 = xn + ξn (2.27)

x¯n+1 = xn + χn. (2.28) where x is the true position, x¯ is the observed position, and noise terms ξn (thermal noise) 2 and χn (observation noise) are assumed to be Gaussian, with hξni = 0 and ξn = 2Dts, 2 2 and hχni = 0 and χn = χ , D is the diﬀusion coeﬃcient of the particle, and ts is the sampling interval. The Z transform is the analog of the Fourier and Laplace transform for analyzing discrete time signals in the frequency domain given by [45]

∞ X −n x(z) = xnz . (2.29) n=0

Let the Z transform of xn be denoted by x, ξn by ξ, and χn by χ. From Eq. (2.29), the Z transform of xn+1 is zx. Equations 2.27 and 2.28 become

(z − 1)x = ξ (2.30) zx¯ = x + χ. (2.31)

Multiplying Eq. (2.31) by (z − 1) and substituting Eq. (2.30),

z(z − 1)¯x = ξ + (z − 1)χ. (2.32)

36 Taking the magnitude squared of both sides,

D E D E |z(z − 1)|2 h|x¯|i2 = |ξ|2 + |z − 1|2 |χ|2 + (z − 1) hξχi + (¯z − 1) hξχi . (2.33)

We substitute z = eiωt to revert to the time domain, where ω is frequency,

iωt iωt 2 2 D 2E iωt 2 D 2E iωt −iωt |e (e − 1)| h|x¯|i = |ξn| + |e − 1| |χn| + (e − 1) hξnχni + (e − 1) hξnχni . (2.34) Evaluating the noise and adding a factor of 2 to the right-hand side to make the power spectrum one-sided,

iωt iωt 2 2 iωt 2 2 |e (e − 1)| h|x¯|i = 2[2Dts + |e − 1| χ ]. (2.35)

Expanding out the terms,

−iωt iωt 2 −iωt iωt 2 (2 − e − e )h|x¯| i(ω) = 2[2Dts + (2 − e − e )χ ]. (2.36)

eiωt+e−iωt Using cos ωts = 2 ,

2 2 2(1 − cos ωts)h|x¯| i(ω) = 2[2Dts + 2(1 − cos ωts)χ ]. (2.37)

Rearranging, 2 2(Dts + χ (1 − cos ωts)) h|x¯|2i(ω) = . (2.38) (1 − cos ωts) Equation 2.38 is the equation for the power spectrum for free diﬀusion with observation noise and without camera exposure.

2.5.3 Implementing the Physical Model

Having given some context to the experimental constraints, this final section of this chapter engages in an extended discussion of implementation of the physical model mentioned in Section 2.4. Figure 2.12a) is a plot of Eq. (2.19) at T = 295 K. As can be seen in Fig. 2.12, the potential extends from 0 to 23 µm. Outside the potential, the probability of finding the particle is supposed to be zero, and this is achieved if the potential goes to infinity at the ends of the domain. Since the feedback trap cannot generate an infinite potential, instead, partial harmonic wells with α = 0.1, where α was defined in Section 2.3.1, are added to ends of the potential described by Eq. (2.19). To avoid discontinuity in the slope of the potential, the following constraints were imposed: F (0−) = F (0+), and F (23−) = F (23+), where F is the negative slope of the potential.

37 In Fig. 2.12 b), the red curve is the potential, and the black dotted curves are harmonic potentials with α = 0.1. Harmonic potentials are also superimposed in the two wells to show that the curvature in the wells does not exceed α = 0.1. The maximum alpha anywhere in the potential should be made exactly equal to α = 0.1 to have the shortest time to do the experiment. The value of α can be tuned a number of ways, such as by adjusting the curvature of the potential, the feedback rate, determined by ts, or changing the particle size. The maximum α in Fig. 2.12 a) is just below 0.1, but since the time needed to run the experiment is only marginally longer than if the maximum α = 0.1 exactly, no changes were made to Eq. (2.19). Figure 2.13 a) plots Eq. (2.19) with harmonic ends (black curve), following the construction in Fig. 2.12. Temperatures Thot = 5295 K (red line), Twarm = 1995 K (orange line), and Tcold = 295 K (blue line) that are used for the experiment are seen as the horizontal lines. Figure 2.13 b) plots the Boltzmann distributions corresponding to the potential and three temperatures shown in a).

We only need to obtain U/kBT , where T = 295 K, 1995 K, or 5295 K to obtain the the Boltzmann distributions corresponding to those temperatures. Rather than changing the actual temperature of the room, we instead adjust the potential that is imposed on the particle. For example, keeping the room temperature at T = 295 K, to get the Boltzmann distribution at twice the temperature, I reduce U by a factor of one half. The Fokker-Planck equation, Eq. (2.10), solved using the method of Grima and Newman [46], involves a spatial discretization. The choice of the spacing in the spatial discretization affects the calculations. As the spacing is made smaller, the solution converges to a limit that is the solution for the continuous potential. The solution in time is not discretized and can be calculated for any value of time. I fill in the details of the derivation of the Grima and Newman method in Chapter A. Shown in Fig. 2.14 is the calculated distance from equilibrium using different spacings of spatial discretization of the potential. The distance from equilibrium versus time curves are plotted for spacings, ∆x =0.5, 0.3, 0.1, 0.05, (purple, pink, green and blue, respectively) going from lowest to highest on the horizontal section. The curve for 0.1 and 0.05 are overlapping, so the solution has converged. Figure 2.15 shows the probability distribution over time, as calculated by solving the

Fokker-Planck equation during a quenching process from Thot = 5295 K to Tcold = 295 K, for the potential given in Eq. (2.19). At t = 0, the probability distribution corresponds to the Boltzmann distribution at the hot temperature, Thot = 5295 K. As time progresses, the probability distribution shifts into the lowest-energy state as the system system cools to the bath temperature at Tcold = 295 K. There are three distinct regions: left, centre, and right. The probability shift into the lowest-energy well from the left region requires hopping out of a well and is slow. The probability shift from the right region can involve sliding down a slight incline and is fast, because deterministic motion dominates, and the time for the to

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Figure 2.14: The step size is varied in the spatial discretization from purple: ∆x = 0.5, pink: ∆x = 0.3, green: ∆x = 0.1, to blue: ∆x = 0.05. The evolution from the initially warm system in a), and from the initially hot system in b) to cold is shown.

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Figure 2.15: Evolution of probability distributions for the potential shown in Fig. 2.13 from 5295 K to 295 K. Lines (green) sloping to the left denote the region where the probability shift into the lowest-energy well is slow. Solid-grey region is the stable energy well to which all the probability is ﬂowing. Lines (cyan) sloping to the right denote the region where the probability shift into the lowest-energy well is fast. travel the width of the potential scales linearly with the width; or it can involve mostly free diﬀusion, and the time for the particle to travel the width of the potential is proportional to the square of the width.

The quenching of both the initially hot system and the initially warm system at Twarm = 1995 K involve the same fast and slow modes of probability shifting because the potential for the quench is the same. The key to the initially hot system cooling faster than the initially warm system is in the right region, by making the potential to be wider while maintaining the slope of the plateau. As we increase the size of the right region, the initially hot system will have greater access to the additional states that are added, and its fast cooling will be prolonged, giving it a chance to catch up and overtake the cooling. Once all the probability from the right region is in the lowest-energy well, the remaining probability that is in the metastable well shifts to the ground state under the same dynamics for both the initially hot system and initially warm system. In Chapter 3, I introduce the experimental apparatus that I use to test the experimental model that was detailed in this section. I present experimental evidence that the apparatus performs as desired. Then I outline the measurements I need to make to observe the Markovian Mpemba eﬀect. In Chapter 4, I present my experimental results.

42 Chapter 3

Experiment

In this chapter, details about obtaining the image, and applying the feedback forces with the feedback trap are given. From the image of the particle, I discuss how the position of the particle is determined. I also explain how trajectories are used to compute probability distributions, and the distance from equilibrium, as well as how uncertainty is estimated. An example for calculating the distance from equilibrium is provided which also illustrates the effect of perturbations of the value of the diffusion coefficient, and consequently the effect on the distance-from-equilibrium curves. This chapter ends with experimental measurements of the system undergoing free diffusion, and diffusion in a harmonic potential over points in space that span the width of the potential that will be used for testing the Markovian Mpemba effect. These tests show that the feedback trap is able to sufficiently accurately produce an arbitrarily-shaped potential that satisfies certain constraints. I ensure these constraints are met with the potential that I will use to measure the Markovian Mpemba effect.

3.1 Experimental Setup

Turning now to the experimental implementation of the Markovian Mpemba effect, I will first introduce the experimental apparatus. The feedback trap will be used to experimentally test the Markovian Mpemba effect. The feedback trap consists of two main parts: an imaging component and a feedback force. The simplest application of the feedback trap is when only the imaging component is active, such that the system is undergoing free diffusion. The simplest nontrivial scenario is a constant rate of change of force with position, in which the system approximately behaves as though it is inside of a harmonic potential. For testing the Markovian Mpemba effect, a potential with an arbitrary shape, consisting of multiple potential wells and sloped regions is required. Figure 3.1 is a schematic of the experimental setup. This setup was made mostly by undergraduate student Robert Löffler, as part of his thesis work [47], and the samples used in this thesis were prepared

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Figure 3.1: Schematic of the experimental setup. The red dot is the particle and the blue rectangle is the sample cell. The particle is located mostly between the dotted lines, near the bottom of the sample cell. Cyan rectangles in the sample cell indicate electrodes. The electrodes are shown as line charges in the top view. The red arrows indicate light rays; the black rectangle is the beam blocker; and the cyan oval represents the lens that forms the image at the camera.

44 according the procedure used by him, which was adapted from the work of Gavrilov et al. [43]. The system consists of a non-functionalized silica microsphere1, shown as the red sphere, diffusing in deionized water2 contained in a sample cell with a glass bottom, shown as the blue rectangle. The sample cell was covered with a glass slide to prevent evaporation of the water bath. The particle vertically fluctuates at an average height, h, above the bottom of the sample cell, because it is vertically confined via a combination of the gravitational potential, and an electrostatic potential due to electrostatic interactions between the particle, the aqueous solution, and the bottom surface of the sample chamber, which will discussed in greater detail in Section 3.3. Forces in the plane parallel to the ground are created by applying voltages at the electrodes, producing an electric field at the particle which moves via electrokinetic forces [48]. The computer sends two control voltages from a multifunction DAQ device3, which are amplified using an amplifier made by SFU’s in-house electronics shop, before being applied to the system via two orthogonal pairs of platinum electrodes4 (cyan rectangles in Fig. 3.1) immersed in the aqueous solution. Using two voltages, the motion of the particle can be controlled in two dimensions. I have measured that the voltage is between ± 60 V at the electrodes. We would like to approximate the electric field at the particle as being from infinite line charges such that the electric field is approximately uniform within the working area. Since the particle is much smaller than the electrodes, this could be possible if the particle were only moving near the middle of the electrodes. In Section 3.4.3, I check that a harmonic trap can be produced over a series of positions which span the desired range of motion. Although the net charges of the water, particle, and sample cell surfaces are zero, at the interfaces between the glass surfaces and the aqueous medium, mobile ions in solution can form layers with net charge [49]. In the presence of an electric field due to the applied voltage difference, the flow of ions induces a drag force acting on the particle [50]. The relation between the force and the applied voltage is determined using real-time software which allows the desired force to be applied [34]. Since the desired force can be chosen corresponding to the position of the particle, those forces can be chosen with a potential of a particular shape in mind, such that behaviour of the particle due to the forces is the same as its behaviour if it were in such a potential. Further, if the distribution of the particle in the potential at some desired temperature is known, the particle can be placed using a harmonic well at specific positions with probabilities given by the distribution. The particle is uniformly illuminated by a high-power red LED5, which is indicated by three parallel red arrows entering from the top of the image. The light rays from the LED are incident on the particle as a plane wave. The light scatters off the particle and

11.49 ± 0.1 µm, SS04N, Bangs Laboratories, Inc. 2Millipore Milli-Q, 18.2 MΩ cm−1, 0.22 µm 3National Instruments PCI e-6353 4Pt80/Ir20, 0.25 mm, Goodfellow Corp. 5Thorlabs, M660L4, λ=660 nm, 1200 mA, 940 mW

45 Figure 3.2: Dark-field image of a 1.49 ± 0.1 µm silica microsphere recorded by the camera at 200 fps. A diffraction pattern is seen because of the small size of the particle. is collected by the microscope objective6, which outputs the light rays from the particle as parallel beams, effectively forming the image infinitely far away. A beam blocker, the black rectangle, blocks the direct and forward-scattering light beam, leaving only high-angle scattered light from the particle, so that the light from the LED does not overpower the scattered-light from the particle. A lens, the cyan oval, is used to form the particle image at a CMOS monochrome camera7. The result is a darkfield image as shown in Fig. 3.2. Further software processing by a computer is done to determine the position of the particle, including applying a intensity threshold on the pixels in the image, and calculating the centroid of the intensity [51]. Pixels with an intensity level below that are set to zero intensity to reduce the measurement error due to random fluctuations in intensity. Figure 3.3 is a colourmap of the intensity of the camera image after thresholding. Cooler colours indicate higher intensity and warmer colours indicate lower intensity, as shown by the colour scale, with the highest intensity near the middle of the bead. The particle position is determined by calculating the centroid of the intensity distribution [51]. When the colours seen in Fig. 3.3 converted to gray-scale, each pixel has a scalar number that corresponds

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Figure 3.3: Colourmap image of a 1.49 ± 0.1 µm silica microsphere recorded by the camera at 200 fps. The colours represent intensity. Purple is highest intensity and red is zero intensity. to the intensity. If the intensity is symmetric about some axis, then the centroid is on the axis; if the intensity is skewed towards side of the axis, the centroid is shifted towards that side. To determine the conversion factor from pixels to microns for the scale bar, an image was taken, without thresholding, of a calibration ruler, which is a glass slide standard, with lines that are spaced a known distance apart. Over several repeated measurements, I obtained a pixel-to-micron conversion of 7.09 ± 0.01 px/µm.

3.2 Experimental Protocol

Having discussed how the experiment works, this next section addresses what data needs to be collected to observe the Markovian Mpemba effect. The Markovian Mpemba effect is experimentally observed by calculating and comparing the distance from equilibrium of two cooling processes over time. The two cooling processes involve two identical systems that differ only in their initial temperature. The quantities that are required for calculating the distance from equilibrium, using Eq. (2.9a), are U,

π(Tb), and p(x, tp), where x is position, Tb is bath temperature, and tp is the time elapsed in the quench. The quantities U, π(Tb) and Tb, are related to equilibrium properties of

47 t0 t1 t2

n=1

n=2

n=3

Figure 3.4: Schematic of data structure. n is experimental runs, and t is the sampling time. the system, and are already thoroughly understood. Although the Boltzmann equation,

Eq. (2.7), can be used to obtain U and π(Tb) directly, measured values of are used instead. The experimentally measured distribution can have some distortions, whose eﬀect on the calculated distance from equilibrium can be reduced if we compare distributions which are both distorted in the same way. Details for the estimation of U and π(Tb) are provided in Section 4.2 in Results.

The non-equilibrium properties are encoded in the last quantity, p(x, tp). The ability to measure p(x, tp) is central to the observation of the Markovian Mpemba eﬀect. The method for estimating p(x, tp) is explained in this section. The technique used for estimating uncertainties of p(x, tp), data resampling, is also described.

3.2.1 Estimating the time-dependent distribution

To determine the distance from equilibrium of a particle at a certain time, tp, during cooling, the probability distribution of the particle position, p(x, tp), must be estimated. This estimation is done by doing repeated experimental runs. In the experiment, 1000 repeated experimental runs are used to calculate each distance-from-equilibrium curve in Fig. 4.8. The number of experimental runs that are done affects how well the experimental Markovian Mpemba effect agrees with the prediction based on the Fokker-Planck equation, Eq. (2.10). The length of time taken for each experimental run was 100 s, which is sufficient for observing that the initially hotter system overtakes the initially warmer system, allowing a sufficient amount of overtaking to be able to see a clear separation of the two curves, and showing that the phenomenon persists for a non-negligible amount of time. In each run, a trajectory, which is a sequence of measurements of the position of the particle recorded at 5 ms intervals, is obtained.

48 Figure 3.4 illustrates the structure of how the data is analyzed. The horizontal rows are trajectories from repeated experimental runs. The vertical columns are the sampling times. Horizontal arrows indicate that the trajectory length is adjusted to increase the time of cooling, and vertical arrows indicate that the number of trajectories is adjusted to improve the estimate of the probability distribution. The data that is used to estimate the probability distribution at a particular time is enclosed by a rectangle. Three rectangles are shown to illustrate the sets of data that are used to estimate probability distributions at t0, t1, and t2.

In this scheme, p(x, tp) is estimated using independent measurements of position. Each experimental run contributes one count to the histogram estimate of the probability distribution at each given time. To estimate p(x, tp) from the experimental data, the measured position x at tp from each trajectory is histogrammed into 100 bins with binwidth 0.24 µm. Since all the data is available for short times once the full length experiment, short trajectories are taken as subsets of the longest trajectories. Using this method, we can extract the maximum amount of information out of the data that is collected. Rather than doing two sets of experiments to measure the particle position after, for example, 1 s and 3 s of cooling, I do the one 3 s set and use the measured position at 1 second to calculate the probability distribution of the particle after 1 second of cooling, and the measured position at 3 s to calculate the probability distribution after 3 s of cooling. The distance-from- equilibrium function from the estimated distributions at times, tp, spaced far apart, with a suﬃcient number to show the shape of the distance-from-equilibrium versus time curve. It can be expected that if the distance is too high or too low in an earlier data point, that later data points will be shifted in the same direction because the points are all taken from the same process, but the correlation time for that has not been studied here.

Having obtained p(x, tp), along with knowing U, π(Tb), and Tb, the distance from equilibrium can be calculated by plugging in these values into Eq. (2.9a). An example for going from trajectory to the distance from equilibrium will given after a brief discussion about estimating uncertainities.

3.2.2 Data resampling

The problem of estimating uncertainties is discussed in the following section. To estimate the uncertainty in our measurements without measuring repeated sets of data, the resampling method was used [52]. The resampling method randomly samples values from the measured set of data as though new samples are being drawn from the entire population. The variance in the resampled data approximates the variance in the population [52]. I check that this is true for the type of data we are measuring using simulations. Figure 3.5 shows simulation data on independent trials (top) and resampled data (bottom). I compare 10 and 100 independent trials, with 5 and 1000 resampled data sets. When the number of repeated data sets and resampled data sets are small, the error bars are dif-

49 ferent sizes on adjacent points on the curve. When the number of data sets is suﬃciently large, the error bars at adjacent points on the curve are similar in size and further increasing the number of data sets yields no further large changes in the size of error bars. Resampling 105 data sets is used, which exceeds the required number of data sets needed to obtain a good estimate of the uncertainty.

3.3 The effect of height fluctuations on the particle diffusion coefficient

In addition, it is important to ask what are the possible effects of assuming a constant diffusion coefficient when it in fact varies and how to tell when the variation in the diffusion constant is too large to be assumed to be constant. The real-time estimation of the particle diffusion coefficient is a direct measurement of the diffusion coefficient of the particle. The diffusion coefficient for a particle that is diffusing infinitely far from any surface is given by Einstein’s relation, Eq. (2.11). However, in the experiment, the particle is diffusing near a wall. The diffusion coefficient of a particle diffusing parallel to the wall is reduced as a function of the distance to the wall as follows [34, 53] D(z) 9 d 1 d3 45 d4 1 d5 = 1 − + − − , (3.1) D∞ 32 z 64 z 4096 z 512 z where z is the actual perpendicular distance of the particle from the wall, and d is the diameter of the particle. Since z is on the order of d, the wall cannot be approximated as being infinitely far away. As we do not measure the distance of the particle to the wall of the sample cell, this will also impact the uncertainty in D. Equation 3.1 is graphed in Fig. 3.6 for a 1.49 µm particle diffusing in water with η = 0.93 mPa s. I compare a plot of Eq. (3.1) (solid red line) with the truncated series up to the 1/z term (dashed black line) . We see that the truncated series converges to Eq. (3.1) when the particle is over 1 µm away from the wall; the additional corrections are required when the particle is closer than 1 µm to the wall. Previously, the real-time estimate of the diffusion coefficient is taken as the average over many feedback cycles [34]. Following this logic, the real-time estimation can be turned off, after the estimate of the diffusion coefficient has converged to a constant value. In practice, the real-time estimation is left on, because the average value is almost constant. As a reference, the real-time diffusion coefficient was 0.227 ± 0.004 µm2/s for the data collected to plot Fig. 3.12. However, care must be taken when the real-time estimate is left on, because when the voltage required exceeds the maximum voltage that can be generated, the real-time estimate of the diffusion coefficient and mobility, which is calculated based on the difference between the intended position of the particle and its measured position [34], quickly deteri-

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Figure 3.6: Diffusion coefficient vs height. Red solid line is from Eq. (3.1) for a 1.49 µm particle diffusing in water with η = 0.93 mPa s. Dotted black line is using up to the 1/z term. orates, causing ‘instability’. Being able to apply the correct restoring forces is the principle of operation of the feedback trap and that requires knowing the diffusion coefficient and mobility of the particle. This is the reason why the particle in a harmonic trap at moderate values of α may not remain trapped, even though the theory predicts that higher values of α are allowed. When the diffusion coefficient and mobility values are wrong, the trap is unable to respond correctly to prevent the particle from going out of bounds. I have argued that the diffusion coefficient of the particle should not be approximated as a constant number. Due to the proximity of the particle to the bottom surface of the sample cell, small variations in height could be large in terms of the relative percentage of the distance to the wall. The importance of this consideration depends on the time scale of the changes in height. Given that the time scale is short compared to the time scale of the dynamics of the experiment, large fluctuations would be averaged out. To capture changes of the diffusion coefficient that are on the relevant time scale, I use an averaging time in the real-time estimate of 10 s, which is short compared to the length of a single run which is 100 s. Using this method, I have at least 10 cycles of variation in my experiment. This real-time diffusion coefficient is only used in the calculation of the voltages. The length scale of the potential is kept fixed at a value of the diffusion coefficient that is estimated at the beginning of the experiment. To obtain an estimate of the variations in height of the particle over time, I simulate a trajectory of the particle in the vertical direction using a potential that includes the gravitational potential and Debye-Huckle potentials [53], which is the electrostatic repulsion

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53 in my simulations of the particle trajectory in the potential shown in Fig. 2.13. This series of steps was used to obtain simulations of trajectories that account for the varying diffusion coefficient. These measured trajectories are processed as described in Section 3.2.1 to obtain the time-dependent distribution. Light and dark green solid lines are solutions to the Fokker-Planck equation, Eq. (2.10) for systems initially at T = 5295 K and T = 1995 K, respectively, quenched to T = 295 K. Squares joined by dashed lines are simulations using varying the diffusion coefficient. Circles joined by dashed lines are simulations using a fixed diffusion coefficient. Triangles joined by solid lines are experimental results. All shades of red denote systems initially at T = 5295 K quenched to T = 295 K, and all shades of orange denote systems initially at T = 1995 K quenched to T = 295 K. Simulation taking into account of a varying diffusion coefficient and simulation using a fixed value of the diffusion coefficient is compared with the Fokker-Planck solution. In a), experimental results is also plotted. The method for determining the error bars is discussed in Section 3.2. We see that there is a shift in the curves away from the Fokker-Planck result when a varying diffusion coefficient is used in the simulation. This is also observed in the experiment, but the experimental result and simulations using a varying diffusion coefficient do not overlap. This could be because the parameters used in the Debye-Huckle potential to do the simulation are inaccurate, and further studies will have to be done.

3.4 Benchmarking the Feedback Trap

In the section that follows, it will be argued that the feedback trap will be able to approximately generate the desired potential for the Markovian Mpemba effect experiment. Prior to commencing the study, the apparatus needs to be checked if it is working. To do this, I measure the distribution of the particle under free diffusion, and in a feedback trap- generated harmonic potential, and compare it to theory for the power spectrum of a particle in a feedback trap [32]. To rule out the possibility that there are systematic variations with position, I measure the average position and standard deviation of a harmonic potential with α = 0.1 placing the harmonic well at a range of positions in the feedback trap and check for trends. Having shown that the above can be done, I can proceed to construct an arbitrary potential that satisfies α < 0.1 at all points of the potential, which is true of the desired potential shown in Fig. 2.13.

3.4.1 Free diﬀusion

When a signal is suspected to be periodic, the power spectrum can be calculated which shows the frequencies that are present, and the amplitude of those frequencies [54]. Dr. Bechhoefer pointed out that the power spectrum for Brownian motion in a harmonic well should overlap with the power spectrum for free diﬀusion when plotted on the same graph at high frequencies. Therefore, I did simulations and collected experimental data for Brownian

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Figure 3.8: Effect of varying diffusion coefficient on the Markovian Mpemba effect. a) is using 1000 sets of trajectories and b) is using 2000 sets of trajectories. Light and dark green solid lines are the cooling curves calculated using the Fokker-Planck equation for systems initially at T = 5295 K, and T = 1995 K, respectively. Triangles joined by solid lines are experimental data; squares joined by dashed lines are simulated data where the diffusion coefficient is from simulating the vertical height; and circles joined by dashed lines are simulated data where the diffusion coefficient is fixed. All shades of red are for systems initially at T = 5295 K being quenched to T = 295 K, and all shades of orange are for systems initially at T = 1995 K being quenched to T = 295 K.

55 motion to check this. Although this is a good check because it decouples the optics from the feedback mechanism, there is neither simulated nor experimental results on this result in previous feedback trap-related publications8 [32, 33]. Upon doing so, I discovered that the code written by Dr. Bechhoefer for processing the feedback trap data does not produce the correct power spectrum for a freely diffusing particle, to which Dr. Bechhoefer asserted that something must be wrong with both my experimental and simulation data. He later claimed that no one from our laboratory has previously collected experimental data for a freely diffusing particle using the feedback trap with feedback forces set to zero. The power spectrum can be obtained by performing a discrete Fourier transform of the data [54]. A property of the power spectrum is that the standard deviation is equal to the value [52]. Thus, calculating the power spectrum a single time gives no information. The standard deviation can be reduced by taking the average over multiple power spectra, which reduces the standard deviation by a factor of the inverse square root of the number of power spectra used in the average [52]. This can be achieved by dividing the data into blocks, and calculating the power spectra for each block of data [52]. The greater the number of blocks, the smaller the standard deviation of the averaged power spectrum can be made. However, for smaller sized blocks, fewer frequencies are included in the power spectrum. The theoretical power spectra in Fig. 3.10 and Fig. 3.11 which are the blue dashed lines, are calculated by inserting measured experimental parameters into Eq. (2.38). For free diffusion, the experimental parameters that need to be measured are the diffusion coefficient D, and the standard deviation of the observation noise χ. Real-time estimation [34] of parameters was not used while taking the measurement for free diffusion, although it could have been, and a more accurate estimate of the diffusion would have been obtained. Instead, I calculate the diffusion without considering the observation noise. The diffusion coefficient D can be calculated using [32]

2 2 h(∆¯xn) i = 2Dts ⇒ D = h(∆¯xn) i/2ts (3.2) where ∆¯xn are measured one-step displacements, x¯n+1 − x¯n, sampled at times t = nts, where ts is the time increment. Brownian motion was observed over about 127 s, and the position of the particle was recorded at 200 Hz. This single trajectory was divided into 7 pieces, each 15 s long. Dis- placements in position ∆¯xn, were calculated as differences in the recorded positions and plugged into Eq. (3.2) to calculate the diffusion coefficient D. The calculated diffusion coefficient D is D = 0.24 ± 0.01 µm2/s. Unless otherwise specified, parameter uncertainties are one standard deviation, about 68% of the probability under the normal distribution. Since the vertical probability distribution of the particle is skewed away from the bottom

8A claim is made in Gavrilov et al.’s [33] experimental paper that harmonic potentials with α < 0.05 cannot be measured, due to being limited by the ﬁeld of view.

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Figure 3.9: One-dimensional trajectory of particle undergoing Brownian motion sampled at 200 Hz for about 127 s. of the sample cell, the actual diffusion should be higher than what I have calculated. The standard deviation of the observation noise is measured to be χ = 6.25 ± 0.01 × 10−3 µm, using the observed positions of a stuck particle. Since I only make one measurement and it is too short to be divided into shorter subsets and analyzed separately, the standard deviation comes from fitting to the data. The observation noise is negligible and can be 2 omitted since from Eq. (2.38), Dts χ (1 − cos ωts). However, I include observation noise as a fitting parameter and compare it with the independently measured observation noise for the stuck particle. For a freely diffusing particle, a single trajectory of one particle about 127 s long was recorded, sampled at 200 Hz, to obtain 25476 experimental data points. The motion of the particle is in three-dimensions; however, the particle motion in each dimension is independent of the motion in the other dimensions. The one-dimensional trajectory of the particle is used, shown in Fig. 3.9. The data set was broken up into 199 blocks consisting of 128 consecutive data points each, and power spectra were calculated for each block over a frequency range of 1.6 Hz to 100 Hz and averaged. Figure 3.10 is a plot of the power spectrum for experimental data for free diffusion, also showing the theoretical curve. As can be seen in Fig. 3.10, the experimental data (red markers) clearly deviates from the theoretical prediction (blue dashed line) calculated from Eq. (2.38). Dr. Bechhoefer initially believed that a mismatch of units was the reason for this discrepancy. I pointed out the analysis code he wrote was using a rectangular window. As discussed in Wu et al., when the data contains noninteger multiples of periods, leakage occurs, in which the power at one frequency spreads into neighbouring frequencies [54]. He shows that using Hanning window reduces the amount of leakage. I developed the revised code that uses a Hanning window in collaboration with Dr. Bechhoefer.

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Figure 3.10: Power spectrum of experimental data for freely diﬀusing particle without windowing. Red markers are experimental data. Blue dashed line is calculated using Eq. (2.38).

Figure 3.11 is the revised power spectrum for the freely diffusing particle. Equation 2.38 was fit to the data, and the result is shown as the light-red line on Fig. 3.11. The point at the first non-zero frequency is not included in the fit because seems to occur due to the Hanning window. For this fit, χ2-statistic = 55 for ν = 61 degrees of freedom for which we expect the χ2-statistic = 61 ± 11. I have calculated the 99.95% confidence interval (light-red shaded region) from the fitted curve. The choice of 99.95% was to make the confidence interval visible in the graph. As explained earlier, the error bars is the standard deviation of the power spectrum is given by the value reduced by a factor of the inverse square root of the number of power spectra used in the averaged. This plot shows the averaged result, so calculation of residuals is not applicable. The fit parameters are diffusion D and the standard deviation of the observation noise χ. The fitted diffusion is D = 0.254 ± 0.004 µm2/s, and the fitted standard deviation of the observation noise is χ = 4 ± 2 × 10−3 µm, which is consistent with previous results to within two standard deviations. The fitted observation noise is expected to differ from the observation noise of a stuck bead because experimental conditions, such as the amount of illumination and proximity to the bottom of the sample chamber, can vary.

3.4.2 Harmonic potential

For particle diﬀusion in a harmonic potential with α = 0.1, 105001 experimental data points are obtained, sampled at 200 Hz. The parameter α is deﬁned as follows. The measured trajectory of the particle in a harmonic well given by U(x) = 1/2kx2, that is sampled

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Figure 3.11: Power spectrum of experimental data for freely diffusing particle with windowing. Red markers are experimental data. Blue dashed line is calculated using Eq. (2.38). Shaded light-red region is the 99.95% confidence interval for the fitted power spectrum, based on the distribution of the estimate.

at a rate of 1/ts, with drag constant γ, has α = kts/γ [32]. The parameter α is a non- dimensionalized result that makes results obtained under experimental conditions with the same α value comparable. In particular, the size of the particle can diﬀer, but the α can still be the same by changing the value of the spring constant k. The power spectrum of observed positions x¯ of a particle diﬀusing in a harmonic well, sampled at a rate of 1/ts for unit delay, td = ts, again omitting the camera exposure because it is chosen to be one order of magnitude smaller than the sampling time ts so that its contribution is negligible, is given by [32]

2 2[2Dts + 2χ (1 − cos ωts)] h|x¯|2i(ω) = , (3.3) |e2iωts − eiωts + α|2 where most constants are already defined in Section 3.4.1, with an additional parameter, α described earlier in this section, which parametrizes the harmonic well. The theoretical power spectrum is calculated for experimental parameters plugged into Eq. (3.3). The real-time values of the diffusion coefficient D are real-time estimates of D produced by the software that is updated with every feedback cycle [34]. For the above data, the real-time estimate of the diffusion yields D = 0.227 ± 0.004 µm2/s. The standard deviation of the observation noise is measured to be χ = 6.25 ± 0.01 × 10−3 µm, using the observed positions of a stuck particle, as was used for free diffusion.

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Figure 3.12: Power spectrum of experimental data for a particle diffusing in a harmonic potential for α = 0.1. Red markers is experimental data. Blue dashed line is calculated using Eq. (3.3). Shaded light-red region is 99.95% confidence interval centered around darker red line that is a fit to the data.

The power spectrum in Fig. 3.12 is calculated for blocks of 2100 consecutive data points, averaged over 50 blocks, over a frequency range of 0.1 Hz to 100 Hz with a Hanning window applied. The standard deviation is obtained by the method described in Section 3.4.1. Figure 3.12 is a plot of the power spectrum of the experimental data of particle motion in a harmonic well, and the theoretical curve. As before, red markers are experimental data, blue dashed line is the theoretical curve, this time from Eq. (3.3). Equation 3.3 was fit to the data, and the result is shown as the light-red line in Fig. 3.12. The light-red region is the 99.95% confidence interval of the fitted curve. The choice of 99.95% was to make the confidence interval visible in the graph. For the fitted curve, χ2-statistic = 1119 for ν = 1049 degrees of freedom from which we expect the χ2-statistic = 1049 ± 46. The fit parameters are the standard deviation of the observation noise χ and diffusion D. The fitted parameter for the diffusion is D = 0.217 ± 0.002 µm2/s. The fitted standard deviation of the observation noise is χ = 1 ± 4 × 10−3 µm, which is consistent with previous results to within two standard deviations. The fitted observation noise is expected to differ from the observation noise of a stuck bead because experimental conditions, such as the amount of illumination and proximity to the bottom of the sample chamber, can vary.

3.4.3 Harmonic potential at diﬀerent positions

This is the ﬁrst study using a potential as wide as the one shown in Fig. 2.13, which is over 20 µm wide. Previous studies using the feedback trap have only dealt with potentials

60 that are about 5 µm wide. There was concern that edge effects would occur once the size of the potential became large compared to the dimensions of the apparatus. This would lead to a breakdown of the assumption of a uniform electric field. Therefore, tests were done moving the harmonic potential with α = 0.1 to different locations in the trap. The positions were selected at evenly spaced intervals spanning the size of the potential chosen for this experiment. Figure 3.13 is a plot of the average and standard deviation of the particle position in a harmonic well that is moved to different locations in the feedback trap. Green markers are from about 3 minutes of data each. The error bar of the red marker is from 21 repeated measurements of 3 minutes each. The repeated measurements are done at a single position to give an estimate of the size of the standard deviation, and single measurements are done at the other positions. Blue dashed line in a) is the setting the output position equal to the input position, and in b) is square root of the position variance [32]

2 1 + α hx i = 2Dts . (3.4) α(1 − α)(2 + α) Shaded blue region is the blue dashed line plus and minus the standard deviation of the red marker. As can be seen in Fig. 3.13, the measured positions of the particle lies on the plotted theoretical line, and the standard deviation of those positions appear to be statistically distributed around the theoretical value. Since systematic variations in the measured parameters were ruled out, I proceeded to construct the wide potential.

61 a)

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0 Output Position ( -20

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0.12

0.10

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Figure 3.13: Measured mean and standard deviation of the position of a particle in a harmonic potential with α = 0.1 placed at diﬀerent locations. a) Plot of programmed location of the position of the minimum of the harmonic well versus measured average position of a particle in that potential. b) Plot of measured standard deviation of the particle. Green markers are single measurements. Red marker is from 21 repeated measurements. The standard deviation of the 21 measurements gives the error bars. Only the red marker in a) and b) have error bars because single measurements are made at the other positions. Blue dashed line in a) is output position equal to input position, and in b) is from Eq. (3.4). Shaded blue region in b) is from adding the error bars of the red marker to the blue dashed line.

62 Chapter 4

Results

In this chapter, results and analysis are presented together. Computer simulation was done as a precursor and follow-up to experiment. The primary purpose of the simulation is to confirm the data-analysis procedure and to estimate uncertainties. The first simulation is to determine the initial position of the particle. The particle is initially in the equilibrium hot or warm state. It is too costly to allow the particle to equilibrate with the potential coupled to a hot bath because the time that is needed for the particle to come to equilibrium must be multiplied by the number of experimental runs. Subsequent simulations are done to show how much data needs to be taken for experimental observation of the Markovian Mpemba Effect for a colloidal particle in the potential given in Eq. (2.19) for bath temperatures 295 K, 1995 K, or 5295 K. The bath that the particle is coupled to is at thermal equilibrium with the room, whose temperature I measured to be 22 ± 2°C with measurements taken at different times over the course of several days. Therefore, value of temperature I used for all my simulations was Tb = 22°C. The temperature of the room when I actually did the experiment was a bit higher, with a maximum of 26°C being observed at one point. As explained in Section 2.4, what the actual temperature is does not change the result of the experiment. I used the value of the diffusion coefficient that I independently measured in Sec- tion 3.4.1, D = 0.24 ± 0.01 µm2/s for the simulations. The real-time measurement of the diffusion coefficient during experiment tended to be a bit higher, because the temperature was a bit higher as well. Fluctuations in the diffusion coefficient which were discussed in Section 3.3 are not taken into account because we do not measure the parameters for the Debye-Huckle potential.

63 4.1 Sampling from a distribution

As was pointed out in the introduction to this thesis, it is not experimentally feasible to directly create a system in thermodynamic equilibrium. Instead, we use the alternate method described below. In the experimental protocol, we require that an ensemble of trajectories be measured whose initial positions are sampled from a system that is at equilibrium at the hot and warm temperatures. The direct method to obtain the initial position is to pick a random time to sample a position from a system at equilibrium. However, it takes time to prepare the system such that it is at equilibrium. To initialize the particle in the hot and warm state, instead of waiting for the system to equilibrate, we place the particle at predetermined positions that are sampled with the help of a software-based random number generator, from the Boltzmann distributions calculated using Eq. (2.7), for which temperature T in Eq. (2.7) corresponds to the hot and warm temperatures. We choose to use a computer- generated random position from the desired Boltzmann distribution. Having generated the position, it becomes a matter of placing the particle at the calculated position and releasing it at the start of the experiment. Considerable work and expertise goes into designing a random number generator to make the numbers that come out seem random. There is no random number generator that is designed specifically to sample from our desired probability distribution; fortunately, a simple transformation method [52] allows us to take a random sample from a known distribution, and transform it into a random sample from our desired distribution. A position is randomly sampled using a computer random generator; a built-in uniform distribution random number generator is used to generate a number between zero and one. This number is interpreted as the cumulative probability of the uniform distribution, the integrated probability density starting from the smallest position where the probability density is nonzero up to the position at which the cumulative probability is being defined. The value of the cumulative probability in the uniform distribution is matched with a similarly defined cumulative probability in the hot or warm distribution, and the corresponding position in the hot or warm distribution is used as the initial position of the particle. The cumulative distribution function (cdf), is the set of cumulative probability defined for all positions. The cdf is a map between values of position and values of the cumulative probability. The inverse of the cumulative distribution function, cdf−1 outputs the corresponding value of the position for a given value of the cumulative probability following the mapping of the cdf, as expressed by the formula

x(u) = cdf−1(u), (4.1) where x is position and u is a random sample from a uniform probability distribution. The built-in uniform random number generator in Matlab is used, which samples from

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Figure 4.1: Cumulative distribution function of Boltzmann distribution for the potential in Eq. (2.19) with bath temperature Tb = 1995 K. u is randomly sampled from a uniform distribution over [0,1], and x is position. a uniform probability distribution with its range set to [0,1]. The probability of a sample being in the range [u, u + du] is given in Eq. (4.2).

 du if 0 < u < 1 p(u)du = (4.2) 0 otherwise

The output x of this transformation is a random sample from the desired probability distribution. Figure 4.1 illustrates how x is determined. The sample u is read on the vertical axis, and the transformed sample x is read on the horizontal axis, and (x, u) is a point on the curve cdf(x). Figure 4.2 shows histograms of computer-generated random samples of position plotted along with the theoretical curves given by Eq. (2.7), that were shown in Fig. 2.13 b). Red and orange markers are from histogramming 10000 randomly sampled points into 100 bins, with binwidth 0.24 µm. Figure 4.2 a) is the Boltzmann distribution for Twarm = 1995 K, and b) is the Boltzmann distribution for Thot = 5295 K. Both ﬁgures exhibit two peaks, corresponding to the wells in the potential shown in Fig. 2.13 a). The potential rises sharply at x = 0 and x = 23, causing the probability density to rapidly go to zero to the left of x = 0 and to the right of x = 23, assuming the particle is initially between x = 0 and x = 23. As can be seen in Fig. 4.2, there is more scatter in the histogram for the hotter temperature.

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Figure 4.2: Sampled Boltzmann distribution for a) Tb = 1995 K and b) Tb = 5295 K for a 1.49 µm particle diﬀusing at 0.24 µm2/s. Red and orange markers are histograms of 10000 computer-generated random samples and black line is theoretical value.

66 4.2 Estimating the equilibrium distribution and potential

The equilibrium distribution and potential cannot be measured perfectly because of finite trajectories and statistical noise. To ensure the measured equilibrium distribution and potential has good resolution of features and data taken over the whole domain, simulations are used to obtain the equilibrium distribution and potential for different experimental protocols, to select the protocol with results closest to the Boltzmann distribution computed using Eq. (2.7). First, we use simulation to choose the temperature at which to sample the distribution. Then, we use simulation to see how long a trajectory is needed. The optimal temperature and trajectory length is determined by plotting the distance from equilibrium between the measured distribution and the Boltzmann distribution against temperature and trajectory length. The optimum temperature is the temperature with the smallest distance from equilibrium, and the optimal trajectory length is some trajectory length after which additional increase of trajectory length does not result in a significant decrease in the distance from equilibrium. The equilibrium distribution is estimated from the measured distribution of particle positions of a particle diffusing in the potential in Eq. (2.19), coupled to a bath at some temperature Tb. The trajectory must be sufficiently long that the particle has time to visit everywhere in its configuration space multiple times. The potential is ideally the same as the imposed potential, but since the experimental potential might differ slightly from the desired potential, we should measure the potential that is actually felt by the particle. The measured potential can be calculated from the measured distribution of particle positions using

U(x) = −kBTb ln π(x) (4.3) where x is position, and π(x) is the Boltzmann distribution, given in Eq. (2.7). Figure 4.3 shows the Boltzmann distributions for the potential in Eq. (2.19) at three bath temperatures: red dotted curve, Thot = 5295 K; orange dashed curve, Twarm = 1995 K, and blue solid curve, Tcold = 295 K. For the warm and hot temperatures, the Boltzmann distribution have much more probability density than the Boltzmann distribution for the cold temperature over much of the potential width. These Boltzmann distributions are potentially easier to measure. For the cold temperature, the Boltzmann distribution has a very low probability density for much of its domain, approaching 10−30. There is effectively zero chance of ever observing the particle anywhere but inside the two energy wells. This would be fine for the estimate of the particle distribution, but we would like to also calculate the potential, and zero probability corresponds with an infinite potential, which is not physically possible, as well as being difficult to deal with mathematically.

Measurement of the Boltzmann distribution at diﬀerent bath temperatures than Tb =

295 K is achieved using the relationship between U(x), π(x), and Tb in Eq. (4.3). We scale

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Figure 4.3: Boltzmann distributions for the potential in Eq. (2.19) on log-scale for three bath temperatures. Red is Thot = 5295 K. Orange is Twarm = 1995 K. Blue is Tcold = 295 K.

the U that we impose, while keeping Tb = 295 K to estimate π(x) at the corresponding scaled temperature. For example, for the imposed scaled potential 2U(x), the corresponding scaled temperature is Tb = 295/2 K. To measure the Boltzmann distribution at the cold temperature, the unscaled potential U(x) is ﬁrst calculated from Eq. (4.3) using the measured π(x) and the scaled temperature. Then, π(x) at Tb is calculated using

1 π(x) = e−U(x)/kBT ,T = 295 K, (4.4) Z where Z is the normalization constant. To summarize, the Boltzmann distribution for the cold temperature can be estimated as follows:

1. Measure the Boltzmann distribution at a higher temperature by scaling U(x).

2. Calculate U(x) from the measured distribution at the higher temperature.

3. Calculate π(x) at 295 K using the potential calculated in the previous step.

4.2.1 Optimum temperature

The choice of temperature has statistical implications on how well the equilibrium distribution and potential can be measured. For low temperatures, there will be few counts in most areas outside of the potential wells. For high temperatures, the position counts will be spread out over a large area and the signal-to-noise ratio at the wells will be low. For some

68 intermediate warm temperature, the Boltzmann distribution has both well-defined features and nonzero counts everywhere so that the potential, calculated using Eq. (4.3), does not go to infinity anywhere. When the bin size on the histogram is widened to improve the signal-to-noise ratio, the tradeoff is that smaller features will be lost. One solution to avoid this tradeoff is to use variable bin size, with wide bins where the distribution is flat, and narrow bins where the distribution is steeply changing. While variable bin size on the histogram will improve the estimate of the probability distribution, the optimal bin size depends on the shape of the distribution. There is no choice of variable bin size that is optimal for every distribution, and it is necessary to use consistent histograms with the same bin sizes for calculating distance from equilibrium; therefore, a fixed bin size is used. Similarly, little is gained in using the method of kernel density estimation either [55, 56]. Through the measured distribution, temperature affects the distance from equilibrium. The metric that is used to study the Markovian Mpemba effect is the distance from equilibrium; therefore, the distance from equilibrium is used to select the temperature at which the Boltzmann distribution is to be measured. Figure 4.4 shows the distance from equilibrium versus temperature for simulated data. On the vertical axis is the calculated distance from equilibrium, calculated from the Boltzmann distribution calculated using Eq. (2.7) to the measured distribution. On the bottom axis is rescaled temperature, Troom/T . The results for two trajectory lengths are shown for comparison. Figure 4.4 a) is calculated using trajectories of length 1 × 106 sampled at 200 Hz, around 83 min of equivalent experiment time, and b) is calculated using trajectories of length 2 × 107 sampled at 200 Hz, around 27.78 h. Each trajectory is histogrammed into 100 bins with bin width 0.27 µm. Uncertainties are from ten repeated simulations. The initial positions of the trajectories are randomly selected from the uniform distribution. We see that for the shorter trajectory in Fig. 4.4 a), the uncertainty is larger than for the longer trajectory in Fig. 4.4 b). Going from left to right, on the temperature axis, the distance from equilibrium is large at high temperatures, low for intermediate temperatures, and again increases approaching room temperature. By this plot, at a temperature of around 1475 K, slightly colder than the warm temperature, the distance from equilibrium is minimized. Although this optimum temperature was found, we preferred to show we can directly measure the equilibrium distribution for one of the temperatures that is actually used in the experiment: Thot = 5295 K, Twarm = 1995 K, or Tcold = 295 K. To illustrate how temperature affects the measured distribution, Fig. 4.5 shows in the

ﬁrst column, the histograms of trajectories at temperatures a) Twarm = 1995 K, b) Thot =

5295 K, and c) Tcold = 295 K; and, in the second column, the histograms rescaled to cold temperature from d) the warm temperature and e) the hot temperature. Black curves are Boltzmann distributions calculated using Eq. (2.7) and red markers are histograms of

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Figure 4.4: Simulated data showing calculated distance from equilibrium distribution calculated using Eq. (2.7) for diﬀerent temperatures. a) Black squares is trajectory length of 1 × 106 time intervals of 5 ms. b) Red circles is trajectory length 2 × 107 time intervals of 5 ms.

70 simulated trajectories 83 minutes long sampled at 200 Hz, using 100 bins with binwidth 0.27 µm. Comparing plots c), d), and e) we see that the measured data points agrees best with the theory in d). Both c) and e) have excess counts measured in the smaller peak on the left. Therefore, the warm temperature is used to measure the equilibrium distribution.

4.2.2 Why there is a minimum allowable barrier height

How high a temperature should we use? By the logic that high barriers prevent a particle from exploring all parts of the potential in a timely manner, as high a temperature as possible. So, should we use Thot and not Twarm? The answer is no, because, as the estimate of the probability distribution away from the two energy wells improves, the estimate of the probability distribution at the two wells becomes worse. To see this, when the temperature is very high, the probability distribution would be effectively an uniform distribution as in a flat potential; the thermal fluctuations of the particle enable it to hop over very low barriers, and we will be unable to measure the shape of the wells at all. Mathematically, we can obtain a bound on how high the barriers must be. For simplicity, we derive it for a discrete Boltzmann distribution. We take the ratio of the Boltzmann distribution for two states, i and j using Eq. (2.7),

πi(Tb) Ni = = e−∆E/kBTb , (4.5) πj(Tb) Nj where Ni and Nj are the number of observations in state i and j, respectively, and ∆E =

Ei − Ej. For small ∆E/kBTb, we can use a Taylor expansion,

−∆E/kBTb Ni = Nje (4.6)

≈ Nj(1 − ∆E/kBTb). (4.7)

Then,

∆N = Ni − Nj (4.8)

≈ −Nj∆E/kBTb, where the negative sign tells us that a positive ∆E corresponds to a negative ∆N, which is what we would expect, because fewer counts are in the location where the energy is higher. We will ignore the negative sign in the following calculations because we are interested in

ﬁnding a bound for Nj, which we know must be positive, since it is the number of counts.

Since ∆E/kBTb is small, the diﬀerence in the energy levels Ei and Ej is small, and therefore, the number of counts Ni and Nj in the two energy levels is about the same. The counts are from a Poisson distribution, so to distinguish a height diﬀerence between Ni and Nj,

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Figure 4.5: Estimating the probability distribution for the potential given by Eq. (2.19) at a bath temperature of Tcold = 295 K. Black curve is theory, calculated from Eq. (2.7). Red markers are simulated data of trajectories over about 83 s long. Column one are the measured probability distributions for temperatures a) 1995 K, b) 5295 K, and c) 295 K. d) is the distribution in a) rescaled to cold. e) is the distribution in b) rescaled to cold.

72 p we need ∆N Nj . Then Eq. (4.8) becomes

q ∆N ≈ Nj∆E/kBTb Nj , (4.9) or, 2 Nj 1/(∆E/kBTb) . (4.10)

The number of counts given by Eq. (4.10)is the minimum number of counts to measure in a particular histogram bin for the energy in that bin to be significantly different from zero. Many more counts are needed to be able to discern the shape of the distribution. To illustrate, for T = 5295 K, and the potential Eq. (2.19), the energy difference of the lowest well and the barrier with the metastable well is 0.56 kBTb, so the number of counts measured in the lowest well must be much greater than 3. In contrast, for T = 295 K, and the potential Eq. (2.19), the energy difference of the lowest well and the barrier with the metastable well is 10 kBTb, and the number of counts measured in the lowest well must be much greater than

0.01. At the same two positions on the potential, the Nj for T = 5295 K and T = 295 K differ by a factor of 300. This suggests that it can be prohibitively cumbersome to measure the probability distribution when the energy difference is smaller, compared to when the energy difference is larger.

4.2.3 Trajectory length

The shortest allowable trajectory length was chosen because of the expected difficulty of obtaining a long uninterrupted trajectory. A minimum trajectory length is necessary for errors in the estimated distribution due to nonergodicity to be small. Namely, the particle should visit all parts of the domain with nonzero probability density. An “excursion” is defined as each time a particle exceeds 20 µm, where the next excursion is not counted until the particle has travelled to a position less than 5 µm before exceeding 20 µm again. Figure 4.6 a), from simulated data, is a plot of how the number of excursions varies with trajectory length measured for T = 1995 K. The error bars are from 50 repeated trajectories. New trajectories are generated for each data point, with initial positions randomly sampled from a uniform distribution. In order to judge the quality of the measurement, the distance from equilibrium, D, as defined in Eq. (2.9a), is calculated using the Boltzmann distribution, as defined in Eq. (2.7), as the equilibrium distribution, for simulated trajectories of different lengths. As seen in Fig. 4.6 b), from simulated data, the calculated distance from equilibrium improves minimally beyond 15 h. If the equilibrium distribution were all that was needed, a trajectory measured for 15 h would be all that is needed. However, the potential is also needed, which is proportional to the negative logarithm of the distribution. If there were zero counts instead of a finite small number, the potential would become infinite. Therefore, the target

73 length was chosen to be 2×107 sampled at 200 Hz, or around 28 h of equivalent experiment time, with which one should expect about two excursions. This section has reviewed some key considerations for successful measurement of the equilibrium Boltzmann distribution and potential.

4.3 Observation of the Markovian Mpemba Eﬀect

From the previous discussion, it can be seen that achievement of the following results was not accidental. The best course of data collection was determined by thorough simulation of all possible scenarios prior to doing the experiment. Having discussed how the measurements are to be made, the ﬁnal section of this chapter presents experimental results and a discussion of those results.

4.3.1 Experimental Boltzmann distribution and potential

This section of the thesis presents the experimental results that were collected using the methods described in the previous section. It then goes on to evaluate the accuracy of the measurement and discuss the impact of the measurement errors on the ability to observe the Markovian Mpemba effect using this data. Figure 4.7 shows the experimentally measured equilibrium distribution at the cold temperature in a), and the corresponding potential scaled by the cold temperature, in b). Blue lines are the theoretical curves. Green markers are calculated using 27.8 h of experimental data, histogrammed into 100 bins with binwidth 0.24 µm. Uncertainty in the experimental data is estimated by assuming that the number of counts in each bin is Poisson distributed. In Fig. 4.7 a), red markers are simulated data equivalent to the number of hours as the experimental data, histogrammed into 100 bins with binwidth 0.27 µm, and the uncertainty is the sample standard deviation from repeating the simulation 1000 times. In Fig. 4.7 b), red line is calculated from the red markers, with uncertainty given by the light-red shaded region. Light-red shaded region is twice the estimated population standard deviation, calculated from the sample standard deviation. For the experimental and simulated Boltzmann distribution, where the probability is low the number of counts is close to zero. This contributes to poor statistics, but the difference between 0 and 1 × 10−20, say, is unimportant for the estimated Boltzmann distribution. The potential, being calculated as the negative logarithm of the Boltzmann distribution, has large deviations from the theory in the areas where the probability is low. For instance, a measured probability of 0 equates to an infinite potential, which is very much different from a finite potential. The grey-shaded region in Fig. 4.7 is 0.9999 of the probability density. It is difficult to estimate the probability density accurately outside that region, where the particle spends very little time. In fact, the particle only reaches the long plateau region once in the

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Figure 4.6: Simulation to choose a target length trajectory to estimate the equilibrium distribution and potential at T = 1995 K. a) Number of excursions. b) Calculated distance from cold equilibrium distribution calculated from Eq. (2.7) of estimated equilibrium distributions at various temperatures rescaled to the cold temperature as a function of trajectory length.

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Figure 4.7: Measured potential and equilibrium distribution. Grey-shaded region is where the area under the theoretical pdf calculated using Eq. (2.7) integrates to 0.99999. Less than 1 × 10−5 probability lies outside the grey-shaded region. Experimental data is in green markers. Theoretical curve is blue line. a) Cold Boltzmann distribution. Simulation data is in red markers. b) Potential. Red line is calculated from red markers in a), with uncertainty the light-red shaded region. Light-red shaded region is twice the estimated population standard deviation (One standard deviation was difficult to see). experimental data set. This is likely the reason that the experimentally measured potential is higher than it should be on most of the plateau: since the particle was not observed in a second excursion up the plateau, the measured number of counts there was low. Moreover, the particle never initiated its own travel up the plateau. By random chance, the initial position of one of the measured trajectories, which was sampled from a uniform distribution, was near the top of the plateau. In the experiment, external disturbances can arise, and the particle can escape before the desired time of the trial is up; therefore, several long trajectories are taken, and the combined data is analyzed as a single long trajectory for Fig. 4.7. Since the distribution being measured is a stationary distribution, the measured distribution does not depend on whether a single trajectory is used, or multiple trajectories. The effect of correlations between neighbouring points in a trajectory is negligible when a sufficiently long trajectory

76 is taken. In the complicated potential, the correlation time depends on the time scale of the experiment. For the purposes of the experiment, a particle that is trapped in the lower- energy well will not escape within the time frame of the experiment. The time it takes for the Markovian Mpemba effect to be observed is about 100 s; therefore, it seems that two points sampled at least 100 s apart will be uncorrelated. Then, the length of the trajectory should be long compared with 100 s to ensure that a sufficient number of uncorrelated points are represented in the sample. The influence of the initial position is kept to a minimum when the trajectory is long such that the number of points that are correlated with the initial position are small compared with the total number of measured positions. The initial position of the particle is randomly selected from a uniform distribution so as not to inject information about the distribution. Of the data that was collected, only some were used. Criteria for selecting the trajectories were as follows: I ranked the trajectories by length and only included in the analysis the longest trajectories. The combined length of the 14 longest measured trajectories, ranging from 32 to 548 min, surpassed the target length of 27.8 h. The remaining shorter trajectories were discarded on the basis of having excess data. An alternative method to analyze the experimental data would be to estimate the probability distribution for each measured trajectory and use the variation in the probability distributions to estimate the uncertainties. This method would require taking multiple long trajectories and is not done because long trajectories are more difficult to acquire, and the trajectories I measured were insufficiently long for the particle to explore the entire potential well at least once.

4.3.2 Noise ﬂoor

The following is a brief report on an experimental feature predicted by the Langevin equation that the Fokker-Planck equation failed to consider. The Fokker-Planck equation deals with probability distributions and does not take into account the effects of measurement noise which can lead to notable differences between the experimental results and the predicted results. Distributions can be estimated using the Fokker-Planck equation, Eq. (2.10), or the Langevin equation, Eq. (2.17). However, while solutions to the Fokker-Planck equation can be expressed with mathematical formulas, in simulations using the Langevin equation, as well as in experiment, there is statistical noise. The difference in estimated distributions leads to differences in the calculated distances from equilibrium. In particular, statistical noise in the estimated distribution yields a calculated distance from equilibrium that approaches but does not reach zero, when comparing the estimated distribution and Boltz- mann distribution given by Eq. (2.7). Further cooling does not decrease the measured distance from equilibrium, under the experimental design used here.

77 An importance consequence is, if the distance-from-equilibrium versus time curves for two processes are somehow known to intersect, but that intersection point occurs at a distance from equilibrium at or below the noise floor, the Markovian Mpemba Effect will not be observed. More data, and more accurate measurements could lower the noise floor, although the gains may be disproportionate to the effort. Before doing the Langevin simulation for the quenching part of the experiment, one should calculate the distance from equilibrium estimated from Langevin simulations to determine if the Markovian Mpemba effect predicted using the Fokker-Planck equation is observable when using the Langevin model. Furthermore, the experimental data can have additional sources of error that have not been included in the Langevin model. One should, additionally, calculate the distance from equilibrium of the experimentally estimated equilibrium distribution to check that the predicted Markovian Mpemba effect can be observed experimentally. The distance from equilibrium is a comparison of two distributions, where one is taken as the equilibrium distribution. Only the potential of the equilibrium distribution is included in the calculation, which makes the two not interchangeable. When comparing a non- equilibrium distribution to an equilibrium distribution, then the equilibrium distribution is the obvious choice because the potential is undefined for the non-equilibrium distribution. In the case that both the estimated distribution and the Boltzmann distribution are equilibrium distributions, it is equally possible to choose either distribution. There are positive and negative reasons in either choice. On the one hand, the Boltzmann distribution is not experimental, which was why the estimated distribution had to be considered in the first place. On the other hand, the experimental equilibrium distribution and potential is estimated from a single measurement. A better estimate would have been to take an average over multiple measurements, which was not done due to time constraints. Since there is no obvious choice of equilibrium distribution to be used to calculate the distance from equilibrium, in Fig. 4.8, I show the distance from equilibrium calculated using both methods: first using the estimated distribution as the reference, and then using the Boltzmann distribution as the reference. This is not meant to be rigorous and is intended to serve as an estimate of the uncertainty in the experimental value of distance from equilibrium. I could also have used simulations to estimate the noise floor.

4.3.3 Measured distance from equilibrium

This next section is a synthesis of the measured quantities from the previous sections and the result is observation of the Markovian Mpemba eﬀect. In Fig. 4.8, the distance from equilibrium over time is shown for two cooling processes. Triangles joined by solid lines are the experimental results. Measurements are made using the protocol in Section 3.2.1. Red is for systems initially at T = 5295 K quenched to T = 295 K, and orange is for systems initially at T = 1995 K quenched to T = 295 K.

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Figure 4.8: Experimental observation of the Markovian Mpemba Eﬀect. Light and dark green denote solutions to the Fokker-Planck equation, Eq. (2.10), for systems initially at T = 5295 K, and T = 1995 K, respectively, quenched to T = 295 K. Triangles joined by solid lines is experimental results. Red is for the system initially at Thot = 5295 K quenched to T = 295 K, and orange is for the system initially at Twarm = 1995 K quenched to T = 295 K. The thick, blue horizontal region compares the experimentally measured distribution, shown in Fig. 4.7, with the Boltzmann distribution, both at Tcold = 295 K. The intersection of the light and dark green lines, and the red and orange lines indicate that the Markovian Mpemba eﬀect is observed.

79 Uncertainties for the red and orange markers are estimated using the method discussed in Section 3.2.2. Light- and dark-green solid lines are solutions to the Fokker-Planck equation, Eq. (2.10), for systems initially at T = 5295 K and T = 1995 K, respectively, quenched to T = 295 K. The blue region is the distance from equilibrium between the experimentally measured distribution, shown in Fig. 4.7, and the Boltzmann distribution, given by Eq. (2.7), both at

Tcold = 295 K, calculated using two methods. In the first method, giving the upper bound, the measured Boltzmann distribution is taken as p(x, t), and πb(x) and U(x) are calculated using Eq. (2.7). In the second method, giving the lower bound, Eq. (2.7) is used for p(x, t), and the measured values are used for πb(x) and U(x). This region depicts the smallest measurable distance from equilibrium, which was discussed in Section 4.3.2. As can be seen in Fig. 4.8, the distance-from-equilibrium curves for the initially hot system start farther from the cold equilibrium distribution than the initially warm system. Experimental observation of the Markovian Mpemba Effect is seen as the crossing over of the curves for the two initial temperatures. We see that although the experimental curves deviate slightly from the Fokker-Planck solution, the initially hot curve starts off higher than the initially warm curve, and at around 40 s, the two curves cross over each other and the initially hot curve remains lower than the initially warm curve up to 100 s. At some very long time, we would predict that eventually both will converge to the blue region, having reached equilibrium. To understand the inconsistency between the experimental curve and curve predicted based on the Fokker-Planck equation, we turn to simulations. Figure 4.9 a), using 1000 trajectories, compares experimental and simulation distance-from-equilibrium curves to the Fokker-Planck solution. This plot repeats the distance-from-equilibrium curves that were shown in Fig. 4.8. In b), using 2000 trajectories, and c), using 10000 trajectories, results from simulations of the Langevin equation is compared with the Fokker-Planck solution. Light and dark green denote solutions to the Fokker-Planck equation, Eq. (2.10), for systems initially at T = 5295 K, and T = 1995 K, respectively, quenched to T = 295 K. Triangles joined by solid lines is experimental results. Circles joined by dashed lines is simulation using a fixed value of the diffusion coefficient. All shades of red are for systems initially at T = 5295 K being quenched to T = 295 K, and all shades of orange are for systems initially at T = 1995 K being quenched to T = 295 K. The uncertainties that are estimated by using data resampling to obtain 105 sets of data. Even though we would expect that large variability in the distributions would be ac- counted for by the error bars, in Fig. 4.9 a), the simulation result appears to deviate one or more standard deviations from the Fokker-Planck result. One possible explanation could be that, since the same process is being measured over time, deviations at earlier times would have residual effects later on; thus, multiple deviations at several times could have a compounding effect. To support this explanation, we see that in Fig. 4.9 b) and c), the sim-

80 ulation results is within one standard deviation of the Fokker-Planck result. These series of plots shows that by increasing the number of trajectories, the simulation results converge to the Fokker-Planck results. Therefore, we would also expect the experimental results to also converge to the Fokker-Planck solution when the number of trajectories used to calculate the distance-from-equilibrium curve is increased. However, the experimental results deviate more than simulation results from the Fokker- Planck result. One explanation for this discrepancy could be the distortions in the measured shape of the distributions, which is one of the reasons that I measure the cold equilibrium distribution and potential instead of calculating the distribution from the Boltzmann equation, Eq. (2.7). We see that deviations are largest for intermediate times in the cooling process. As explained in Section 4.2.2, the measurement of the distribution improves when the energy barriers are increased; therefore, the distance from equilibrium is more accurate when the distribution is close to the equilibrium at T = 295 K, which is at large times. At early times, the distributions are also well-approximated because of the method of initial- izing the particle at the starting position of the trajectory, as explained in Section 4.1. Another effect that was mentioned in Section 3.3 is that the particle resides close to a surface, so it is possible, therefore, that height fluctuations have a large effect on the diffusion coefficient of the particle as it moves along the surface. While the simulations shown in Fig. 4.9 use a fixed diffusion coefficient, the effect of a varying diffusion coefficient was explored using simulation, as shown in Fig. 3.8. Figure 3.8 shows a deviation similar to that observed in the experimental data in Fig. 4.9; however, this is only a qualitative comparison because the vertical probability distribution was not experimentally determined. In summary, the experiment was successful as it was able to show that a Markovian Mpemba effect occurred. The next chapter provides some final comments and recommen- dations for future work.

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82 Chapter 5

Conclusion

The Mpemba effect refers to the phenomenon that the freezing of initially hotter water can sometimes occur before the freezing of initially cooler water, when all other variables are seemingly identical. Its cause remains a mystery because it is a far-from-equilibrium cooling process for which classical thermodynamics does not apply. Far-from-equilibrium cooling process, such as rapid cooling, can be described by the theory of stochastic thermodynamics. Using stochastic thermodynamics, a Mpemba-like effect can be studied in small systems, which can be more readily isolated from competing mechanisms for cooling. With better isolation of the process, stochastic thermodynamic experiments are much more re- peatable than experiments done on water. This repeatability in stochastic thermodynamic experiments is the distinguishing feature from experiments done on water that makes the Mpemba-like effect in a small system a good candidate for studying the elusive mechanisms that are the driving force for the Mpemba effect in water. The Mpemba-like effect in a small system is referred to as the Markovian Mpemba effect because it is possible to give a Markovian description of the small system. The Markovian Mpemba effect is, additionally, a term used to refer to the theory that predicts whether a Mpemba-like effect will occur in a system with a given set of parameters, and suggests guidelines to follow when seeking to design an experiment in which the effect can be observed. This thesis has set out to determine whether the Markovian Mpemba effect can occur in the small system of a colloidal particle diffusing in a water bath in a potential landscape, where the Markovian Mpemba effect’s predicted behaviour is determined using the Fokker- Planck equation. This study has found that generally the predicted behaviour is aligned with the behaviour observed with a computer simulation of the experiment using the Langevin equation before carrying out the experiment itself. A limitation of the using the Fokker- Planck equation is that there is no guarantee that the chosen parameters are ones that could ever be encountered. In the same vein, it does not address whether the Markovian Mpemba effect may not occur at all in conditions that can be encountered. The results of this thesis indicate that imposing certain constraints on the allowable set of parameters, determined by

83 specific material properties of the system and the ability of the experimental apparatus to create the desired cooling process, does not create a scenario where the Markovian Mpemba effect is never observed. The analysis of this thesis has confirmed the findings of Lu and Raz [27] that the theory enables them to discover Mpemba-like effects in systems not previously known to exhibit this behaviour. A further finding of Lu and Raz was that an inverse Markovian Mpemba effect involving heating instead of cooling can occur. Although this study is based on Markovian Mpemba effect for a cooling process, the findings suggest that the inverse effect, the Markovian Mpemba effect for a heating process, can also occur in the same system. While this study did not require a modification of the existing experimental apparatus, it did involve verification that the apparatus could work sufficiently accurately. The accuracy of the apparatus had been investigated for narrower potentials in earlier work [30, 43, 44, 47] than was required for this work. Prior to commencing experiments on the Marko- vian Mpemba effect, I investigated the functioning of the apparatus using a harmonic well placed at a number of evenly spaced locations spanning the width of the potential. Having confirmed that there is no substantial systematic variation in the harmonic well with position, I also prepared for the experiment by doing computer simulations to determine the amount of data required to observe the Markovian Mpemba effect. The amount of data needed was minimized by considering how different quantities are related, an by making the measurement on the quantity that takes the least amount of data to resolve at a satisfactory level. For example, by measuring the equilibrium distribution of the particle, the potential could be calculated using Boltzmann’s equation. Another example is that by choosing a temperature for which features in the corresponding equilibrium distribution were sufficiently high contrast but not too extreme, less data was required to obtain a decently close approximation to the Boltzmann distribution calculated using Eq. (2.7), from which, using Boltzmann’s equation, the equilibrium distribution associated with the desired temperature could be calculated. Further reductions in the required amount of measurements were attained by using the technique of data resampling to determine uncertainties in the measured quantities rather than doing repeated trials. With the same goal of minimizing the time needed to do the experiment, implementation of the process was expedited wher- ever possible. For example, the initialization of the particle in an equilibrium state would require a significant amount of time if done in the direct method of allowing the particle to come to equilibrium within the potential. Instead, what was done was that the particle was positioned at points randomly selected from the desired equilibrium distribution associated with baths at the initial temperatures using a harmonic well, after which the cooling process was manifested by replacing the harmonic well with the desired potential and allowing the particle to explore that potential. By making measurements strategically and implementing the process cleverly, I was able to complete my experimental measurements in a short amount of time. The nominal time required to complete all measurements was about

84 3.5 days of good data, although the actual time was a couple of weeks, which includes the time to set up the experiment, and to reset the experiment when an external disturbance ejected the particle from the potential. The scope of this study was limited in terms of the accuracy of the measurements. The current study has only examined whether a system that was initially farther from equilibrium ends up being closer to equilibrium some time later than a system that was initially closer to equilibrium. While the measurements were sufficiently accurate for the Markovian Mpemba effect to be observed, there are large deviations in the experimental results from the Fokker-Planck results. These deviations are even larger than what is expected from simulations with a similar sample size and have not been explained. One possible explanation is a breakdown of the assumption that the diffusion coefficient remains constant in the experiment and instead fluctuates with changes in height of the particle above the bottom surface of the sample chamber. It is unfortunate that the study did not include measurements of the height of the particle above the bottom surface of the sample chamber because estimated changes in the height of the particle were found to produce a sufficiently large impact the value of the diffusion coefficient to cause deviations in results obtained using simulations of the Langevin equation that were similar to the deviations observed in the experimental results. The estimate of the amount of fluctuation in particle height was arrived at using parameters for the vertical potential found in the literature for the same size and type of particle, and same treatment of the water to simulate the height of the particle over time. Despite the similarities between that system and our system, it is possible that differences in the vertical potential could be caused by some other factors, affecting the height of the particle and the value of the diffusion coefficient that would exist in our experimental setup. Further work needs to be done to establish whether, in our experimental setup, a constant diffusion coefficient is a valid assumption. What is now needed is an experimental measurement of the vertical potential experienced by the particle in our experimental setup. I would recommend that the height of the particle be monitored while doing the same experiment over again. By using the measured height to calculate the variation in the diffusion coefficient, a more realistic simulation can be done that takes into account of the varying diffusion coefficient rather than assuming it is constant. If the simulation results are un- changed, which would be an indication that the assumption of constant diffusion is valid, then further research might investigate a different explanation for the deviation of the experimental result from the Fokker-Planck result. If it turns out that a constant diffusion coefficient is not a valid assumption, a natural progression of this work is to make a change in the experimental setup to reduce the amount of variation in the diffusion coefficient; or, alternatively, extend the theory of the Markovian Mpemba effect to the case where diffusion varies. While the findings of the study of the Markovian Mpemba effect offers some insight into anomalous behaviour associated with rapid cooling of small systems, it does

85 not address the ultimate goal of explaining the Mpemba effect in water and Mpemba-like effects in other systems which was the initial motivation of this study. Considerably more work will need to be done to determine whether a connection can be drawn between the mechanism for cooling in the Markovian Mpemba effect, the mechanism for cooling in the Mpemba effect in water, and the mechanism for relaxation in Mpemba-like effects in other systems.

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90 Appendix A

Derivation of Master equation discretization

The method of Grima and Newman writes the Fokker-Planck equation as a function of only 2 second derivatives in space, ∂x, without ﬁrst derivatives, ∂x. The following expansions of derivatives are used:

γ0U 0 γ0U ∂xe = γ e ∂xU 2 γ0U 0 γ0U 2 02 γ0U 2 ∂xe = γ e ∂xU + γ e (∂xU) (A.1) −γ0U 0 −γ0U ∂xe = −γ e ∂xU 2 −γ0U 0 −γ0U 2 02 −γ0U 2 ∂xe = −γ e ∂xU + γ e (∂xU)

2 −γ0U −γ0U −γ0U ∂x(ρe ) = ∂x(e ∂xρ + ρ∂xe ) −γ0U −γ0U 2 −γ0U 2 −γ0U = ∂xe ∂xρ + e ∂xρ + ∂xρ∂xe + ρ∂xe −γ0U −γ0U 2 2 −γ0U = 2 × ∂xρ∂xe + e ∂xρ + ρ∂xe 0 −γ0U −γ0U 2 0 −γ0U 2 02 −γ0U 2 = 2 × ∂xρ(−γ e ∂xU) + e ∂xρ + ρ(−γ e ∂xU + γ e (∂xU) ) −γ0U 0 2 0 2 02 2 = e (−2γ ∂xU∂xρ + ∂xρ − γ ρ∂xU + γ ρ(∂xU) ) (A.2)

91 On the left-hand side of Eq. (A.2) is only a second derivative, and on the right hand side is a combination of ﬁrst and second derivatives. Using Eq. (A.1) and Eq. (A.2), we have that

γ0U 2 −γ0U −γ0U 2 γ0U ∂tρ = D[e ∂x(ρe ) − e ρ∂x(e )] (A.3) γ0U −γ0U 0 2 0 2 02 2 = D[e (e (−2γ ∂xU∂xρ + ∂xρ − γ ρ∂xU + γ ρ(∂xU) )) −γ0U 0 γ0U 2 02 γ0U 2 − e ρ(γ e ∂xU + γ e (∂xU) )] 0 2 0 2 = −2Dγ ∂xU∂xρ + D∂xρ − 2Dργ ∂xU 2 0 = D∂xρ − 2Dγ ∂x(ρ∂xU) 1 = D∂2ρ − ∂ (ρ∂ U) x γ x x 1 = (k T ∂2ρ − ∂ (ρ∂ U)) γ B x x x 1 = (k T ∂2ρ − ∂ (ρ∂ U)) (A.4) γ B x x x where a change of variables, 1/γ = 2Dγ0 is used to obtain the Fokker-Planck equation in the last line. Next, the central spatial diﬀerence is used for the discretization of Eq. (A.3). For the 0 function f = eγ U , we have

γ0U γ0U γ0U 0 e i+1 − 2e i + e i−1 ∂2eγ Ui = . (A.5) x h2

0 For f = ρe−γ U , we have

−γ0U −γ0U −γ0U 0 ρi+1e i+1 − 2ρie i + ρi+1e i−1 ∂2ρ e−γ Ui = . (A.6) x i h2 Substituting Eq. (A.5) and Eq. (A.6) into Eq. (A.3),

−γ0U −γ0U −γ0U 0 ρi+1e i+1 − 2ρie i + ρi−1e i−1 ∂ ρ = D[eγ Ui ( ) t i h2 γ0U γ0U γ0U 0 e i+1 − 2e i + e i−1 − e−γ Ui ρ ( )] i h2 D 0 0 = [(ρ e−γ (Ui+1−Ui) − 2ρ + ρ e−γ (Ui−1−Ui)) h2 i+1 i i−1 0 0 γ (Ui+1−Ui) γ (Ui−1−Ui) + (−ρie + 2ρi − ρie )]

D 0 0 = [(ρ e−γ (Ui+1−Ui) + ρ e−γ (Ui−1−Ui)) h2 i+1 i−1 0 0 −γ (Ui−Ui+1) −γ (Ui−Ui−1) − ρi(e + e )] (A.7) X = [Wijpj(t) − Wjipi(t)] (A.8) i

Comparing Eq. (A.7) to the Eq. (A.8), the master equation, we obtain that the transition- 0 D γ (Ui−Uj ) rate matrix is Wij = h2 e and ρi = pi, which matches the result in [46].

92 Appendix B

Testing the distance-from-equilibrium function

As described in Section 2.2, the Markovian Mpemba effect occurs in a discrete Markov h c h c process when |a2 | < |a2|, and it does not occur when |a2 | < |a2|. It is not obvious from the proof given by Lu and Raz [27] that the Markovian Mpemba effect result for the discrete Markov process and the continuous Markov process coincide c h for all magnitude and sign combinations ((+, +), (+, −), (−, +), (−, −)) of a2 and a2 . In particular, in their proof, the sign, at large times, of the distance from equilibrium of the initially hotter system minus the distance from equilibrium of the initially cooler system h c appears to depend on the sign of (a2 − a2). h c Consider the case that a2 can always be made positive, but the sign of a2 can be positive or negative, and that the overall sign of the other factors is positive. Other cases might be h c possible, but will not be discussed here. Then, for |a2 | > |a2|, the Markovian Mpemba effect h c is never observed, as expected, since (a2 − a2) is always positive, as will be the difference h c in the distance from equilibrium at large times. However, for |a2 | < |a2|, the Markovian h c h c Mpemba effect should always be observed, but for a2 > 0 and a2 < 0, (a2 −a2) is positive, as will be the difference in the distance from equilibrium at large times, and it would seem that, in this case, the Markovian Mpemba effect is not detected. Here, I construct an example using the three-state model and show, using simulations, that the Markovian Mpemba effect h c and the |a2 | > |a2| test still agree, even for this special case. h c In the examples of a2(T ) versus T plots given in the Lu and Raz’s [27]paper, a2 and a2 always have the same sign. After the temperature when the a2 coefficient is maximized, Tm, for T > Tm, the positive a2 coefficient decreases towards zero, but does not become negative, or, the negative a2 coefficient increases towards zero, but does not become positive. Here, I have chosen parameters for the three-state model such that a2 changes sign for some h c h c T > Tm, specifically designed to test the case that |a2 | < |a2| with a2 > 0 and a2 < 0. All h c possible combinations of signs and magnitudes of a2 and a2 for this particular example, are tested.

93 The Markovian Mpemba eﬀect occurs when the following conditions are satisﬁed [27]: 1 U2 < ; 2 (B.1) B13 − U3 < B12 − U2;

B13 < B23.

The three-state system is depicted in Fig. B.1. U1 = 0 is the ground state energy. The energies of the other two states are U2 = 0.05 and U3 = 1, and the energy barriers are B12 = 2.5, B23 = 3.5, and B13 = 1.1.

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Figure B.1: Discrete energies and energy barriers for three-state model.

So the three conditions are satisfied and there should be a Markovian Mpemba effect. We observe in Fig. B.2 that the coefficient in the eigenfunction expansion associated with the second largest eigenvalue has a change in the sign of the slope that is characteristic of the h c Markovian Mpemba effect. In this example, Th and Tc can be chosen such that a2 and a2 are not necessarily positive, and do not necessarily have the same sign. h c h c h c The combinations that are possible here are: 0 > a2 > a2, a2 > 0 > a2, a2 > a2 > 0. Other h c combinations such as a2 < 0 < a2 are possible in other examples, but not this specific example. For the first possibility, the Markovian Mpemba effect occurs. For the second h c possibility, depending on the relative magnitude of a2 and a2, the Markovian Mpemba effect may or may not occur. For the third possibility, the Markovian Mpemba effect does h c not occur. We check using specific values of a2 and a2 to see whether the Markovian Mpemba effect prediction is detected using the entropic distance metric for each of these cases. A small bath temperature, Tb = 0.1 is used. h c From Fig. B.2, for Th = 1 and Tc = 0.5, we get a2 = −0.05 and a2 = −0.09, and the Markovian Mpemba effect is expected to occur. We see in Fig. B.3 that the Markovian Mpemba effect is observed.

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Figure B.3: Plot of Distance versus time for Tb = 0.1, and Th = 1 and Tc = 0.5.

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Figure B.4: Plot of Distance versus time for Tb = 0.1, and Th = 3 and Tc = 0.5.

h c For Th = 3 and Tc = 0.5, we have a2 = 0.02 and a2 = −0.09, and the Markovian Mpemba eﬀect is expected to occur. We see in Fig. B.4 that the Markovian Mpemba eﬀect is observed.

h c For Th = 5 and Tc = 3, we have a2 = 0.03 and a2 = 0.02, and the Markovian Mpemba effect is not expected to occur. We see in Fig. B.5 that the Markovian Mpemba effect is not observed. h c For Th = 10 and Tc = 1.4, we have a2 = 0.05 and a2 = −0.02, and the Markovian Mpemba effect is not expected to occur. We see in Fig. B.6 that the Markovian Mpemba effect is not observed.

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Figure B.5: Plot of Distance versus time for Tb = 0.1, and Th = 5 and Tc = 3.

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Figure B.6: Plot of Distance versus time for Tb = 0.1, and Th = 10 and Tc = 1.4.