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 © Lisa Zhang 2019 SIMON FRASER UNIVERSITY Spring 2019 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 eigen- function 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.

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