When the Hotter Cools More Quickly: Mpemba Effect in Granular Fluids

week ending PRL 119, 148001 (2017) PHYSICAL REVIEW LETTERS 6 OCTOBER 2017

Antonio Lasanta,1,2 Francisco Vega Reyes,2 Antonio Prados,3 and Andrés Santos2 1Gregorio Millán Institute of Fluid Dynamics, Nanoscience and Industrial Mathematics, Department of Materials Science and Engineering and Chemical Engineering, Universidad Carlos III de Madrid, 28911 Leganés, Spain 2Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, 06006 Badajoz, Spain 3Física Teórica, Universidad de Sevilla, Apartado de Correos 1065, 41080 Sevilla, Spain (Received 16 November 2016; revised manuscript received 20 March 2017; published 4 October 2017) Under certain conditions, two samples of fluid at different initial temperatures present a counterintuitive behavior known as the Mpemba effect: it is the hotter system that cools sooner. Here, we show that the Mpemba effect is present in granular fluids, both in uniformly heated and in freely cooling systems. In both cases, the system remains homogeneous, and no phase transition is present. Analytical quantitative predictions are given for how differently the system must be initially prepared to observe the Mpemba effect, the theoretical predictions being confirmed by both molecular dynamics and Monte Carlo simulations. Possible implications of our analysis for other systems are also discussed.

DOI: 10.1103/PhysRevLett.119.148001

Let us consider two identical beakers of water, initially at In a general physical system, the study of the ME implies two different temperatures, put in contact with a thermal finding those additional variables that control the temper- reservoir at subzero (on the Celsius scale) temperature. ature relaxation and determining how different they have to While one may intuitively expect that the initially cooler be initially in order to facilitate its emergence. In order to sample would freeze first, it has been observed that this is quantify the effect with the tools of nonequilibrium not always the case [1]. This paradoxical behavior named statistical mechanics, a precise definition thereof is man- the Mpemba effect (ME) has been known since antiquity datory. An option is to look at the relaxation time to the and discussed by philosophers like Aristotle, Roger Bacon, final temperature as a function of the initial temperature Francis Bacon, and Descartes [2,3]. Nevertheless, physi- [1,2,4,5,9,11,24,32,34]. Alternatively, one can analyze the cists only started to analyze it in the second part of the past relaxation curves of the temperature: if the curve for the century, mainly in popular science or education journals initially hotter system crosses that of the initially cooler one [1–23]. and remains below it for longer times, the ME is present There is no consensus on the underlying physical [3,9,18,20,27–30,33]. mechanisms that bring about the ME. Specifically, water In this Letter, we combine both alternatives above and evaporation [4,5,9,24], differences in the gas composition investigate the ME in a prototypical case of intrinsically out- of water [11,17,25], natural convection [6,23,26], or the of-equilibrium system: a granular fluid [52–55], i.e., a (dilute influence of supercooling, either alone [14,27] or combined or moderately dense) set of mesoscopic particles that do not with other causes [28–31], have been claimed to have an preserve energy upon collision. As a consequence, the mean impact on the ME. Conversely, the own existence of the kinetic energy, or granular temperature TðtÞ [53], decays ME in water has been recently put in question [32]. monotonically in time unless an external energy input is Notwithstanding, Mpemba-like effects have also been applied. The simplicity of the granular fluid makes it an ideal observed in different physical systems, such as carbon benchmark for other, more complex, nonequilibrium sys- nanotube resonators [33] or clathrate hydrates [34]. tems. We analyze the time evolution of the granular fluid The ME requires the evolution equation for the temper- starting from different initial preparations and quantitatively ature to involve other variables, which may facilitate or investigate how the ME appears. This is done for both the hinder the temperature relaxation rate. The initial values of homogeneous heated and freely cooling cases. those additional variables depend on the way the system Our granular fluid is composed of smooth inelastic hard has been prepared, i.e., “aged,” before starting the relax- spheres. Therefore, the component of the relative velocity ation process. Typically, aging and memory effects are along the line joining the centers of the two colliding associated with slowly evolving systems with a complex particles is reversed and shrunk by a constant factor α [52], energy landscape, such as glassy [35–43] or dense granular the so-called coefficient of normal restitution. In addition, systems [44–46]. However, these effects have also been the particles are assumed to be subject to random forces in observed in simpler systems, like granular gases [47–50] or, the form of a white-noise thermostat with variance m2ξ2, very recently, crumpled thin sheets and elastic foams [51]. where m is the mass of a particle. Thus, the velocity

ðv Þ 1 2 ð1Þ distribution function (VDF) f ;t obeys an Enskog- st −λ−τ δθ ¼ ½ðλþ − μ Þδθ0 − μ δa2 0e Fokker-Planck kinetic equation [56–58]. γ 2 3 2 ;

The granular temperature and the excess kurtosis 1 2 ð1Þ − ½ðλ − μstÞδθ − μ δ −λþτ ð Þ (or second Sonine coefficient)R are defined as TðtÞ¼ − 2 0 2 a2;0 e ; 3 2 2 3 4 γ 3 ðm=3Þhv ðtÞi ≡ ðm=3nÞ dvv fðv;tÞ Rand a2 ¼ 5 hv i= 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hv i − 1, respectively, where n ¼ dvfðv;tÞ is the 1 2 where λ ¼ ½Λ11 þ Λ22 ðΛ11 − Λ22Þ þ 4Λ12Λ21 are number density. From the kinetic equation for the VDF, 2 Λ γ ≡ λ − λ 0 one readily finds [56] the eigenvalues of the matrix and þ − > . Let us imagine two initial states f0 ¼ fA and fB, with ðθ0;a2 0Þ¼ðθ ;a2 Þ and ðθ ;a2 Þ, respectively. We 2κ ; A A B B dT 3=2 θ θ ¼ − ðμ2T − χÞ; ð1aÞ assume that A > B and a2A >a2B. Both cooling dt 3 (θA > θB > 1) and heating (θB < θA < 1) processes may be considered. From Eq. (3), the time τ for the possible c 1 3=2 crossing of the two relaxation curves satisfies d lnð1 þ a2Þ 4κ μ4T − χ ¼ μ 3=2 − χ − 5 ð Þ 3 2T 1 þ ; 1b dt T a2 1 2μð1Þ − 3ðλ − μstÞΔθ Δ τ ¼ 2 − 2 0= a2;0 ð Þ pffiffiffiffiffiffiffiffiffi c ln ð1Þ ; 4 γ 2μ − 3ðλ − μstÞΔθ Δ where κ ≡ 2ngðnÞσ2 π=m, σ and gðnÞ are the sphere 2 þ 2 0= a2;0 diameter and the pair correlation function at contact [59], Δθ ¼ θ − θ Δ ¼ − χ ≡ ð3 2κÞξ2 μ μ where 0 A B and a2;0 a2A a2B. For a given respectively, m= , and 2 and 4 are dimension- α τ less collisional rates. , in this simplified description the crossover time c depends on ðθ ;a2 Þ and ðθ ;a2 Þ (or, more generally, Note that Eqs. (1) are formally exact, but (a) T and a2 A A B B on the details of the two initial VDFs fA and fB) only are coupled, and (b) the equations are not closed in those Δθ Δ μ through the single control parameter 0= a2;0. two variables since n are functionals of the whole VDF. τ However, if inelasticity is not too large, the nonlinear Figure 1(a) displays c as a function of the ratio Δθ0=Δa2;0 for α ¼ 0.9. Equation (4) implies that there is contributions of a2 and the complete contributions Δθ Δ of higher order cumulants can be neglected. This is the a maximum of the control parameter 0= a2;0 for which so-called first Sonine approximation [56,60], which yields the ME can be observed, namely, ð0Þ ð1Þ ð0Þ 2 ð1Þ 3 ð0Þ μn ≃ μn þ μn a2, with μ2 ¼ 1 − α , μ2 ¼ 16 μ2 , ð1Þ Δθ0 2 μ2 ð0Þ 9 2 ð0Þ ð1Þ 3 2 ¼ : ð5Þ μ4 ¼ð2 þ α Þμ2 , μ4 ¼ð1þαÞ½2þ32ð69þ10α Þð1−αÞ. Δ 3 λ − μst a2;0 max þ 2 Using the first Sonine approximation above in Eqs. (1), they become a closed set, but the T-a2 coupling still This quantity determines the phase diagram for the occur- remains. Taking into account this coupling, and since μ2 rence of the ME, as shown in Fig. 1(b). is an increasing function of a2, it turns out that the Equation (5) can be read in two alternative ways. First, it relaxation of the granular temperature T from an initially means that for a given difference Δa2;0 of the initial “ ” cooler (smaller T) sample could possibly be overtaken by kurtosis, the ME appears when the difference Δθ0 of the that of an initially “hotter” one, if the initial excess kurtosis scaled initial temperatures is below a maximum value ðΔθ Þ Δ of the latter is larger enough. We build on and quantify the 0 max, proportional to a2;0. Second, for a given value implications of this physical idea in the following. of Δθ0, the ME is observed only for a large enough First, we consider the uniformly heated case (i.e., χ ≠ 0) and prepare the granular fluid in an initial state that is close to the steady one, in the sense that Eqs. (1) can be linearized 2 ¼ðχ μstÞ3 around the stationary values [56,57] Tst = 2 and st ð0Þ ð0Þ ð1Þ ð1Þ st ð0Þ ð1Þ st a2 ¼½5μ2 − μ4 =½μ4 − 5μ2 , where μn ¼ μn þ μn a2 . θ ¼ Let us use a dimensionless temperature T=Tpffiffiffiffiffiffist and δθ ¼ θ − 1 δ ¼ − st τ ¼ κ define , a2 a2 a2 , and Tstt.A straightforward calculation gives d δθ δθ ¼ −Λ × ; ð2Þ dτ δa2 δa2

ð1Þ st 2 FIG. 1. (a) Crossover time τ as a function of the ratio Δθ0=Δa2 0 where the matrix Λ has elements Λ11 ¼ μ2 , Λ12 ¼ 3 μ2 , c ; for α ¼ 0.9. (b) Phase diagram in the plane Δθ0=Δa2 0 vs α.The Λ ¼ −2μst st Λ ¼ 4 ½μð1Þ − 5μð1Þð1 þ stÞ ; 21 2 a2 , and 22 15 4 2 a2 . Thus, regions of the plane inside which there appears or does not appear ðΔθ Δ Þ the relaxation of the temperature reads the ME are separated by the curve 0= a2;0 max. 148001-2 week ending PRL 119, 148001 (2017) PHYSICAL REVIEW LETTERS 6 OCTOBER 2017

−4 FIG. 3. Relaxation time τϵ (with ϵ ¼ 10 ) as a function of the initial scaled temperature θ0 for α ¼ 0.9. Three values of the initial excess kurtosis are considered, a2;0 ¼ 0.5 (solid line), a2;0 ¼ 0 (dotted line), and a2;0 ¼ −0.35 (dashed line). The horizontal (grey) segments join values of initial temperatures FIG. 2. Relaxation of the scaled temperature to the steady state that share the same value of the relaxation time and, thus, mark for α ¼ 0.9. The upper and the lower curves correspond to the the onset of either the ME (θ0 > 1) or the inverse ME (θ0 < 1). ME for the cooling and heating processes, respectively (see text). The direct simulation Monte Carlo (DSMC) (open symbols) and molecular dynamics (MD) (filled symbols) data show an ex- Figure 3 shows τϵ as a function of the initial temperature θ0 cellent agreement with the theoretical prediction (lines). for ϵ ¼ 10−4 and the same values of the initial excess kurtosis as considered in Fig. 2. In this diagram, for a given difference of the initial kurtosis, i.e., Δa2 0 > ðΔa2 0Þ , ; ; min pair of a2 0, the range of initial temperatures for which ðΔ Þ Δθ ; with a2;0 min proportional to 0. This quantitatively the ME emerges is clearly visualized. Note that this range measures how different the initial conditions of the system does not change if the value of the bound ϵ is changed must be in order to have the ME. to ϵ0, since the diagram is only shifted vertically by an In order to check the accuracy of our theoretical results, 0 amount ð1=λ−Þ lnðϵ=ϵ Þ. we compare them in Fig. 2 with MD simulations (at a σ3 ¼ 0 02 A relevant question is whether or not the ME keeps density n . ) and with the numerical integration of appearing in the zero driving limit. In the undriven case the Enskog-Fokker-Planck equation by means of the χ ¼ 0 α ¼ 0 9 ( ), the granular fluid relaxes to the so-called homo- DSMC method [61]. In all our simulations, . and geneous cooling state (HCS) [52], which is the reference the initial VDF is assumed to have a gamma-distribution state for deriving the granular hydrodynamics [64]. If the v2 form [62] in the variable with parameters adjusted linear relaxation picture developed above remained valid in to reproduce the chosen values of θ0 and a2 0. First, ; the nonlinear relaxation regime, at least qualitatively, the three different initial conditions (A, B, and C) with temper- θ ¼ 1 04 θ ¼ 1 035 answer would be negative. Note that the maximum temperatures above the stationary, A . , B . , and ðΔ Þ θ ¼ 1 03 ¼ 0 5 ¼ 0 ature difference T0 max would vanish in the limit as C . , and excess kurtosis a2A . , a2B , and χ → 0 ∝ χ2=3 → 0 ðΔθ Þ (Tst ), as a consequence of 0 max a2 ¼ −0.35, are considered. The ME is clearly observed C being independent of χ. Interestingly, we show below that as a crossover of the relaxation curves of the temperature this simple scenario does not hold, and the ME is also [see, also, Fig. 1(a)]. Second, we analyze a “heating” observed for very small driving: indeed, ðΔT0Þ remains protocol by choosing initial temperatures below the steady max finite in this limit. value, namely, θ0 ¼ 0.97, θ0 ¼ 0.965, and θ0 ¼ 0.96, A B C For very small driving, there is a wide initial time with the same values of the excess kurtosis as in the region inside which the system evolves as if it were cooling “cooling” case. Again, a crossover in the temperature freely. Therefore, for the sake of simplicity, we now take relaxation curves appears, signaling the granular analog χ ¼ 0 in the evolution Eq. (1). While the system freely of the inverse ME proposed in a recent work [63].It cools for all times (limt→∞T ¼ 0), the excess kurtosis is interesting to note that the evolution curves correspond- HCS ð0Þ ð0Þ 0 tends to a constant value [56,57] a2 ¼½5μ2 − μ4 = ing to θ ¼ 1.03 and θ ¼ 0.97 are nonmonotonic. This ð1Þ ð0Þ ð1Þ C A ½μ − μ − 5μ . Since there is no natural temperature peculiar behavior is predicted by Eq. (3) to take place 4 4 2 2 ð1Þ st scale in the free cooling case, we can make use of if − μ =μ < δθ0=δa2 0 < 0. 3 2 2 ; dimensionless variables by scaling temperature and time Alternatively, we can characterize the system celerity for ¼ with an arbitrary reference value Tref, i.e., T T=Tref cooling (or heating) by defining a relaxation time τϵ as the pffiffiffiffiffiffiffiffi and t ¼ κ T t. time after which jθðτϵÞ − 1j < ϵ, with ϵ ≪ 1. From Eq. (3), ref If present at all, we expect the ME to occur for relatively ð1Þ short times, more specifically, before a2 has relaxed to its 1 3ðλ − μstÞδθ − 2μ δ þ 2 0 2 a2;0 HCS τϵ ¼ ln : ð6Þ stationary value a2 . So as to look for a possible crossover λ− 3ϵγ of the cooling curves, we linearize the equations around 148001-3 week ending PRL 119, 148001 (2017) PHYSICAL REVIEW LETTERS 6 OCTOBER 2017 ¼ 1 T (by choosing Tref such that the initial temperatures neglected, the system freely cools, and the ME is observed HCS verify jT0 − 1j ≪ 1) and a2 ¼ a2 . Therefrom, the evo- provided that the condition (9) is fulfilled. Afterwards, the lution of T is obtained as initially hotter system remains below the initially cooler one forever. When approaching the steady state, both the 2 2 μð1Þδ temperature and the excess kurtosis start to evolve towards 2 a2;0 −μHCS δT ¼ δT þ − e 2 t st 0 3 3 λ − μHCS their stationary values Tst and a2 , but in both curves one a 2 HCS has a2;0 ¼ a2 , and Eq. (5) tells us that no further crossing ð1Þ Δ ¼ 0 2 μ2 δa2;0 −λ 2 of the curves takes place ( a2;0 ). þ at − ð Þ HCS e ; 7 In summary, we have shown by means of a simple 3 λ − μ2 3 a analytical theory that the ME naturally appears in granular HCS fluids, as a consequence of the relevance of non- where δT ¼ T − 1, δa2 0 ¼ a2 0 − a2 , and λ ¼ ; ; a Gaussianities in the time evolution of T. Specifically, this 4 ½μð1Þ − 5μð1Þ − 2μð0Þ þ 5μð0Þ 15 4 2 4 2 . In turn, a2 decays expo- allows us to (i) prove that the ME is to be expected on quite HCS λ−1 nentially to a2 with a characteristic time a . a general basis and for a wide range of systems, as long as Similar to the thermostatted case, we consider two non-Gaussianities are present and (ii) quantitatively predict ð Þ¼ð Þ ð Þ initial states T0;a2;0 TA;a2A and TB;a2B , with the region of parameters within which the ME is present. Δ ¼ − 0 Δ ¼ − 0 T0 TA TB > , a2;0 a2A a2B > . Logically, Moreover, we have also predicted the existence of an only the cooling case makes sense. In Fig. 4, we plot inverse ME: when the system is heated instead of cooled, two relaxation curves of the temperature for α ¼ 0.9, the initially cooler sample may heat sooner [63]. In this ¼ 1 ¼ 0 99 ¼ 0 5 ¼ −0 35 with TA , TB . , a2A . , a2B . , with way, we have provided a general theoretical framework for ¼ the choice Tref TA. The ME is clearly observed, and the understanding of the ME. the crossover time tc is The main assumptions in our theory are (i) the validity of the kinetic description, (ii) the system remaining 1 3 λ − μHCS ΔT −1 homogeneous for all times, and (iii) the first Sonine t ¼ ln 1 − a 2 0 ; ð8Þ c HCS 2 ð1Þ Δ λa − μ2 μ a2;0 approximation within the kinetic description. All these 2 assumptions have been validated in the paper. First, the see inset in Fig. 4. Therefore, there is a maximum value of numerical integration of the Enskog equation provided by the DSMC simulations have been successfully compared the ratio ΔT0=Δa2 0 for which the ME appears, . with MD simulations. Second, we have also checked that ð1Þ the system remains homogeneous in the MD simulations, ΔT 2 μ 0 ¼ 2 : ð9Þ both for the heated and undriven cases. Concretely, in the Δ 3 λ − μHCS a2;0 max a 2 latter, the system size has been chosen to be well below the clustering instability threshold [55,65]. Third, the accuracy Thus, the ME actually survives in the zero driving limit. of the first Sonine approximation has been confirmed by Had we considered a small value of the driving χ instead of the excellent agreement between our analytical results and χ ¼ 0, Eqs. (7)–(9) would characterize the strongly non- the DSMC simulations, even for the not-so-small values of linear regime, in which the initial scaled temperature the excess kurtosis a2 considered throughout. θ ¼ ≫ 1 0 T0=Tst . In a first stage of the relaxation, as long Finally, we stress that non-Gaussianities may have a ≫ as the granular temperature T Tst, the driving can be leading role in the emergence of the ME in other systems, even when there is no inelasticity. For example, the temperature of a molecular fluid, which is basically the mean kinetic energy per particle, does not remain constant if the system interacts with a thermal reservoir. Let us assume that the coupling with the reservoir brings about a nonlinear drag, as considered, for instance, in Refs. [66–69]. Then, the evolution equation of the temperature would involve higher moments of the transient nonequilibrium VDF. In this quite general situation, the ME would also stem from those non-Gaussianities [70]. This work has been supported by the Spanish Ministerio de Economía y Competitividad Grants No. FIS2013- 42840-P (A. L., F. V. R., and A. S.), No. FIS2016-76359- FIG. 4. Evolution of the temperature in the free cooling case. P (F. V. R. and A. S.), No. MTM2014-56948-C2-2-P Again, the agreement between the theory (lines) and both DSMC (A. L.), and No. FIS2014-53808-P (A. P.), and also by (open symbols) and MD (filled symbols) simulation data is the Junta de Extremadura Grant No. GR15104, partially excellent. The inset shows tc as a function of ΔT0=Δa2;0. financed by the European Regional Development Fund 148001-4 week ending PRL 119, 148001 (2017) PHYSICAL REVIEW LETTERS 6 OCTOBER 2017

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