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

week ending PRL 119, 148001 (2017) PHYSICAL REVIEW LETTERS 6 OCTOBER 2017 When the Hotter Cools More Quickly: Mpemba Effect in Granular Fluids 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 0031-9007=17=119(14)=148001(6) 148001-1 © 2017 American Physical Society week ending PRL 119, 148001 (2017) PHYSICAL REVIEW LETTERS 6 OCTOBER 2017 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 λÆ ¼ ½Λ11 þ Λ22 Æ 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.

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