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 two different temperatures, put in contact with a thermal reservoir at subzero (on the Celsius scale) temperature. While one may intuitively expect that the initially cooler sample would freeze first, it has been observed that this is not always the case [1]. This paradoxical behavior named the Mpemba effect (ME) has been known since antiquity and discussed by philosophers like Aristotle, Roger Bacon, Francis Bacon, and Descartes [2,3]. Nevertheless, physicists only started to analyze it in the second part of the past century, mainly in popular science or education journals .There is no consensus on the underlying physical mechanisms that bring about the ME. Specifically, water evaporation [4,5,9,24], differences in the gas composition of water [11,17,25], natural convection [6,23,26], or the influence of supercooling, either alone [14,27] or combined with other causes [28][29][30][31], have been claimed to have an impact on the ME. Conversely, the own existence of the ME in water has been recently put in question [32]. Notwithstanding, Mpemba-like effects have also been observed in different physical systems, such as carbon nanotube resonators [33] or clathrate hydrates [34].The ME requires the evolution equation for the temperature to involve other variables, which may facilitate or hinder the temperature relaxation rate. The initial values of those additional variables depend on the way the system has been prepared, i.e., "aged," before starting the relaxation process. Typically, aging and memory effects are associated with slowly evolving systems with a complex energy landscape, such as glassy [35][36][37][38][39][40][41][42][43] or dense granular systems [44][45][46]. However, these effects have also been observed in simpler systems, like granular gases [47][48][49][50] or, very recently, crumpled thin sheets and elastic foams [51].In a general physical system, the study of the ME implies finding those additional variables that control the temperature relaxation and determining how different they have to be initially in order to facilitate its emergence. In order to quantify the effect with the tools of nonequilibrium statistical mechanics, a precise definition thereof is mandatory. An option is to look at the relaxat...

We analyze the fluctuations of the dissipated energy in a simple and general model where dissipation, diffusion, and driving are the key ingredients. The full dissipation distribution, which follows from hydrodynamic fluctuation theory, shows non-Gaussian tails and no negative branch, thus violating the fluctuation theorem as expected from the irreversibility of the dynamics. It exhibits simple scaling forms in the weak- and strong-dissipation limits, with large fluctuations favored in the former case but strongly suppressed in the latter. The typical path associated with a given dissipation fluctuation is also analyzed in detail. Our results, confirmed in extensive simulations, strongly support the validity of hydrodynamic fluctuation theory to describe fluctuating behavior in driven dissipative media.

While memory effects have been reported for dense enough disordered systems such as glasses, we show here by a combination of analytical and simulation techniques that they are also intrinsic to the dynamics of dilute granular gases. By means of a certain driving protocol, we prepare the gas in a state where the granular temperature T coincides with its long time limit. However, T does not subsequently remain constant but exhibits a nonmonotonic evolution before reaching its nonequilibrium steady value. The corresponding so-called Kovacs hump displays a normal behavior for weak dissipation (as observed in molecular systems) but is reversed under strong dissipation, where it, thus, becomes anomalous. DOI: 10.1103/PhysRevLett.112.198001 PACS numbers: 45.70.-n, 02.70.-c, 05.20.Dd, 51.10.+y At equilibrium, the response of a system to an external sudden perturbation, like a temperature jump, depends only on the macroscopic variables characterizing the state under study. On the other hand, in nonequilibrium situations, the observed response depends not only on the instantaneous value of the macroscopic variables but also on the previous history. Memory effects are, consequently, ubiquitous out of equilibrium. A classic experiment in this context bears the name of Kovacs [1,2]. A polymer sample, initially at equilibrium at a high temperature T 0 , is rapidly quenched to a low temperature T 1 , at which it evolves for a given waiting time t w . Afterwards, the bath temperature is suddenly increased to T, with T 0 > T > T 1 , such that the instantaneous polymer volume V equals its equilibrium value at T. The sample volume then does not remain constant for t > t w : it first increases, displays a maximum, and returns to equilibrium for longer times only. This simple experiment shows that the macroscopic variables ðP; V; TÞ (the pressure P being kept constant throughout the whole procedure) do not completely characterize the macroscopic state of the system: Its response depends also on the previous thermal history.This kind of crossover, or Kovacs memory effect, has been extensively investigated in glassy and other complex systems starting from the phenomenological theory presented by Kovacs himself [2]. It is displayed by polymers, structural and spin glasses, compacting dense granular media, kinetically constrained models, classical and quantum spin models, distributions of two-level systems, etc. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The quantity displaying the hump may be different from the volume: In several of the previous studies, the energy is the relevant quantity. Interestingly, most of the observed behavior can be understood within a linear response theory approach, although the temperature jumps are usually not small in the experiments [14,16,17].Whereas the Kovacs effect has previously been reported for dense media or systems exhibiting complex energy landscape, we focus here on a low density granular gas [18,19] where the effect is a priori less expected. Because of inelastic collis...

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We consider a general class of nonlinear diffusive models with bulk dissipation and boundary driving, and derive its hydrodynamic description in the large size limit. Both the average macroscopic behavior and the fluctuating properties of the hydrodynamic fields are obtained from the microscopic dynamics. This analysis yields a fluctuating balance equation for the local energy density at the mesoscopic level, characterized by two terms: (i) a diffusive term, with a current that fluctuates around its average behavior given by nonlinear Fourier's law, and (ii) a dissipation term which is a general function of the local energy density. The quasi-elasticity of microscopic dynamics, required in order to have a nontrivial competition between diffusion and dissipation in the macroscopic limit, implies a noiseless dissipation term in the balance equation, so dissipation fluctuations are enslaved to those of the density field. The microscopic complexity is thus condensed in just three transport coefficients, the diffusivity, the mobility and a new dissipation coefficient, which are explicitly calculated within a local equilibrium approximation. Interestingly, the diffusivity and mobility coefficients obey an Einstein relation despite the fully nonequilibrium character of the problem. The general theory here presented is applied to a particular albeit broad family of systems, the simplest nonlinear dissipative variant of the so-called KMP model for heat transport. The theoretical predictions are compared to extensive numerical simulations, and an excellent agreement is found.

Abstract. The Kovacs or crossover effect is one of the peculiar behaviours exhibited by glasses and other complex, slowly relaxing systems. Roughly it consists in the nonmonotonic relaxation to its equilibrium value of a macroscopic property of a system evolving at constant temperature, when starting from a non-equilibrium state. Here, this effect is investigated for general systems whose dynamics is described by a master equation. To carry out a detailed analysis, the limit of small perturbations in which linear response theory applies is considered. It is shown that, under very general conditions, the observed experimental features of the Kovacs effect are recovered. The results are particularized for a very simple model, a two-level system with dynamical disorder. An explicit analytical expression for its non-monotonic relaxation function is obtained, showing a resonant-like behaviour when the dependence on the temperature is investigated.

We report the emergence of a giant Mpemba effect in the uniformly heated gas of inelastic rough hard spheres: The initially hotter sample may cool sooner than the colder one, even when the initial temperatures differ by more than one order of magnitude. In order to understand this behavior, it suffices to consider the simplest Maxwellian approximation for the velocity distribution in a kinetic approach. The largeness of the effect stems from the fact that the rotational and translational temperatures, which obey two coupled evolution equations, are comparable. Our theoretical predictions agree very well with molecular dynamics and direct simulation Monte Carlo data.Let us consider two beakers of water at different temperatures. Mpemba and Osborne showed that the initially hotter sample cools sooner under certain conditions [1], i.e., the curve giving the time evolution of its temperature crosses that of the initially cooler sample and stays below it for longer times. This is called the Mpemba memory effect, which is known since antiquity in cultures for which water in the form of ice and snow is common [2]. Later, the Mpemba effect has been clearly identified in different physical systems [3][4][5][6][7], although there is still some debate about its existence in water [8].From a physical point of view, one would like to answer how different the initial preparation of two samples of the system under study must be so that the Mpemba effect arises. This is the main-currently unresolved in general-question, although there has been some recent progress in this respect [5,6]. Lu and Raz [5] analyzed the Mpemba effect in a generic Markovian system by monitoring the relaxation of an entropy-like variable that measures the distance to the steady state. This makes it possible to define and investigate Mpemba-like effects in systems for which there is not an obvious definition of a nonequilibrium temperature, but makes the comparison with the usual experimental setup described above difficult.A different approach was carried out by some of us in the study of the Mpemba effect for a granular fluid of smooth hard spheres [6]. Therein, the granular temperature-basically the average kinetic energy per particle-is the physical quantity monitored to investigate the Mpemba effect. In the smooth-sphere case, the angular velocities play no role since there is no energy transfer between the translational and the rotational degrees of freedom, and the kinetic energy is thus purely * prados@us.es translational. We showed that the Mpemba effect stems from the coupling of the granular temperature and the kurtosis, which measures the deviation of the velocity distribution function from the Maxwellian shape at the lowest order. More specifically, it is the difference between the initial values of the kurtosis of the two samples that controls the appearance of the Mpemba effect.In the granular fluid of smooth hard spheres, the kurtosis is typically small. On the one hand, this facilitates the theoretical analysis, because it makes it possi...

We look into the Mpemba effect—the initially hotter sample cools sooner—in a molecular gas with nonlinear viscous drag. Specifically, the gas particles interact among them via elastic collisions and with a background fluid at equilibrium. Thus, within the framework of kinetic theory, our gas is described by an Enskog–Fokker–Planck equation. The analysis is carried out using the first Sonine approximation, in which the evolution of temperature is coupled to that of excess kurtosis. This coupling leads to the emergence of the Mpemba effect, which is observed at an early stage of relaxation and when the initial temperatures of the two samples are close enough. This allows for the development of a simple theory, linearizing the temperature evolution around a reference temperature, namely, the initial temperature closer to the asymptotic equilibrium value. The linear theory provides a semiquantitative description of the effect, including expressions for crossover time and maximum temperature difference. We also discuss the limitations of our linearized theory.