While we were working on the extension of our results to binary mixtures, we have realized that the scaled function ϕ defined in Eq. (46) also depends on the dimensionless parameterwhere the bath temperature T b is defined in Eq. (45) as γ b T b = βm 2 ξ 2 b . Note that θ = 0 for the stochastic thermostat (β = 0). The dependence of ϕ on θ changes only some expressions derived in the first-order approximation. Therefore, the last line of Eqs. (54) and (B15) should readThe difference is in the inclusion of the term proportional to ∂ζ (0) /∂θ . Moreover, Eq. (B10) should be replaced byand the third line of Eq. (B18) should readAll the above changes affect in principle the expressions of the coefficients μ and ζ 11 . However, given that the above new contributions to those coefficients are in general very small, they can be neglected in the final forms of μ and ζ 11 . Consequently, the main conclusions of the paper detailed in Sec. V remain unchanged. The authors are indebted to Nagi Khalil for pointing out the new dependence to us.059906-1 1539-3755/2013/87(5)/059906 (1)

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 describe a series of experiments and computer simulations on vibrated granular media in a geometry chosen to eliminate gravitationally induced settling. The system consists of a collection of identical spherical particles on a horizontal plate vibrating vertically, with or without a confining lid. Previously reported results are reviewed, including the observation of homogeneous, disordered liquid-like states, an instability to a 'collapse' of motionless spheres on a perfect hexagonal lattice, and a fluctuating, hexagonally ordered state. In the presence of a confining lid we see a variety of solid phases at high densities and relatively high vibration amplitudes, several of which are reported for the first time in this article. The phase behavior of the system is closely related to that observed in confined hard-sphere colloidal suspensions in equilibrium, but with modifications due to the effects of the forcing and dissipation. We also review measurements of velocity distributions, which range from Maxwellian to strongly non-Maxwellian depending on the experimental parameter values. We describe measurements of spatial velocity correlations that show a clear dependence on the mechanism of energy injection. We also report new measurements of the velocity autocorrelation function in the granular layer and show that increased inelasticity leads to enhanced particle selfdiffusion.

The transport coefficients of a granular fluid driven by a stochastic bath with friction are obtained by solving the inelastic Enskog kinetic equation from the Chapman-Enskog method. The heat and momentum fluxes as well as the cooling rate are determined to first order in the deviations of the hydrodynamic field gradients from their values in the homogeneous steady state. Since the collisional cooling cannot be compensated locally for the heat produced by the external driving force, the reference distribution f (0) (zeroth-order approximation) depends on time through its dependence on temperature. This fact gives rise to conceptual and practical difficulties not present in the undriven case. On the other hand, to simplify the analysis and given that we are interested in computing transport in the first order of deviations from the reference state, the steady-state conditions are considered to get explicit forms for the transport coefficients and the cooling rate. A comparison with recent Langevin dynamics simulations for driven granular fluids shows an excellent agreement for the kinematic viscosity although some discrepancies are observed for the longitudinal viscosity and the thermal diffusivity at large densities. Finally, a linear stability analysis of the hydrodynamic equations with respect to the homogeneous steady state is performed. As expected, no instabilities are found thanks to the presence of the external bath.

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...