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.

Stochastic thermodynamics lays down a broad framework to revisit the venerable concepts of heat, work and entropy production for individual stochastic trajectories of mesoscopic systems. Remarkably, this approach, relying on stochastic equations of motion, introduces time into the description of thermodynamic processes—which opens the way to fine control them. As a result, the field of finite-time thermodynamics of mesoscopic systems has blossomed. In this article, after introducing a few concepts of control for isolated mechanical systems evolving according to deterministic equations of motion, we review the different strategies that have been developed to realize finite-time state-to-state transformations in both over and underdamped regimes, by the proper design of time-dependent control parameters/driving. The systems under study are stochastic, epitomized by a Brownian object immersed in a fluid; they are thus strongly coupled to their environment playing the role of a reservoir. Interestingly, a few of those methods (inverse engineering, counterdiabatic driving, fast-forward) are directly inspired by their counterpart in quantum control. The review also analyzes the control through reservoir engineering. Besides the reachability of a given target state from a known initial state, the question of the optimal path is discussed. Optimality is here defined with respect to a cost function, a subject intimately related to the field of information thermodynamics and the question of speed limit. Another natural extension discussed deals with the connection between arbitrary states or non-equilibrium steady states. This field of control in stochastic thermodynamics enjoys a wealth of applications, ranging from optimal mesoscopic heat engines to population control in biological systems.

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.

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.

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.

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.