Protocols for the Mpemba effect in granular fluids

Abstract

The so-called Mpemba effect (ME) is a counterintuitive memory phenomenon according to which, given two samples of a fluid, the initially hotter one may cool more rapidly than the initially cooler one to a common steady state. Although initially reported in the case of water, its existence for that liquid is still questioned. On the other hand, the ME has been observed to emerge (both theoretically and computationally) in the case of granular fluids, where the role of temperature (T) is played by the mean kinetic energy per particle and the steady state is tuned by a certain set of control parameters. The key point responsible for the ME in granular fluids is the observation that the relaxation of T(t) to its stationary value is governed not only by its initial value T(0) but also by the initial values of one or more additional variables. In some of the works in the literature on the ME for granular fluids, the state at t=0 was assumed ad hoc, without a description of the prior preparation protocols for t<0. In this talk, I will discuss two classes of granular systems where protocols for the ME can be clearly devised. In the first class, the system is an inertial suspension made of inelastic and smooth hard spheres under shear [1], the control parameters being the shear rate and the bath temperature. One of the samples (A) is prepared in a quasi-equilibrium unsheared steady state, while the other sample (B) is prepared in a sheared steady state with a bath temperature smaller than that of sample A. Then, at t=0, a common bath temperature (equal to that of sample B) and a common shear rate (smaller than that of sample B) are applied and both systems are allowed to relax to the new steady state. The second class of systems is defined by a dilute two-dimensional gas of hard and rough hard disks driven by a stochastic force and a stochastic torque, which inject translational and rotational energy, respectively [2]. In this case, the control parameters are the noise temperature associated with the total noise intensity and the fraction of rotational noise. In the preparation protocol for the standard ME, sample A is prepared with a higher noise temperature than sample B, the fraction of rotational noise being 0 and 1 for samples A and B, respectively. At t=0, both samples are subjected to a common noise temperature (smaller than that of sample B) and a common fraction of rotational noise (different from 0 and 1), and their relaxation to the new steady state is analyzed.