We study in this work a dilute granular gas immersed in a thermal bath made of smaller particles with nonnegligible masses as compared with the granular ones. The kinetic theory for this system is developed and described by an Enskog--Fokker--Planck equation for the one-particle velocity distribution function. Granular particles are assumed to have inelastic and hard interactions, losing energy in collisions as accounted by a constant coefficient of normal restitution. The interaction with the thermal bath is based on a nonlinear drag force plus a white-noise stochastic force. To get explicit results of the temperature aging and steady states, Maxwellian and first Sonine approximations are developed. The latter takes into account the coupling of the excess kurtosis with the temperature. Theoretical predictions are compared with direct simulation Monte Carlo and event-driven molecular dynamics simulations. While good results for the granular temperature are obtained from the Maxwellian approximation, a much better agreement, especially as inelasticity and drag nonlinearity increase, is found when using the first Sonine approximation. The latter approximation is, additionally, crucial to account for memory effects like Mpemba and Kovacs-like ones.

Finding the proper entropy functional associated with the inelastic Boltzmann equation for a granular gas is a yet unsolved challenge. The original H-theorem hypotheses do not fit here and the H-functional presents some additional measure problems that are solved by the Kullback–Leibler divergence (KLD) of a reference velocity distribution function from the actual distribution. The right choice of the reference distribution in the KLD is crucial for the latter to qualify or not as a Lyapunov functional, the “homogeneous cooling state” (HCS) distribution of the freely cooling system being a potential candidate. Due to the lack of a formal proof, the aim of this work is to support this conjecture aided by molecular dynamics simulations of inelastic hard disks and spheres in a wide range of values for the coefficient of restitution (α). Our results reject the Maxwellian distribution as a possible reference, whereas reinforce the HCS one. Moreover, the KLD is used to measure the amount of information lost on using the former rather than the latter, and reveals a nonmonotonic dependence with α. Additionally, a Maxwell-demon-like velocity-inversion experiment highlights the microscopic irreversibility of the granular gas dynamics.

The residual multiparticle entropy (RMPE) of a fluid is defined as the difference, Δs, between the excess entropy per particle (relative to an ideal gas with the same temperature and density), s_ex, and the pair-correlation contribution, s₂. Thus, the RMPE represents the net contribution to s_ex due to spatial correlations involving three, four, or more particles. A heuristic “ordering” criterion identifies the vanishing of the RMPE as an underlying signature of an impending structural or thermodynamic transition of the system from a less ordered to a more spatially organized condition (freezing is a typical example). Regardless of this, the knowledge of the RMPE is important to assess the impact of non-pair multiparticle correlations on the entropy of the fluid. Recently, an accurate and simple proposal for the thermodynamic and structural properties of a hard-sphere fluid in fractional dimension 1 < d < 3 has been proposed [Santos, A.; López de Haro, M. Phys. Rev. E 2016, 93, 062126]. The aim of this work is to use this approach to evaluate the RMPE as a function of both d and the packing fraction ϕ. It is observed that, for any given dimensionality d, the RMPE takes negative values for small densities, reaches a negative minimum Δs_min at a packing fraction ϕ_min, and then rapidly increases, becoming positive beyond a certain packing fraction ϕ₀. Interestingly, while both ϕ_min and ϕ₀ monotonically decrease as dimensionality increases, the value of Δs_min exhibits a nonmonotonic behavior, reaching an absolute minimum at a fractional dimensionality d ≃ 2.38. A plot of the scaled RMPE Δs/|Δs_min| shows a quasiuniversal behavior in the region −0.14 ≲ ϕ − ϕ₀ ≲ 0.02.

New proposals for the equation of state of four- and five-dimensional hard-hypersphere mixtures in terms of the equation of state of the corresponding monocomponent hard-hypersphere fluid are introduced. Such proposals (which are constructed in such a way so as to yield the exact third virial coefficient) extend, on the one hand, recent similar formulations for hard-disk and (three-dimensional) hard-sphere mixtures and, on the other hand, two of our previous proposals also linking the mixture equation of state and the one of the monocomponent fluid but unable to reproduce the exact third virial coefficient. The old and new proposals are tested by comparison with published molecular dynamics and Monte Carlo simulation results and their relative merit is evaluated.