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The formulation of the First and Second Principles of thermodynamics for a particle in contact with a heat bath and submitted to an external force is analyzed, by means of the Boltzmann-Lorentz kinetic equation. The possible definitions of the thermodynamic quantities are discussed in the light of the H theorem verified by the distribution of the particle. The work fluctuation relations formulated by Bochkov and Kuzovlev, and by Jarzynski, respectively, are derived from the kinetic equation. In addition, particle simulations using both the direct simulation Monte Carlo method and Molecular Dynamics, are used to investigate the practical accuracy of the results. Work distributions are also measured, and they turn out to be rather complex. On the other hand, they seem to depend very little, if any, on the interaction potential between the intruder and the bath.

The self-diffusion coefficient of a granular gas in the homogeneous cooling state is analyzed near the shearing instability. Using mode-coupling theory, it is shown that the coefficient diverges logarithmically as the instability is approached, due to the coupling of the diffusion process with the shear modes. The divergent behavior, which is peculiar of granular gases and disappears in the elastic limit, does not depend on any other transport coefficient. The theoretical prediction is confirmed by molecular dynamics simulation results for two-dimensional systems.

The hydrodynamic part of the velocity autocorrelation function of a granular fluid in the homogeneous cooling state has been calculated by using mode-coupling theory for a finite system with periodic boundary conditions. The existence of the shearing instability, leading to a divergent behavior of the velocity flow fluctuations, is taken into account. A time region in which the velocity autocorrelation function exhibits a power law decay, when time is measured by the number of collisions per particle, has been been identified. Also the explicit form of the exponential asymptotic long time decay has been obtained. The theoretical prediction for the power law decay is compared with molecular dynamics simulation results, and a good agreement is found, after taking into account finite size corrections. The effects of approaching the shearing instability are also explored.

We construct a novel class of exact solutions to the Boltzmann equation, in both its classical and quantum formulation, for arbitrary collision laws. When the system is subjected to a specific external forcing, the precise form of which is worked out, non equilibrium damping-less solutions are admissible. They do not contradict the $H$-theorem, but are constructed from its requirements. Interestingly, these solutions hold for time-dependent confinement. We exploit them, in a reverse-engineering perspective, to work out a protocol that shortcuts any adiabatic transformation between two equilibrium states in an arbitrarily short time-span, for an interacting system. Particle simulations of the direct Monte Carlo type fully corroborate the analytical predictions., Comment: 5 pages, 1 figure

A granular gas composed of inelastic hard spheres or disks in the homogeneous cooling state is considered. Some of the particles are labeled and their number density exhibits a time-independent linear profile along a given direction. As a consequence, there is a uniform flux of labeled particles in that direction. It is shown that the inelastic Boltzmann-Enskog kinetic equation has a solution describing this self-diffusion state. Approximate expressions for the transport equation and the distribution function of labeled particles are derived. The theoretical predictions are compared with simulation results obtained using the direct Monte Carlo method to generate solutions of the kinetic equation. A fairly good agreement is found.

A steady self-diffusion process in a gas of hard spheres at equilibrium is analyzed. The system exhibits a constant gradient of labeled particles. Neither the concentration of these particles nor its gradient are assumed to be small. It is shown that the Boltzmann-Enskog kinetic equation has an exact solution describing the state. The hydrodynamic transport equation for the density of labeled particles is derived, with an explicit expression for the involved self-diffusion transport coefficient. Also an approximated expression for the one-particle distribution function is obtained. The system does not exhibit any kind of rheological effects. The theoretical predictions are compared with numerical simulations using the direct simulation Monte Carlo method and a quite good agreement is found.

The shearing instability of a dilute granular mixture composed of smooth inelastic hard spheres or disks is investigated. By using the Navier-Stokes hydrodynamic equations, it is shown that the scaled transversal velocity mode exhibits a divergent behaviour, similarly to what happens in one-component systems. The theoretical prediction for the critical size is compared with direct Monte Carlo simulations of the Boltzmann equations describing the system, and a good agreement is found. The total energy fluctuations in the vicinity of the transition are shown to scale with the second moment of the distribution. The scaling distribution function is the same as found in other equilibrium and non-equilibrium phase transitions, suggesting the existence of some kind of universality.

A peculiarity of the hydrodynamic Navier-Stokes equations for a granular gas is the modification of the Fourier law, with the presence of an additional contribution to the heat flux that is proportional to the density gradient. Consequently, the constitutive relation involves, in the case of a one-component granular gas, two transport coefficients: the usual (thermal) heat conductivity and a diffusive heat conductivity. A very simple physical interpretation of this effect, in terms of the mean free path and the mean free time is provided. It leads to the modified Fourier law with an expression for the diffusive Fourier coefficient that differs in a factor of the order of unity from the expression obtained by means of the inelastic Boltzmann equation. Also, some aspects of the Chapman-Enskog computation of the new transport coefficients as well as of the comparison between simulation results and theory are discussed.

The volume fluctuations in the steady state reached by a vibrated granular gas of hard particles confined by a movable piston on the top are investigated by means of event driven simulations. Also, a compressibility factor, measuring the response in volume of the system to a change in the mass of the piston, is introduced and measured. From the second moment of the volume fluctuations and the compressibility factor, an effective temperature is defined, by using the same relation as obeyed by equilibrium molecular systems. The interpretation of this effective temperature and its relationship with the granular temperature of the gas, and also with the velocity fluctuations of the movable piston, is discussed. It is found that the ratio of the temperature based on the volume fluctuations to the temperature based on the piston kinetic energy, obeys simple dependencies on the inelasticity and on the piston-particle mass ratio.

The influence of the heating mechanism on the kinetic energy densities of the components of a vibrated granular mixture is investigated. Collisions of the particles with the vibrating wall are inelastic and characterized by two coefficients of normal restitution, one for each of the two species. By means of molecular dynamics simulations, it is shown that the non-equipartition of kinetic energy is not affected by the differential mechanism of energy injection, aside the usual boundary layer around the wall. The macroscopic state of the mixture in the bulk is defined by intensive variables that do not include the partial granular temperatures of the components.

The existence of two stationary solutions of the nonlinear Boltzmann equation for inelastic hard spheres or disks is investigated. They are restricted neither to weak dissipation nor to small gradients. The one-particle distribution functions are assumed to have an scaling property, namely that all the position dependence occurs through the density and the temperature. At the macroscopic level, the state corresponding to both is characterized by uniform pressure, no mass flow, and a linear temperature profile. Moreover, the state exhibits two peculiar features. First, there is a relationship between the inelasticity of collisions, the pressure, and the temperature gradient. Second, the heat flux can be expressed as being linear in the temperature gradient, i.e. a Fourier-like law is obeyed. One of the solutions is singular in the elastic limit. The theoretical predictions following from the other one are compared with molecular dynamics simulation results and a good agreement is obtained in the parameter region in which the Fourier state can be actually observed in the simulations, namely not too strong inelasticity., Comment: 24 pages 9 figures

The steady state of a vibrated granular gas confined by a movable piston on the top is discussed. Particular attention is given to the hydrodynamic boundary conditions to be used when solving the inelastic Navier-Stokes equations. The relevance of an exact general condition relating the grain fluxes approaching and moving away from each of the walls is emphasized. It is shown how it can be used to get a consistent hydrodynamic description of the boundaries. The obtained expressions for the fields do not contain any undetermined parameter. Comparison of the theoretical predictions with molecular dynamics simulation results is carried out, and a good agreement is observed for low density and not too large inelasticity. A practical way of introducing small finite density corrections to the dilute limit theory is proposed, to improve the accuracy of the theory.

The steady state velocity fluctuations of a movable piston located on the top of a vibrated granular gas are studied by means of molecular dynamics simulations. From the second moment of the distribution, a temperature parameter for the piston is defined and compared with the granular temperature of the gas located just below it. Then the two temperature parameters refer to two interacting macroscopic systems. The equipartition of energy valid in usual molecular systems is strongly violated, and the temperature of the piston can be larger or smaller than the one of the gas, depending of the parameters defining the system at the particle level. The simulation results for the ratio of temperatures are in agreement with some theoretical predictions from kinetic theory assuming the validity of a hydrodynamic description in the limit of weak inelasticity of the gas.

Hydrodynamic equations are used to identify the final state reached by a freely evolving granular gas above but close to its shear instability. The theory predicts the formation of a two bands shear state with a steady density profile. There is a modulation between temperature and density profiles as a consequence of the energy balance, the density fluctuations remaining small, without producing clustering. Moreover, the time dependence of the velocity field can be scaled out with the squared root of the average temperature of the system. The latter follows the Haff law, but with an effective cooling rate that is smaller than that of the free homogeneous state. The theoretical predictions are compared with numerical results for inelastic hard disks obtained by using the direct Monte Carlo simulation method, and a good agreement is obtained for low inelasticity.

The spontaneous symmetry breaking in a vibro-fluidized low-density granular gas in three connected compartments is investigated. When the total number of particles in the system becomes large enough, particles distribute themselves unequally among the three compartments. Particles tend to concentrate in one of the compartments, the other two having the (relatively small) same average number of particles. A hydrodynamical model that accurately predicts the bifurcation diagram of the system is presented. The theory can be easily extended to the case of an arbitrary number of connected compartments.

The behavior of an isolated dilute granular gas near the threshold of its clustering instability is investigated by means of fluctuating hydrodynamics and the direct simulation Monte Carlo method. The theoretical predictions from the former are shown to be in good agreement with the simulation results. The total energy of the the system is renormalized by the fluctuations of the vorticity field. Moreover, the scaled second moment of the energy fluctuations exhibits a power-law divergent behavior

The hydrodynamic state of an impurity immersed in a low density granular gas is analyzed. Explicit expressions for the temperature and density fields of the impurity in terms of the hydrodynamic fields of the gas are derived. It is shown that the ratio between the temperatures of the two components, measuring the departure from energy equipartition, only depends on the mechanical properties of the particles, being therefore constant in the bulk of the system. This ratio plays an important role in determining the density profile of the intruder and its position with respect to the gas, since it determines the sign of the pressure diffusion coefficient. The theoretical predictions are compared with molecular dynamics simulation results for the particular case of the steady state of an open vibrated granular system in absence of macroscopic fluxes, and a satisfactory agreement is found.

The difference of temperatures between an impurity and the surrounding gas in an open vibrated granular system is studied. It is shown that, in spite of the high inhomogeneity of the state, the temperature ratio remains constant in the bulk of the system. The lack of energy equipartition is associated to the change of sign of the pressure diffusion coefficient for the impurity at certain values of the parameters of the system, leading to a segregation criterium. The theoretical predictions are consistent with previous experimental results, and also in agreement with molecular dynamics simulation results reported in this paper., Comment: To appear in Phys. Rev. Lett

The global energy fluctuations of a low density gas granular gas in the homogeneous cooling state near its clustering instability are studied by means of molecular dynamics simulations. The relative dispersion of the fluctuations is shown to exhibit a power-law divergent behavior. Moreover, the probability distribution of the fluctuations presents data collapse as the system approaches the instability, for different values of the inelasticity. The function describing the collapse turns out to be the same as the one found in several molecular equilibrium and non-equilibrium systems, except for the change in the sign of the fluctuations.

It is shown that the hydrodynamic modes of a dilute granular gas of inelastic hard spheres can be identified, and calculated in the long wavelength limit. Assuming they dominate at long times, formal expressions for the Navier-Stokes transport coefficients are derived. They can be expressed in a form that generalizes the Green-Kubo relations for molecular systems, and it is shown that they can also be evaluated by means of $N$-particle simulation methods. The form of the hydrodynamic modes to zeroth order in the gradients is used to detect the presence of inherent velocity correlations in the homogeneous cooling state, even in the low density limit. They manifest themselves in the fluctuations of the total energy of the system. The theoretical predictions are shown to be in agreement with molecular dynamics simulations. Relevant related questions deserving further attention are pointed out.

The properties of a dilute granular gas in the homogeneous cooling state are mapped to those of a stationary state by means of a change in the time scale that does not involve any internal property of the system. The new representation is closely related with a general property of the granular temperature in the long time limit. The physical and practical implications of the mapping are discussed. In particular, simulation results obtained by the direct simulation Monte Carlo method applied to the scaled dynamics are reported. This includes ensemble averages and also the velocity autocorrelation function, as well as the self-diffusion coefficient obtained from the latter by means of the Green-Kubo representation. In all cases, the obtained results are compared with theoretical predictions.

The departure of a granular gas in the instable region of parameters from the initial homogeneous cooling state is studied. Results from Molecular Dynamics and from Direct Monte Carlo simulation of the Boltzmann equation are compared. It is shown that the Boltzmann equation accurately predicts the low density limit of the system. The relevant role played by the parallelization of the velocities as time proceeds and the dependence of this effect on the density is analyzed in detail.

The linearized Boltzmann equation is considered to describe small spatial perturbations of the homogeneous cooling state. The corresponding macroscopic balance equations for the density, temperature, and flow velocity are derived from it as the basis for a hydrodynamic description. Hydrodynamics is defined in terms of the spectrum of the generator for the dynamics of the linearized Boltzmann equation. The hydrodynamic eigenfunctions and eigenvalues are calculated in the long wavelength limit. The results allow identification of the hydrodynamic part of the solution to the linearized Boltzmann equation. This contribution is used to calculate the fluxes in the macroscopic balance equations, leading to the Navier-Stokes equations and associated transport coefficients. The results agree with those obtained earlier by the Chapman-Enskog method. The implications of this analysis for application of methods of linear response to granular fluids and derivation of Green-Kubo expressions for transport coefficients are discussed.

The velocity distribution of a fluidized dilute granular gas in the direction perpendicular to the gravitational field is investigated by means of Molecular Dynamics simulations. The results indicate that the velocity distribution can be exactly described neither by a Gaussian nor by a stretched exponential law. Moreover, it does not exhibit any kind of scaling. In fact, the actual shape of the distribution depends on the number of monolayers at rest, on the restitution coefficient and on the height at what it is measured. The role played by the number of particle-particle collisions as compared with the number of particle-wall collisions is discussed.

The spontaneous symmetry breaking taking place in the direction perpendicular to the energy flux in a dilute vibrofluidized granular system is investigated, using both a hydrodynamic description and simulation methods. The latter include molecular dynamics and direct Monte Carlo simulation of the Boltzmann equation. A marginal stability analysis of the hydrodynamic equations, carried out in the WKB approximation, is shown to be in good agreement with the simulation results. The shape of the hydrodynamic profiles beyond the bifurcation is discussed.