Steady self-diffusion in classical gases

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

Uniform self-diffusion in a granular gas

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

The shearing instability of a dilute granular mixture

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

Vibrated granular gas confined by a piston

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

Power-law decay of the velocity autocorrelation function of a granular fluid in the homogeneous cooling state

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