Analysis of density fluctuations of a rarefied granular gas from Boltzmann equation

The density fluctuations of a rarefied granular gas are analyzed, within the framework of the Boltzmann equation. The one particle-distribution function is perturbed from a time-dependent Maxwellian and the perturbed distribution function – which is considered a linear combination of the thirteen moments of the distribution function – is represented as a plane wave of small time-dependent amplitudes and constant wave number vector. From an equation for the perturbed distribution function, a system of differential equations for the time-dependent amplitudes is obtained. The density fluctuations are determined from the solution of the system of differential equations. From the time evolution of the density fluctuations two distinct cases arise: (i) for short wavelengths, the density fluctuations of the inelastic and elastic cases represent damped waves that go to zero for large dimensionless time; (ii) for large wavelengths, the density fluctuations of the elastic case represent damped waves, whereas the density fluctuations of the inelastic case grow with the dimensionless time, so that an instability occurs. For the perturbed distribution function represented with five moments, the behaviors of the density fluctuations corresponding to large and small wavelengths are: (i) for elastic collisions the density fluctuations evolve as harmonic waves without dissipation; (ii) for inelastic collisions there were always growing solutions related with instabilities.