Which is the temperature of granular systems? A mean field model of free cooling inelastic mixtures

Abstract

We consider a mean field model describing the free cooling process of a two component granular mixture, a generalization of so called Maxwell model. The cooling is viewed as an ordering process and the scaling behavior is attributed to the presence of an attractive fixed point at $v=0$ for the dynamics. By means of asymptotic analysis of the Boltzmann equation and of numerical simulations we get the following results: 1)we establish the existence of two different partial granular temperatures, one for each component, which violates the Zeroth Law of Thermodynamics; 2) we obtain the scaling form of the two distribution functions; 3) we prove the existence of a continuous spectrum of exponents characterizing the inverse-power law decay of the tails of the velocity, which generalizes the previously reported value 4 for the pure model; 4) we find that the exponents depend on the composition, masses and restitution coefficients of the mixture; 5) we also remark that the reported distributions represent a dynamical realization of those predicted by the Non Extensive Statistical Mechanics, in spite of the fact that ours stem from a purely dynamical approach.