Mean-field model of free-cooling inelastic mixtures

We consider a mean-field model describing the free-cooling process of a two-component granular mixture, a generalization of the so called scalar 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 $v=0\mathrm{}$ 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 of 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 nonextensive statistical mechanics, in spite of the fact that ours stem from a purely dynamical approach.