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This paper deals with a one--dimensional model for granular materials, whichboils down to an inelastic version of the Kac kinetic equation, withinelasticity parameter $p>0$. In particular, the paper provides bounds forcertain distances -- such as specific weighted $\chi$--distances and theKolmogorov distance -- between the solution of that equation and the limit. Itis assumed that the even part of the initial datum (which determines theasymptotic properties of the solution) belongs to the domain of normalattraction of a symmetric stable distribution with characteristic exponent$\a=2/(1+p)$. With such initial data, it turns out that the limit exists and isjust the aforementioned stable distribution. A necessary condition for therelaxation to equilibrium is also proved. Some bounds are obtained withoutintroducing any extra--condition. Sharper bounds, of an exponential type, areexhibited in the presence of additional assumptions concerning either thebehaviour, near to the origin, of the initial characteristic function, or thebehaviour, at infinity, of the initial probability distribution function.

Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model