Complete characterization of convergence to equilibrium for an inelastic Kac model

Abstract

Pulvirenti and Toscani introduced an equation which extends the Kac caricature of a Maxwellian gas to inelastic particles. We show that the probability distribution, solution of the relative Cauchy problem, converges weakly to a probability distribution if and only if the symmetrized initial distribution belongs to the standard domain of attraction of a symmetric stable law, whose index $\alpha$ is determined by the so-called degree of inelasticity, $p>0$, of the particles: $\alpha =\frac{2}{1+p}$. This result is then used: (1) To state that the class of all stationary solutions coincides with that of all symmetric stable laws with index $\alpha$. (2) To determine the solution of a well-known stochastic functional equation in the absence of extra-conditions usually adopted.

References in corpus (3)

• Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem
• Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model
• Central limit theorem for the solution of the Kac equation

Cited by in corpus (3)

• Local stability of perfect alignment for a spatially homogeneous kinetic model
• Characterization of weak convergence of probability-valued solutions of general one-dimensional kinetic equations
• Probabilistic View of Explosion in an Inelastic Kac Model