Mathematical Analysis Seminars @ DISIM

Title: The inelastic Lorentz gas: derivation, and long-time behaviour

Abstract: Understanding the Boltzmann equation for non-conservative particle systems, where kinetic energy is dissipated at each collision, is challenging and its derivation has remained an open problem. The main difficulty arises from the singularities that both the particle system and the associated kinetic equation may develop in finite time. The inelastic Lorentz gas, composed of light particles undergoing inelastic collisions with infinitely heavy scatterers and evolving according to the inelastic linear Boltzmann equation, provides a non-trivial yet tractable model in which many questions can be addressed.

In this talk, we present a rigorous derivation of the inelastic Lorentz gas from the deterministic dynamics of tagged particles colliding with inelastic scatterers distributed according to a Poisson point process. The proof relies on demonstrating the convergence of the series expansion of the solution in suitable weighted spaces, together with a weak-convergence approach that allows us to handle the singularities of the backward flow. We will also discuss the long-time behaviour of the inelastic Lorentz gas in the case of Maxwell molecules, and in the presence of a uniform gravitational field. In particular, we show the existence of an out-of-equilibrium steady state that attracts all solutions within the considered functional framework.

This is a joint work with Nicola Miele and Alessia Nota (GSSI) (arXiv:2504.02155, arXiv:2511.02934).