Théophile Dolmaire

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).