Equilibration, generalized equipartition, and diffusion in dynamical Lorentz gases

TL;DR

This paper proves that particles in a dynamical Lorentz gas approach a Maxwell-Boltzmann distribution at a specific temperature, with their motion becoming diffusive, even when scatterers are not initially in thermal equilibrium.

Contribution

It establishes the approach to thermal equilibrium and generalized equipartition in a Hamiltonian Lorentz gas with non-thermal scatterer states.

Findings

Particles reach Maxwell-Boltzmann distribution at temperature T

Particle motion becomes diffusive in equilibrium

Temperature T relates to scatterers' average kinetic energy

Abstract

We prove approach to thermal equilibrium for the fully Hamiltonian dynamics of a dynamical Lorentz gas, by which we mean an ensemble of particles moving through a $d$-dimensional array of fixed soft scatterers that each possess an internal harmonic or anharmonic degree of freedom to which moving particles locally couple. We establish that the momentum distribution of the moving particles approaches a Maxwell-Boltzmann distribution at a certain temperature $T$, provided that they are initially fast and the scatterers are in a sufficiently energetic but otherwise arbitrary stationary state of their free dynamics--they need not be in a state of thermal equilibrium. The temperature $T$ to which the particles equilibrate obeys a generalized equipartition relation, in which the associated thermal energy $k_{\mathrm B}T$ is equal to an appropriately defined average of the scatterers' kinetic energy. In the equilibrated state, particle motion is diffusive.