Equilibration, Generalized Equipartition, and Diffusion in Dynamical Lorentz Gases

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 MaxwellBoltzmann 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 B T is equal to an appropriately defined average of the scatterers' kinetic energy. In the equilibrated state, particle motion is diffusive. * Stephan.De-Bievre@math.univ-lille1.fr † parris@mst.edu

Equilibration and Diffusion for a Dynamical Lorentz Gas

“…We view the states of the scatterer that the particle successively visits as independent and identically distributed random variables and fix the distance between two consecutive scatterers met by the particle. In [DBP11], it was shown, for a Markov chain which is a cut version of the one we consider in Section 2, that it captures the essential features of the behaviour of the original system very well, both in terms of the evolution of the momentum distribution of the particle and of its diffusive spatial displacement.…”

“…The leading coefficients of this expansion were computed in [DBP11]. In the following proposition, we provide detailed control on the error terms, in particular their behaviour in M and in α * .…”

“…These models describe the motion of a single particle through an array of independent scatterers, each possessing an internal degree of freedom to which the particle is locally coupled. In [DBP11], it is argued that the particle momentum distribution ρ(t) will converge, asymptotically in time, to a Maxwell-Boltzmann thermal equilibrium distribution characterized by a temperature that is determined by the energy distribution of the individual scatterers . This convergence holds in a suitable parameter range corresponding to a weak coupling limit and for an arbitrary distribution ρ 0 (t) of sufficiently large average mean speed.…”

“…Further more, we assume that the particle is always fast which means that it always crosses the interaction region in a time of order p −1 (see [DBP11]), where · is for the Euclidean norm in R d . We will assume moreover that its kinetic energy p(t) 2 /2 is well above the typical interaction potential αQσ(q(t) − r) that it encounters in any scattering event.…”

“…This means we are dealing with a very complex nonlinear infinite dimensional dynamical system. To simplify the problem, we will follow [DBP11] and instead describe the trajectory of the particle by a Markov chain, eliminating both the complexity induced by recollisions and by the geometry of the family (r i ) i∈Z d provided the horizon is finite. In this manner, we concentrate on the essential dynamical phenomena induced by the individual scattering events.…”