In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse tem
perature $beta$. The system admits the canonical distribution at inverse temperature $beta$ as the unique equilibrium state. We prove that any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one-particle marginal, in the large-system limit, satisfies an effective Boltzmann-type equation.
The steady state for a system of N particle under the influence of an external field and a Gaussian thermostat and colliding with random virtual scatterers can be obtained explicitly in the limit of small field. We show the sequence of steady state d
istribution, as N varies, forms a chaotic sequence in the sense that the k particle marginal, in the limit of large N, is the k-fold tensor product of the 1 particle marginal. We also show that the chaoticity properties holds in the stronger form of entropic chaoticity.
We investigate analytically and numerically the spatial structure of the non-equilibrium stationary states (NESS) of a point particle moving in a two dimensional periodic Lorentz gas (Sinai Billiard). The particle is subject to a constant external el
ectric field E as well as a Gaussian thermostat which keeps the speed |v| constant. We show that despite the singular nature of the SRB measure its projections on the space coordinates are absolutely continuous. We further show that these projections satisfy linear response laws for small E. Some of them are computed numerically. We compare these results with those obtained from simple models in which the collisions with the obstacles are replaced by random collisions.Similarities and differences are noted.
We consider a system of particles subjected to a uniform external force E and undergoing random collisions with virtual fixed obstacles, as in the Drude model of conductivity. The system is maintained in a nonequilibrium stationary state by a Gaussia
n thermostat. In a suitable limit the system is described by a self consistent Boltzmann equation for the one particle distribution function f. We find that after a long time f(v,t) approaches a stationary velocity distribution f(v) which vanishes for large speeds, i.e. f(v)=0 for |v|>vmax(E), with vmax(E)~1/|E| as |E| -> 0. In that limit f(v)~exp(-c|v|^3) for fixed v, where c depends on mean free path of the particle. f(v) is computed explicitly in one dimension.
