Properties of stationary nonequilibrium states in the thermostatted periodic Lorentz gas: The multiparticle system

Abstract

We study the stationary nonequilibrium states of N-point particles moving under the influence of an electric field E among fixed obstacles (disk) in a two-dimensional torus. The total kinetic energy of the system is kept constant through a Gaussian thermostat that produces a velocity dependent mean field interaction between the particles. The current and the particle distribution functions are obtained numerically and compared for small /E/ with analytic solutions of a Boltzmann-type equation obtained by treating the collisions with the obstacles as random independent scatterings. The agreement is surprisingly good for both small and large N. The latter system in turn agrees with a self-consistent one-particle evolution expected to hold in the N-->infinity limit.

Absolute continuity of projected SRB measures of coupled Arnold cat map lattices

“…Of course to make the thermostatted dynamical systems appropriate for modeling physical situations one would need to show that these reduced distributions are equal, in the bulk, to those obtained from stochastic boundary drives or from considering infinite system with Hamiltonian dynamics. This is in fact what appears to be the case when the Moran-Hoover model is extended to many particles [9]. In this paper we prove the absolute continuity of the reduced distributions or induced measure for a very idealized dynamical system made up of an infinite collection of Arnold cat maps of the two torus, indexed by a d-dimensional lattice.…”

On differentiability of SRB states for partially hyperbolic systems

“…The main difference between our approach and the previous work on differentiability [3,4,17,18,31,32,48,6] is that we do not make any assumptions on the dynamics of the perturbed system. This extends greatly the range of applicability of our method.…”

“…This paper is devoted to differentiability of SRB measures for partially hyperbolic systems. The question of differentiability plays important role in averaging, rigidity theory and statistical physics ( [35,31,8,50]) but not much is known beyond the uniformly hyperbolic case (in [17,6] very interesting results about the differentiability of SRB measures for uniformly hyperbolic systems with singularities were obtained). In [48,51] some explicit formulas for derivatives of SRB measures were proposed which should hold for a large class of dynamical systems, however the question of their applicability remains open.…”

Nonequilibrium Stationary Solutions of Thermostated Boltzmann Equation in a Field

“…To get some analytical handle on the form of the NESS we also investigated numerically a model system in which the deterministic collisions with the obstacles are replaced by a stochastic process in which particle velocities get their orientations changed at random times, independently for each particle [5]. This yields, in the limit N → ∞, a self consistent Boltzmann equation.…”

“…This has led us to continue our study of the NESS in current carrying thermostated systems. In our previous work, see [3,5] we carried out extensive numerical and analytical investigations of the dependence of the current on the electric field for a model system consisting of N particles with unit mass, moving among a fixed periodic array of discs in a two dimensional square Λ with periodic boundary conditions, see Fig. 1.…”