H. Spohn

Nonequilibrium steady states of stochastic lattice gas models of fast ionic conductors

We investigate theoretically and via computer simulation the stationary nonequilibrium states of a stochastic lattice gas under the influence of a uniform external fieldE. The effect of the field is to bias jumps in the field direction and thus produce a current carrying steady state. Simulations on a periodic 30 × 30 square lattice with attractive nearest-neighbor interactions suggest a nonequilibrium phase transition from a disordered phase to an ordered one, similar to the para-to-ferromagnetic transition in equilibriumE=0. At low temperatures and largeE the system segregates into two phases with an interface oriented parallel to the field. The critical temperature is larger than the equilibrium Onsager value atE=0 and increases with the field. For repulsive interactions the usual equilibrium phase transition (ordering on sublattices) is suppressed. We report on conductivity, bulk diffusivity, structure function, etc. in the steady state over a wide range of temperature and electric field. We also present rigorous proofs of the Kubo formula for bulk diffusivity and electrical conductivity and show the positivity of the entropy production for a general class of stochastic lattice gases in a uniform electric field.

A NOTE ON THE EIGENVALUE DENSITY OF RANDOM MATRICES

Abstract:The distribution of eigenvalues of N×N random matrices in the limit N→∞ is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges. This fact is a consequence of a more general theorem, proven here, in the statistical mechanics of unstable interactions. Our result establishes the eigenvalue density of some ensembles of random matrices which were not covered by previous theorems.

Microscopic basis for Fick's law for self-diffusion

We investigate self-diffusion in a classical fluid composed of two species which are distinguished through the color of their particles, either black or white, but are identical as regards their mechanical properties. Disregarding color the fluid is in thermal equilibrium. We show that if a single “test particle” in the one-component fluid moves asymptotically as Brownian motion, then the color density and current in certain classes of nonequilibrium states are related, on the appropriate macroscopic scale, through Ficks law, and the former is governed by the diffusion equation. If in addition several test particles move asymptotically as independent Brownian motions, then the colored fluid is, on a macroscopic scale, in local equilibrium with parameters governed by the solution of the diffusion equation.

The fast rate limit of driven diffusive systems

We study the stationary nonequilibrium states of the van Beijeren/Schulman model of a driven lattice gas in two dimensions. In this model, jumps are much faster in the direction of the driving force than orthogonal to it. Van Kampens Ω-expansion provides a suitable description of the model in the high-temperature region and specifies the critical temperature and the spinodal curve. We find the rate dependence ofTc and show that independently of the jump rates the critical exponents of the transition are classical, except for anomalous energy fluctuations. We then study the stationary solution of the deterministic equations (zeroth-orderΩ-expansion). They can be obtained as trajectories of a dissipative dynamical system with a three-dimensional phase space. Within a certain temperature range belowTc, these equations have a kink solution whose asymptotic densities we identify with those of phase coexistence. They appear to coincide with the results of the “Maxwell construction.” This provides a dynamical justification for the use of this construction in this nonequilibrium model. The relation of the Freidlin-Wentzell theory of small random perturbations of dynamical systems to the steady-state distribution belowTc is discussed.

Steady state self-diffusion at low density

We prove that the motion of a test particle in a hard sphere fluid in thermal equilibrium converges, in the Boltzmann-Grad limit, to the stochastic process governed by the linear Boltzmann equation. The convergence is in the sense of weak convergence of the path measures. We use this result to study the steady state of a binary mixture of hard spheres of different colors (but equal masses and diameters) induced by color-changing boundary conditions. In the Boltzmann-Grad limit the steady state is determined by the stationary solution of the linear Boltzmann equation under appropriate boundary conditions.