H. van Beijeren

Exactly solvable model for the roughening transition of a crystal surface

An exactly solvable model of the crystal-vacuum interface is constructed which exhibits a roughening transition. The model is obtained as a special limit of a ferromagnetic Ising model and it is isomorphic to the symmetric six-vertex model. Some of the thermodynamic properties of the system are discussed.

The modified Enskog equation

Arguments are presented for a modified form of the nonlinear Enskog equation, which describes the time dependence of the single-particle distribution function in a dense gas of hard spheres. Unlike the usual Enskog equation it is not restricted to small spatial non-uniformities, and may thus be used to derive Burnett and higher-order hydrodynamic equations. In a single-component system both equations are equivalent at the level of the Navier-Stokes equations. The main importance of the modified Enskog equation becomes manifest when it is extended to mixtures of hard spheres. It is shown that the existing versions of Enskogs theory for mixtures lead to results which are in conflict with irreversible thermodynamics (more specifically, the Onsager symmetry relations do not hold), whereas the present modified Enskog theory gives results in complete agreement with irreversible thermodynamics.

Interface sharpness in the Ising system

A simple proof is given for the existence of a sharp interface in three-dimensional Ising systems, at least up to the critical temperature of the corresponding two-dimensional system.

Kinetic theory of nonlinear viscous flow in two and three dimensions

On the basis of a nonlinear kinetic equation for a moderately dense system of hard spheres and disks it is shown that shear and normal stresses in a steady-state, uniform shear flow contain singular contributions of the form ¦X¦3/2 for hard spheres, or ¦X¦ log ¦X¦ for hard disks. HereX is proportional to the velocity gradient in the shear flow. The origin of these terms is closely related to the hydrodynamic tails t−d/2 in the current-current correlation functions. These results also imply that a nonlinear shear viscosity exists in two-dimensional systems. An extensive discussion is given on the range ofX values where the present theory can be applied, and numerical estimates of the effects are given for typical circumstances in laboratory and computer experiments.

The Modified Enskog Equation for Mixtures

Abstract In a previous paper it was shown that a modified form of the Enskog equation, applied to mixtures of hard spheres, should be considered as the correct extension of the usual Enskog equation to the case of mixtures. The main argument was that the modified Enskog equation leads to linear transport coefficients which are in complete agreement with the laws of irreversible thermodynamics. The existing extensions of the Enskog equation to the case of mixtures, on the other hand, yield results which do not satisfy the Onsager reciprocity relations. In this paper the detailed derivation of these results is presented, i.e. , the normal solution of the modified Enskog equation for mixtures is obtained, the Navier-Stokes equations are derived and the explicit expressions for the linear transport coefficients which follow, are shown to satisfy the laws of irreversible thermodynamics.

Kinetic theory of hard spheres

Kinetic equations for the hard-sphere system are derived by diagrammatic techniques. A linear equation is obtained for the one-particle-one particle equilibrium time correlation function and a nonlinear equation for the one-particle distribution function in nonequilibrium. Both equations are nonlocal, noninstantaneous, and extremely complicated. They are valid for general density, since statistical correlations are taken into account systematically. This method derives several known and new results from a unified point of view. Simple approximations lead to the Boltzmann equation for low densities and to a modified form of the Enskog equation for higher densities.

Long time tails in stationary random media. I. Theory

Diffusion of moving particles in stationary disordered media is studied using a phenomenological mode-coupling theory. The presence of disorder leads to a generalized diffusion equation, with memory kernels having power law long time tails. The velocity autocorrelation function is found to decay like t−(d/2+1), while the time correlation function associated with the super-Burnett coefficient decays liket−d/2 for long times. The theory is applicable to a wide variety of dynamical and stochastic systems including the Lorentz gas and hopping models. We find new, general expressions for the coefficients of the long time tails which agree with previous results for exactly solvable hopping models and with the low-density results obtained for the Lorentz gas. Finally we mention that if the moving particles are charged, then the long time tails imply that there is an ωd/2 contribution to the low-frequency part of the frequency-dependent electrical conductivity.

Roughening transition, surface tension and equilibrium droplet shapes in a two-dimensional Ising system

The exact surface tension for all angles and temperatures is given for the two-dimensional square Ising system with anisotropic nearest-neighbour interactions. Using this in the Wulff construction, droplet shapes are computed and illustrated. Letting temperature approach zero allows explicit study of the roughening transition in this model. Results are compared with those of the solid-on-solid approximation.

Kinetic mean field theories: Results of energy constraint in maximizing entropy

Structure of liquids and solids; crystallography Classical, semiclassical, and quantum theories of liquid structure Statistical theories of liquid structure - Kinetic and transport theory of fluids; physical properties of gases Kinetic and transport theory

Equilibrium time correlation functions in the low density limit

We consider a system of hard spheres in thermal equilibrium. Using Lanfords result about the convergence of the solutions of the BBGKY hierarchy to the solutions of the Boltzmann hierarchy, we show that in the low-density limit (Boltzmann-Grad limit): (i) the total time correlation function is governed by the linearized Boltzmann equation (proved to be valid for short times), (ii) the self time correlation function, equivalently the distribution of a tagged particle in an equilibrium fluid, is governed by the Rayleigh-Boltzmann equation (proved to be valid for all times). In the latter case the fluid (not including the tagged particle) is to zeroth order in thermal equilibrium and to first order its distribution is governed by a combination of the Rayleigh-Boltzmann equation and the linearized Boltzmann equation (proved to be valid for short times).