Klaus W. Kehr

Simulation of Diffusion in Lattice Gases and Related Kinetic Phenomena

This chapter reviews various Monte Carlo studies of dynamical properties of lattice gas models, which serve to simulate self-diffusion of tagged particles in interstitial and substitutional alloys, surface diffusion of adsorbate atoms in adsorbed monolayers, etc. These systems serve as archetypical models of order-disorder and unmixing phase transitions, and are well suited to study the basic aspects of associated kinetic phenomena near equilibrium as well as far from equilibrium, such as nucleation of ordered domains from a disordered phase, their diffusion-controlled growth, coarsening of domain structures by diffusion of domain walls, etc. Earlier Monte Carlo work on related problems, such as the kinetics of nucleation and phase separation [6.1], or the kinetics of crystal growth [6.2], has been a unique tool for checking analytical theories on this subject, and has stimulated beautiful new experiments. We feel that the more recent work reviewed here will again be very stimulating for a variety of fields, from the statistical thermodynamics of irreversible processes to materials science. The main emphasis of this article is on kinetic phenomena near equilibrium, i.e., on simulations of diffusion in lattice gases, which have not yet been reviewed, except for [6.3].

Random walk on a random walk

The authors investigate the random walk of a particle on a one-dimensional chain which has been constructed by a random-walk procedure. Exact expressions are given for the mean-square displacement and the fourth moment after n steps. The probability density after n steps is derived in the saddle-point approximation, for large n. These quantities have also been studied by numerical simulation. The extension to continuous time has been made where the particle jumps according to a Poisson process. The exact solution for the self-correlation function has been obtained in the Fourier and Laplace domain. The resulting frequency-dependent diffusion coefficient and incoherent dynamical structure factor have been discussed. The model of random walk on a random walk is applied to self-diffusion in the concentrated one-dimensional lattice gas where the correct asymptotic behavior is found.

Spin Depolarization of Particles in Tunnelling States

The spin depolarization of a particle in quantum states is considered using a model of two sites, having different transverse magnetic fields, and between which the particle can tunnel. The ensemble average is performed with random distributions of the magnetic fields. A Gaussian decay law is obtained with the exponent reduced by a factor 2, for all times when the system is prepared in suitable eigenstates and for longtimes with arbitrary preparation. The apparent contradiction to van Vlecks theorem is resolved.

Transport in a disordered medium: Analysis and Monte Carlo simulation

We consider a classical stochastic model describing particle transport on a lattice with randomly distributed nearest-neighbor transition rates. Applying an effective medium theory to the model, we determine average properties related to the particles dynamics ind-dimensions. In particular, we calculate the mean-square displacement, and the fourth moment of the displacement in one-, two- and three dimensions. The results compare favorably with Monte Carlo simulations of the model. We also present preliminary results for the velocity autocorrelation function.An aspect of the bond percolation problem, which is a special case of the stochastic model is investigated; the average inverse cluster size, , is calculated. In one dimension the expression for this quantity is exact and in higher dimensions our results are very accurate not too close to the percolation concentration.

Mean number of distinct sites visited by correlated walks. I. Perfect lattices

The mean number of distinct sites visited by correlated walks on one‐, two‐, and three‐dimensional lattices is studied by numerical simulations and by generating‐function techniques. The random walks include correlations over two consecutive steps. The asymptotic behavior is derived analytically in d=1, and in d=2, 3 for the model with restricted reversals, and good agreement with the simulations is found. The model with increased probability for forward steps is studied numerically in d=2, 3 and analyzed. It is found in all cases that the mean number of visited sites cannot be simply obtained by rescaling the step number n with the correlation factor f, but there are additional correction terms that do not obey scaling.

TOY MODEL FOR MOLECULAR MOTORS

A hopping model for molecular motors is presented consisting of a state with asymmetric hopping rates with period 2 and a state with uniform hopping rates. State changes lead to a stationary unidirectional current of a particle. The current is explicitly calculated as a function of the rate of state changes, including also an external bias field. The Einstein relation between the linear mobility of the particle and its diffusion coefficient is investigated. The power input into the system is derived, as well as the power output resulting from the work performed against the bias field. The efficiency of this model is found to be rather small.

Collective and tracer diffusion of lattice gases in lattices with site-energy disorder

Collective and tracer diffusion of lattice gases of arbitrary concentrations were investigated in one-, two-, and three-dimensional lattices with randomly distributed site energies. A model with two different site energies and varying concentrations of the trap sites was employed. A vectorizable computer code has been used for the simulation of the lattice gas with site exclusion. For low particle concentrations, the diffusion coefficient is given by the wellknown single-particle value. For larger particle concentrations and moderate concentrations of deep traps, reduced diffusion of the lattice gas in a background of randomly blocked sites is observed. For larger trap concentrations, when the concentration of the non-trap sites is smaller than the corresponding percolation threshold, the diffusion coefficient is practically independent of the particle concentration and given by the single-particle value in the disordered lattice. In one-dimensional systems one finds this behavior approximately at all trap concentrations.

Connection between dispersive transport and statistics of extreme events

A length dependence of the effective mobility in the form of a power law, B∼L1−1/α, is observed in dispersive transport in amorphous substances, with 0<α<1. We deduce this behavior as a simple consequence of the statistical theory of extreme events. We derive various quantities related to the largest value in samples of n trials, for the exponential and power-law probability densities of the individual events.

Spin Depolarization of Particles in Tunneling Models with Dynamical Disorder

The spin depolarization of a spin-carrying particle which can tunnel between two sites is investigated. The sites have different Zeeman energies and dynamical fluctuations are included in the frame of the Haken-Strobl model. Different behavior of the polarization decay is found depending on the strength of the site-diagonal and offdiagonal fluctuations. Also the ensemble average over distributions of the Zeeman energies is studied.

Collective diffusion of lattice gases on linear chains with site-energy disorder

Abstract Collective diffusion of lattice gases in linear chains with site-energy disorder is studied. The mean-field theory of the diffusivity is formulated in terms of effective transition rates. An exact expression is derived for the diffusion coefficient of single vacancies, in the limit of large particle concentration. Numerical simulations were performed and compared with the predictions of the mean-field theory and a correction to it.