Ultra-Slow Vacancy-Mediated Tracer Diffusion in Two Dimensions: The Einstein Relation Verified

We study the dynamics of a charged tracer particle (TP) on a two-dimensional lattice all sites of which except one (a vacancy) are filled with identical neutral, hard-core particles. The particles move randomly by exchanging their positions with the vacancy, subject to the hard-core exclusion. In case when the charged TP experiences a bias due to external electric field ${\bf E}$, (which favors its jumps in the preferential direction), we determine exactly the limiting probability distribution of the TP position in terms of appropriate scaling variables and the leading large-N ($n$ being the discrete time) behavior of the TP mean displacement $\bar{{\bf X}}_n$; the latter is shown to obey an anomalous, logarithmic law $|\bar{{\bf X}}_n| = \alpha_0(|{\bf E}|) \ln(n)$. On comparing our results with earlier predictions by Brummelhuis and Hilhorst (J. Stat. Phys. {\bf 53}, 249 (1988)) for the TP diffusivity $D_n$ in the unbiased case, we infer that the Einstein relation $\mu_n = \beta D_n$ between the TP diffusivity and the mobility $\mu_n = \lim_{|{\bf E}| \to 0}(|\bar{{\bf X}}_n|/| {\bf E} |n)$ holds in the leading in $n$ order, despite the fact that both $D_n$ and $\mu_n$ are not constant but vanish as $n \to \infty$. We also generalize our approach to the situation with very small but finite vacancy concentration $\rho$, in which case we find a ballistic-type law $|\bar{{\bf X}}_n| = \pi \alpha_0(|{\bf E}|) \rho n$. We demonstrate that here, again, both $D_n$ and $\mu_n$, calculated in the linear in $\rho$ approximation, do obey the Einstein relation.Comment: 25 pages, one figure, TeX, submitted to J. Stat. Phy

Stochastic processes with finite correlation time: modeling and application to the generalized Langevin equation

The kangaroo process (KP) is characterized by various forms of the covariance and can serve as a useful model of random noises. We discuss properties of that process for the exponential, stretched exponential and algebraic (power-law) covariances. Then we apply the KP as a model of noise in the generalized Langevin equation and simulate solutions by a Monte Carlo method. Some results appear to be incompatible with requirements of the fluctuation-dissipation theorem because probability distributions change when the process is inserted into the equation. We demonstrate how one can construct a model of noise free of that difficulty. This form of the KP is especially suitable for physical applications.Comment: 22 pages (RevTeX) and 4 figure