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

Generalized Langevin equations for a driven tracer in dense soft colloids: construction and applications

We describe a tracer in a bath of soft Brownian colloids by a particle coupled to the density field of the other bath particles. From the Dean equation, we derive an exact equation for the evolution of the whole system, and show that the density field evolution can be linearized in the limit of a dense bath. This linearized Dean equation with a tracer taken apart is validated by the reproduction of previous results on the mean-field liquid structure and transport properties. Then, the tracer is submitted to an external force and we compute the density profile around it, its mobility and its diffusion coefficient. Our results exhibit effects such as bias enhanced diffusion that are very similar to those observed in the opposite limit of a hard core lattice gas, indicating the robustness of these effects. Our predictions are successfully tested against molecular dynamics simulations.Comment: 21 pages, 7 figure

Molecular diffusion between walls with adsorption and desorption

The time dependency of the diffusion coefficient of particles in porous media is an efficient probe of their geometry. The analysis of this quantity, measured e.g. by nuclear magnetic resonance (PGSE-NMR), can provide rich information pertaining to porosity, pore size distribution, permeability and surface-to-volume ratio of porous materials. Nevertheless, in numerous if not all practical situations, transport is confined by walls where adsorption and desorption processes may occur. In this article, we derive explicitly the expression of the time-dependent diffusion coefficient between two confining walls in the presence of adsorption and desorption. We show that they strongly modify the time-dependency of the diffusion coefficient, even in this simple geometry. We finally propose several applications, from sorption rates measurements to the use as a reference for numerical implementations for more complex geometries.Comment: 4 pages, 2 figures, 1 supplementary material of 3 page