Directed random walk in adsorbed monolayer

TL;DR

This paper investigates the movement of a tracer particle in an adsorbed monolayer, analyzing the density profiles and velocity using a mean-field approach, and explores asymptotic behaviors in different diffusion regimes.

Contribution

It introduces a mean-field model for tracer dynamics in an adsorbed monolayer, providing explicit formulas for density profiles and velocity in various diffusion limits.

Findings

Density profile is inhomogeneous, with higher density in front of the tracer and lower behind.

Terminal velocity is determined implicitly via a transcendental equation.

Explicit asymptotic forms of velocity are derived for slow and fast diffusion limits.

Abstract

We study the dynamics of a tracer particle, which performs a totally directed random walk in an adsorbed monolayer composed of mobile hard-core particles undergoing continuous exchanges with a vapour phase. In terms of a mean-field-type approach, based on the decoupling of the tracer-particle-particle correlation functions into the product of pairwise, tracer-particle correlations, we determine the density profiles of the monolayer particles, as seen from the stationary moving tracer, and calculate its terminal velocity, V_{tr}. In the general case the latter is determined implicitly, as the solution of a certain transcendental equation. In two extreme limits of slow and fast monolayer particles diffusion, we obtain explicit asymptotic forms of V_{tr}. We show next that the density profile in the monolayer is strongly inhomogeneous: In front of the stationary moving tracer the local density is higher than the average value, \rho_L, and approaches \rho_L as an exponential function of the distance from the tracer. Past the tracer the local density is lower than \rho_L and the approach to \rho_L may proceed differently depending whether the particles number in the monolayer is not or is explicitly conserved. In the former case the approach is described by an exponential dependence with a different characteristic length, compared to the behavior in front of the tracer; in the latter case, the density tends to \rho_L algebraically. The characteristic lengths and the amplitudes of the density relaxation functions are also determined explicitly