Directed random walk in adsorbed monolayer

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