Time-dependent dynamics in the confined lattice Lorentz gas

Abstract We study a lattice model describing the nonequilibrium dynamics arising from the pulling of a tracer particle through a disordered medium occupied by randomly placed obstacles. The model is considered in a restricted geometry to investigate confinement-induced effects. We derive exact analytical results for the characteristic function of the displacement moments, valid to first order in the obstacle density. By calculating the velocity autocorrelation function and its long-time tail, we find that even in equilibrium the system exhibits a dimensional crossover. This picture is further confirmed by the approach of the drift velocity to its terminal value attained in the nonequilibrium stationary state. At long times, the diffusion coefficient is affected by both driving and confinement in a way that we quantify analytically. The force-induced diffusion coefficient depends sensitively on confinement, which qualitatively modifies the nonanalytic force dependence observed in the unbounded model. We then examine the fluctuations of the tracer particle along the driving direction. We show that in the intermediate regime superdiffusive behavior persists even in the presence of confinement. We use stochastic simulations to test the validity of the analytical results, exact to first order in the obstacle density and valid for arbitrary force and confinement.