Dimensional crossover via confinement in the lattice Lorentz gas

TL;DR

This paper studies how a tracer particle's movement in a lattice with obstacles transitions from two-dimensional to one-dimensional behavior over time due to confinement effects, providing analytical and simulation insights.

Contribution

It introduces an analytical framework for the dimensional crossover in a lattice Lorentz gas under confinement, valid for large forces and obstacle densities.

Findings

System exhibits a dimensional crossover from 2D to 1D over time.

Analytical results are validated by stochastic simulations.

Results hold for large forces and confinement sizes.

Abstract

We consider a lattice model in which a tracer particle moves in the presence of randomly distributed immobile obstacles. The crowding effect due to the obstacles interplays with the quasi-confinement imposed by wrapping the lattice onto a cylinder. We compute the velocity autocorrelation function and show that already in equilibrium the system exhibits a dimensional crossover from two- to one-dimensional as time progresses. A pulling force is switched on and we characterize analytically the stationary state in terms of the stationary velocity and diffusion coefficient. Stochastic simulations are used to discuss the range of validity of the analytic results. Our calculation, exact to first order in the obstacle density, holds for arbitrarily large forces and confinement size.