Dimensional crossover via confinement in the lattice Lorentz gas

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.

💡 Research Summary

In this work the authors investigate a lattice Lorentz gas in which a single tracer particle performs a continuous‑time random walk on a two‑dimensional square lattice populated by immobile hard obstacles of density n. The lattice is wrapped periodically in the transverse (y) direction, forming a cylindrical strip of width L (the number of parallel lanes) that is infinite along the axial (x) direction. A constant pulling force F is applied along the axis, biasing the forward and backward hopping rates according to detailed balance: W(±eₓ)=exp(±F/2)/