Dimensional crossover via confinement in the lattice Lorentz gas

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.

Figures (3)

• Figure 1: Tracer moving on a lattice strip with identified edges (size $L$) under the action of a force $F$ along the strip axis. The tracer, scattered by quenched impurities (yellow circles), hops only on accessible sites; rejected transitions are indicated with a star.
• Figure 2: The negative VACF for increasing confinement width $L$. Dashed black lines correspond to the long-time tail asymptotically proportional to $t^{-3/2}$ [Eq. (\ref{['ltt_confined']})]. The dot-dashed black line indicates the long-time tail $\sim (\pi/8) t^{-2}$ for the unconfined two-dimensional lattice Lorentz gas LF_2013.
• Figure 3: Obstacle-induced velocity response in the stationary state $v(t \rightarrow \infty)$ as a function of the force $F$ for increasing obstacle density $n$ and system size $L$. Symbols correspond to stochastic simulations, solid lines represent the theory for arbitrary $F$, dashed lines indicate the linear response. The inset shows the function $\mathcal{V}_{L}(F;0)$.