Driven Lorentz model in discrete time

TL;DR

This paper analyzes a discrete-time driven Lorentz model with obstacles, revealing non-linear force dependence, superdiffusive behavior, and deviations from linear response, validated by simulations.

Contribution

It provides an exact first-order in obstacle density analysis of displacement moments under strong driving force in discrete time, including superdiffusion and non-analytic force effects.

Findings

Displacement approaches terminal velocity as ~N^{-1} exp(-N F^2 / 16) for small F.

Variance exhibits superdiffusion (~N^3) at intermediate steps for large F.

Superdiffusion begins immediately at N=1, unlike continuous-time models.

Abstract

We consider a tracer particle performing a random walk on a two-dimensional lattice in the presence of immobile hard obstacles. Starting from equilibrium, a constant force pulling on the particle is switched on, driving the system to a new stationary state. Our study calculates displacement moments in discrete time (number of steps $N$) for an arbitrarily strong constant driving force, exact to first order in obstacle density. We find that for fixed driving force $F$, the approach to the terminal discrete velocity scales as $\sim N^{-1} \exp(- N F^2 / 16)$ for small $F$, differing significantly from the $\sim N^{-1}$ prediction of linear response. Besides a non-analytic dependence on the force and breakdown of Einstein's linear response, our results show that fluctuations in the directions of the force are enhanced in the presence of obstacles. Notably, the variance grows as $\sim N^3$ (superdiffusion) for $F \to \infty$ at intermediate steps, reverting to normal diffusion ($\sim N$) at larger steps, a behavior previously observed in continuous time but demonstrated here in discrete steps for the first time. Unlike the exponential waiting time case, the superdiffusion regime starts immediately at $N=1$. The framework presented allows considering any type of waiting-time distribution between steps and transition to continuous time using subordination methods. Our findings are also validated through computer simulations.