Driven Lorentz gas model in the discrete time domain

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 ∼N^{-1}exp(-NF^{2}/16) for small F, differing significantly from the exponential decay expected of linear response. Besides a nonanalytic dependence on the force and breakdown of Einstein's linear response, our results show that fluctuations around the mean displacement in direction of the force are enhanced in the presence of obstacles. Notably, the variance grows as ∼N^{3} (superdiffusion) for F→∞ at intermediate steps, reverting to normal diffusion (∼N) at larger steps, a behavior previously observed in continuous time and verified in this work in the discrete domain of the number of steps. Unlike the exponential waiting time case, the superdiffusion regime starts already for N=1. The presented framework 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.