2024/09/04 by Dan Shafir, Alessio Squarcini, Shafir, Dan +5
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #FOS: Physical sciences #Noncommutative and Quantum Gravity Theories #Quantum Mechanics and Applications #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2409.02696
openalex publication_date 2024/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(- N F2 / 16) for small F, differing significantly from the ∼ 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 ∼ N3 (superdiffusion) for F → ∞ at intermediate steps, reverting to normal diffusion (∼ 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.