Active Random Walks in One and Two Dimensions

Abstract

We investigate active lattice walks: biased continuous time random walks which perform orientational diffusion between lattice directions in one and two spatial dimensions. We study the occupation probability of an arbitrary site on the lattice in one and two dimensions, and derive exact results in the continuum limit. Next, we compute the large deviation free energy function in both one and two dimensions, which we use to compute the moments and the cumulants of the displacements exactly at late times. Our exact results demonstrate that the cross-correlations between the motion in the $x$ and $y$ directions in two dimensions persist in the large deviation function. We also demonstrate that the large deviation function of an active particle with diffusion displays two regimes, with differing diffusive behaviors. We verify our analytic results with kinetic Monte Carlo simulations of an active lattice walker in one and two dimensions.

17 pages, 11 figures, +Supplemental Material

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Cited by in corpus (10)

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• Current fluctuations in an interacting active lattice gas
• Generalized disorder averages and current fluctuations in run and tumble particles
• First passage statistics of active random walks on one and two dimensional lattices
• Persistent and anti-Persistent Motion in Bounded and Unbounded Space: Resolution of the First-Passage Problem
• Martingale approach for first-passage problems of time-additive observables in Markov processes
• Effect of initial conditions on current fluctuations in non-interacting active particles
• Mpemba effect on non-equilibrium active Markov chains
• Counter-flow induced clustering: Exact results
• Current fluctuations in finite-sized one-dimensional non-interacting passive and active systems