Correlations and transport in exclusion processes with general finite memory

Abstract

We consider the correlations and the hydrodynamic description of persistent random walkers on a lattice. We derive a drift-diffusion equation and identify a memory-dependent critical density. Above the critical density, the effective diffusion coefficient decreases with the particles’ propensity to move forward and below the critical density it increases with their propensity to move forward. If the correlations are neglected the critical density is exactly 1/2. We also derive a low-density approximation for the same time correlations between different sites. We perform simulations on a one-dimensional system with one-step memory and find good agreement between our analytical derivation and the numerical results. We also consider the previously unexplored special case of totally anti-persistent particles. Generally, the correlation length converges to a finite value. However in the special case of totally anti-persistent particles and density 1/2, the correlation length diverges with time. Furthermore, connecting a system of totally anti-persistent particles to external particle reservoirs creates a discontinuity in the density between the bulk and the reservoir. We find a qualitative description of this phenomenon which agrees reasonably well with the numerics.