Correlations and transport in exclusion processes with general finite memory

Abstract

We consider the correlations and the hydrodynamic description of random walkers with a general finite memory moving on a d dimensional hypercubic 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 new phenomenon: In almost all systems, regardless of the precise details of the microscopic dynamics, when a system is connected to a reservoir, the mean density of particle at the edge is the same as the reservoir following the zeroth law of thermodynamics. In a totally anti-persistent system, however, the density at the edge is always higher than in the reservoir. We find a qualitative description of this phenomenon which agrees reasonably well with the numerics.Correlations and transport in exclusion processes with general finite memory

Persistent exclusion processes: Inertia, drift, mixing, and correlation

“…Whilst our analysis yields a diffusivity linear in the density u for both one and two dimensions, the analogous result for diffusivity in one dimension obtained by Teomy and Metzler [26] varies quadratically with density. In particular, the authors reported a critical density (1/2 in the mean-field approximation), below which the diffusivity increases with increasing persistence, and above which the diffusivity instead decreases with increasing persistence.…”

“…For the basic case of a single species of motion-persistent agents, we find that such systems are approximately governed by nonlinear diffusion equations. Although our analysis yields results resembling those presented by Teomy and Metzler [26], key differences are revealed regarding the descriptions of population-level behaviour. We extend our analysis to consider generalisations of the basic model involving multiple species of interacting agents as well as a superimposed global drift effect, showing that in general such systems are approximately governed by systems of nonlinear advection-diffusion equations.…”

“…It is important to note here that in our agent-based model, agents only remember successful moves, with aborted move attempts being 'forgotten'. This is in contrast with the model presented by Teomy and Metzler [26], in which the memory is updated regardless of the outcome of a move attempt. The resulting differences for bulk system behaviour are immediately appreciable.…”

“…Movement and repolarisation are driven by independent Poisson processes, and in the continuum limit this model yields a system of four nonlinear PDEs describing the dynamics of agents in each of the four possible orientations. Even more recently, Teomy and Metzler [26,27] reported results for transport in a general model of exclusion processes in which agents possess a finite memory of attempted moves and move in a memory-dependent manner. The authors showed that in the continuum limit such a system may be described by nonlinear advectiondiffusion equations, verifying their results for the case of steady-state transport in one dimension.…”

“…In particular, it is known that the non-interacting persistent random walk is governed by a linear diffusion equation [23,24]. However, current work considering multiple interacting persistent random walks remains sparse [25][26][27].…”