An anti-symmetric exclusion process for two particles on an infinite 1D lattice

TL;DR

This paper analyzes a two-particle exclusion process on an infinite 1D lattice with opposite biases, providing exact solutions for propagator, mean displacement, and MSD, revealing two distinct behavioral regimes.

Contribution

It introduces an exactly solvable model of two biased, mutually exclusive particles with opposite biases, including explicit propagator and MSD expressions.

Findings

Exact propagator for the system derived

Identification of two behavioral regimes based on bias

Continuous limit leads to a Fokker-Planck equation for biased Brownian particles

Abstract

A system of two biased, mutually exclusive random walkers on an infinite 1D lattice is studied whereby the intrinsic bias of one particle is equal and opposite to that of the other. The propogator for this system is solved exactly and expressions for the mean displacement and mean square displacement (MSD) are found. Depending on the nature of the intrinsic bias, the system's behaviour displays two regimes, characterised by (i) the particles moving towards each other and (ii) away from each other, both qualitatively different from the case of no bias. The continuous-space limit of the propogator is found and is shown to solve a Fokker-Planck equation for two biased, mutually exclusive Brownian particles with equal and opposite drift velocity.