A simple symmetric exclusion process driven by an asymmetric tracer particle

Abstract

We consider an exclusion process on a periodic one-dimensional lattice where all particles perform simple symmetric exclusion at rate $1$ except for a single tracer particle, which performs partially simple asymmetric exclusion with rate $p$ to the right and rate $q$ to the left. This model was first considered by Ferrari, Goldstein and Lebowitz (1985) as a test for the validity of the Einstein relation in microscopic systems. The main thrust of this work is an exact solution for the steady state of this exclusion process. We show that the stationary probabilities factorize and give an exact formula for the nonequilibrium partition function. We provide formulas for the current and two-point correlations. When the tracer particle performs totally asymmetric exclusion ($q=0$), the results are shown to simplify significantly and we find an unexpected connection with the combinatorics of set partitions. Finally, we study the system from the point of view of the tracer particle, the so-called environment process. In the environment process, we show that the density of particles decays with the scaled position in front of the tracer particle in the thermodynamic limit.