The goal of statistical physics is to study collective behaviors of interacting many-particle systems. In equilibrium statistical physics, the simplest interaction is exclusion – for example, hard spheres that cannot overlap. This model depends on a single dimensionless parameter, the volume fraction; the temperature is irrelevant since the interaction energy is zero when the spheres are non-overlapping and infinite otherwise. Despite its apparent simplicity, the hard-sphere gas is incompletely understood except in one dimension. A similar state of affairs holds for the lattice version of hard spheres; there is little analytical understanding of its unusual liquid–gas transition when the spatial dimension d ≥ 2.

In this chapter we explore the role of exclusion on the simplest non-equilibrium models that are known as exclusion processes. Here particles occupy single lattice sites, and each particle can hop to a neighboring site only if it is vacant (see Fig. 4.1). There are many basic questions we can ask: What is the displacement of a single particle? How does the density affect transport properties? How do density gradients evolve with time? In greater than one dimension, exclusion does not qualitatively affect transport properties compared to a system of independent particles. Interestingly, exclusion leads to fundamentally new transport phenomena in one dimension.