Phase transitions in one-dimensional nonequilibrium systems

The phenomenon of phase transitions in one-dimensional systems is discussed . Equilibrium systems are reviewed and some properties of an energy functio n which may allow phase transitions and phase ordering in one dimension are identified. We then give an overview of the one-dimensional phase transiti ons which have been studied in nonequilibrium systems. A particularly simpl e model, the zero-range process, for which the steady state is known exactl y as a product measure, is discussed in some detail. Generalisations of the model for which a product measure still holds, are also discussed. We anal yse in detail a condensation phase transition in the model and show how con ditions under which it may occur may be related to the existence of an effe ctive long-range energy function. It is also shown that even when the condi tions for condensation are not fulfilled one can still observe very sharp c rossover behaviour and apparent condensation in a finite system. Although t he zero-range process is not well known within the physics community, sever al nonequilibrium models have been proposed that are examples of a zero-ran ge process, or closely related to it, and we review these applications here .