Nonequilibrium Statistical Mechanics of the Zero-Range Process and Application to Networks

Statistical mechanics is concerned with the study of systems with a large number of interacting constituents. Equilibrium statistical mechanics, originally introduced as a theoretical approach for thermodynamics, is well understood and a general theoretical framework exists. Nonequilibrium statistical mechanics is less well understood and no general theoretical framework currently exists. The study of nonequilibrium statistical mechanics often consists of analysing the behaviour of simple models which can exhibit interesting phenomena not expected from equilibrium systems of such simplicity. Recently, statistical mechanics is finding application in areas that are not often strongly associated with physics, such as biology and the social sciences. These fields contain systems with many interacting constituents, but for which the concepts of equilibrium statistical mechanics are often not intuitive or applicable. Thus the study of such systems through simple nonequilibrium models is commonplace. Many such models have been proposed to represent complex networks, such as the Internet, metabolic pathway networks and social acquaintance networks. Networks represent the connections between the components of a system; the structure of these connections is often crucial to the function of the system. The recent analysis of large data sets taken from real networks has revealed high levels of organisation. The study of simple models is being used as a first step to gain insight into how and why this organisation comes about. In this thesis a simple, stochastic, interacting particle system the zero-range process (ZRP) is studied with various analytical and numerical methods. In particular, the application of the ZRP and some of is generalisations to complex networks is focussed upon. The ZRP is a hopping particle model where particles hop between sites of a lattice under certain rules that depend only on the properties of the site from which the particles hop hence the name zero-range. Through its simplicity the steady state of the ZRP can be solved, even for nonequilibrium dynamics, and yet despite its simplicity it can exhibit interesting phenomena such as condensation transitions, where a finite fraction of the total particles in the system will condense onto a single site of the lattice. Firstly, interesting finite-size effects surrounding the condensation transition in a one-dimensional, driven version of the ZRP are studied. These take the form of discrepancies in the current-density diagram between finite and infinite systems, with