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Abstract. Unlike equilibrium statistical mechanics, with its well-established foundations, a similar widely-accepted framework for non-equilibrium statistical mechanics (NESM) remains elusive. Here, we review some of the many recent activities on NESM, focusing on some of the fundamental issues and general aspects. Using the language of stochastic Markov processes, we emphasize general properties of the evolution of configurational probabilities, as described by master equations. Of particular interest are systems in which the dynamics violate detailed balance, since such systems serve to model a wide variety of phenomena in nature. We next review two distinct approaches for investigating such problems. One approach focuses on models sufficiently simple to allow us to find exact, analytic, non-trivial results. We provide detailed mathematical analyses of a onedimensional continuous-time lattice gas, the totally asymmetric exclusion process (TASEP). It is regarded as a paradigmatic model for NESM, much like the role the Ising model played for equilibrium statistical mechanics. It is also the starting point for the second approach, which attempts to include more realistic ingredients in order to be more applicable to systems in nature. Restricting ourselves to the area of biophysics and cellular biology, we review a number of models that are relevant for transport phenomena. Successes and limitations of these simple models are also highlighted.

The process of protein synthesis in biological systems resembles a one dimensional driven lattice gas in which the particles have spatial extent, covering more than one lattice site. We expand the well studied Totally Asymmetric Exclusion Process (TASEP), in which particles typically cover a single lattice site, to include cases with extended objects. Exact solutions can be determined for a uniform closed system. We analyze the uniform open system through two approaches. First, a continuum limit produces a modified diffusion equation for particle density profiles. Second, an extremal principle based on domain wall theory accurately predicts the phase diagram and currents in each phase. Finally, we briefly consider approximate approaches to a non-uniform open system with quenched disorder in the particle hopping rates and compare these approaches with Monte Carlo simulations.

One of the key features of nonequilibrium steady states (NESS) is the presence of nontrivial probability currents. We propose a general classification of NESS in which these currents play a central distinguishing role. As a corollary, we specify the transformations of the dynamic transition rates which leave a given NESS invariant. The formalism is most transparent within a continuous time master equation framework since it allows for a general graph-theoretical representation of the NESS. We discuss the consequences of these transformations for entropy production, present several simple examples, and explore some generalizations, to discrete time and continuous variables.

We study the phenomenon of real space condensation in the steady state of a class of mass transport models where the steady state factorises. The grand canonical ensemble may be used to derive the criterion for the occurrence of a condensation transition but does not shed light on the nature of the condensate. Here, within the canonical ensemble, we analyse the condensation transition and the structure of the condensate, determining the precise shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as L 1/2 where L is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. Our results are asymptotically exact and can also be interpreted within the framework of sums of random variables. We further analyse two additional cases: one where the condensation transition is somewhat different from the usual second order phase transition and one where there is no true condensation transition but instead a pseudocondensate appears at superextensive densities.

We study the phenomenon of real space condensation in the steady state of a class of one dimensional mass transport models. We derive the criterion for the occurrence of a condensation transition and analyse the precise nature of the shape and the size of the condensate in the condensed phase. We find two distinct condensate regimes: one where the condensate is gaussian distributed and the particle number fluctuations scale normally as L 1/2 where L is the system size, and a second regime where the particle number fluctuations become anomalously large and the condensate peak is non-gaussian. We interpret these results within the framework of sums of random variables.PACS numbers: 02.50.Ey, Condensation transitions are ubiquitous in nature. For systems in thermal equilibrium, clustering is well understood in terms of the competition between entropy and energy (typically associated with attractive interactions). More exotic are condensations in systems with no interactions, e.g., Bose-Einstein's free (quantum) particles. Less understood are such transitions in non-equilibrium systems, in some of which even the concept of energy is dubious. For example, condensation is known to occur in many mass transport models, defined only by a set of rules of evolution, with no clear 'attraction' between the masses [1][2][3][4][5][6]. The relevance of these models lies in their applicability to a broad variety of phenomena, e.g., traffic flow [7]force propagation through granular media How such transitions arise is especially intriguing for one dimensional (d = 1) systems with local dynamical rules. A well known example is the Zero-Range Process (ZRP) [2,12,13] in which masses hop from site to (the next) site according to some transfer rule. In the steady state, a finite fraction of the total mass 'condenses' onto a single site when ρ, the global mass density, is increased beyond a certain critical value: ρ c . The system goes from a fluid phase, where the mass at each site hovers around ρ, to a condensed phase, where a fluid of density ρ c coexists with a condensate containing all the 'excess' mass.Though condensation in these systems share interesting analogies [3,6] with the traditional Bose-Einstein condensation, there are important differences. For example, here condensation occurs in real space and in all dimensions. Moreover these systems are non-equilibrium in the sense that they are defined by the dynamics, generally lack a Hamiltonian and the stationary state is not specified by the usual Gibbs-Boltzmann distribution. There are two major problems that one faces in the analysis of condensation phenomenon in these systems. First, the stationary state itself often is very difficult to determine and secondly, even if it is known such as in ZRP, the analysis of condensation has so far been possible only within a grand canonical enemble (GCE) where one is already in the thermodynamic (L → ∞) limit. While the GCE approach correctly predicts when a condensation transition can happen and even the value of the critical densit...

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