Abstract:
the phenomenon of phase transitions in one-dimensional systems is discussed. equilibrium systems are reviewed and some properties of an energy function which may allow phase transitions and phase ordering in one dimension are identified. we then give an overview of the one-dimensional phase transitions which have been studied in nonequilibrium systems. a particularly simple model, the zero-range process, for which the steady state is known exactly as a product measure, is discussed in some detail. generalisations of the model, for which a product measure still holds, are also discussed. we analyse in detail a condensation phase transition in the model and show how conditions under which it may occur may be related to the existence of an effective long-range energy function. it is also shown that even when the conditions for condensation are not fulfilled one can still observe very sharp crossover behaviour and apparent condensation in a finite system. although the zero-range process is not well known within the physics community, several nonequilibrium models have been proposed that are examples of a zero-range process, or closely related to it, and we review these applications here.

Abstract:
The phenomenon of phase transitions in one-dimensional systems is discussed. Equilibrium systems are reviewed and some properties of an energy function which may allow phase transitions and phase ordering in one dimension are identified. We then give an overview of the one-dimensional phase transitions which have been studied in nonequilibrium systems. A particularly simple model, the zero-range process, for which the steady state is known exactly as a product measure, is discussed in some detail. Generalisations of the model, for which a product measure still holds, are also discussed. We analyse in detail a condensation phase transition in the model and show how conditions under which it may occur may be related to the existence of an effective long-range energy function. It is also shown that even when the conditions for condensation are not fulfilled one can still observe very sharp crossover behaviour and apparent condensation in a finite system. Although the zero-range process is not well known within the physics community, several nonequilibrium models have been proposed that are examples of a zero-range process, or closely related to it, and we review these applications here.

Abstract:
I review a simple method, recently introduced to convert rheological compliance measurements into frequency-dependent moduli. New experimental data are presented, and the scientific implications of various data conversion methods discussed.

Abstract:
A disordered version of the one dimensional asymmetric exclusion model where the particle hopping rates are quenched random variables is studied. The steady state is solved exactly by use of a matrix product. It is shown how the phenomenon of Bose condensation whereby a finite fraction of the empty sites are condensed in front of the slowest particle may occur. Above a critical density of particles a phase transition occurs out of the low density phase (Bose condensate) to a high density phase. An exponent describing the decrease of the steady state velocity as the density of particles goes above the critical value is calculated analytically and shown to depend on the distribution of hopping rates. The relation to traffic flow models is discussed.

Abstract:
The width of the distribution of species in a polydisperse system is employed in a small-variable expansion, to obtain a well-controlled and compact scheme by which to calculate phase equilibria in multi-phase systems. General and universal relations are derived, which determine the partitioning of the fluid components among the phases. The analysis applies to mixtures of arbitrarily many slightly-polydisperse components. An explicit solution is approximated for hard spheres.

Abstract:
A theoretical study of vesicles of topological genus zero is presented. The bilayer membranes forming the vesicles have various degrees of intrinsic (tangent-plane) orientational order, ranging from smectic to hexatic, frustrated by curvature and topology. The field-theoretical model for these `$n$-atic' surfaces has been studied before in the low temperature (mean-field) limit. Work presented here includes the effects of thermal fluctuations. Using the lowest Landau level approximation, the coupling between order and shape is cast in a simple form, facilitating insights into the behaviour of vesicles. The order parameter contains vortices, whose effective interaction potential is found, and renormalized by membrane fluctuations. The shape of the phase space has a counter-intuitive influence on this potential. A criterion is established whereby a vesicle of finite rigidity may be burst by its own in-plane order, and an analogy is drawn with flux exclusion from a type-I superconductor.

Abstract:
A theoretical study of toroidal membranes with various degrees of intrinsic orientational order is presented at mean-field level. The study uses a simple Ginzburg-Landau style free energy functional, which gives rise to a rich variety of physics and reveals some unusual ordered states. The system is found to exhibit many different phases with continuous and first order phase transitions, and phenomena including spontaneous symmetry breaking, ground states with nodes and the formation of vortex-antivortex quartets. Transitions between toroidal phases with different configurations of the order parameter and different aspect ratios are plotted as functions of the thermodynamic parameters. Regions of the phase diagrams in which spherical vesicles form are also shown.

Abstract:
Just as transition rates in a canonical ensemble must respect the principle of detailed balance, constraints exist on transition rates in driven steady states. I derive those constraints, by maximum information-entropy inference, and apply them to the steady states of driven diffusion and a sheared lattice fluid. The resulting ensemble can potentially explain nonequilibrium phase behaviour and, for steady shear, gives rise to stress-mediated long-range interactions.

Abstract:
A one-dimensional driven lattice gas with disorder in the particle hopping probabilities is considered. It has previously been shown that in the version of the model with random sequential updating, a phase transition occurs from a low density inhomogeneous phase to a high density congested phase. Here the steady states for both parallel (fully synchronous) updating and ordered sequential updating are solved exactly and the phase transition shown to persist in both cases. For parallel dynamics and forward ordered sequential dynamics the phase transition occurs at the same density but for backward ordered sequential dynamics it occurs at a higher density. In both cases the critical density is higher than that for random sequential dynamics. In all the models studied the steady state velocity is related to the fugacity of a Bose system suggesting a principle of minimisation of velocity. A generalisation of the dynamics where the hopping probabilities depend on the number of empty sites in front of the particles, is also solved exactly in the case of parallel updating. The models have natural interpretations as simplistic descriptions of traffic flow. The relation to more sophisticated traffic flow models is discussed.

Abstract:
The phenomenon of phase transitions in one-dimensional systems is discussed. Equilibrium systems are reviewed and some properties of an energy function which may allow phase transitions and phase ordering in one dimension are identified. We then give an overview of the one-dimensional phase transitions which we have been studied in nonequilibrium systems. A particularly simple model, the zero-range process, for which the steady state is know exactly as a product measure, is discussed in some detail. Generalisations of the model, for which a product measure still holds, are also discussed. We analyse in detail a condensation phase transition in the model and show how conditions under which it may occur may be related to the existence of an effective long-range energy function. Although the zero-range process is not well known within the physics community, several nonequilibrium models have been proposed that are examples of a zero-range process, or closely related to it, and we review these applications here.