Nonequilibrium phase transitions to collective behavior in stochastic spatially extended systems

We explore nonequilibrium collective behavior in large, spatially extended stochastic systems. In Part I, we present a model of discrete, active, noisy phase oscillators sufficiently simple to be characterized in complete detail in a host of diverse settings. In Chapter 2, we introduce the model and detail its utility for the study of both continuous and discontinuous phase transitions to macroscopic oscillations. In Chapter 3 we analyze a locally coupled version of the model undergoing a continuous transition and provide strong evidence that the inherently nonequilibrium system shows qualitative and quantitative characteristics of a known class of equilibrium phase transitions. Chapter 4 offers an analysis of the discrete model in the face of quenched transition rate disorder, where synchronization still occurs and depends on the degree of disorder in the population. The final chapter of Part I, Chapter 5, details the microscopic underpinnings of synchronization above threshold, including the counterintuitive relationship between time-averaged frequency and the mean field oscillations. The second part of the dissertation begins with an introduction to generic relaxational models with field-dependent kinetic coefficients, highlighting in particular the role of this class of models in the study of noise-induced phase transitions. Using these systems as prototypical models of noise-induced phenomena in spatially extended systems, we offer a comprehensive study of static pattern formation in Chapter 7, while Chapter 8 provides a numerical study of coherent oscillatory dynamics induced purely by noise. Finally, Chapter 9 details an analytical framework for studying the dynamics of these systems which is capable of approximately describing both static and time-dependent phases associated with the nonequilibrium transitions in these relaxational model