Large deviations and metastability in condensing stochastic particle systems
Chleboun, P. I. (Paul I.) (2011) Large deviations and metastability in condensing stochastic particle systems. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2560692~S1
Condensation, or jamming transitions, are observed throughout the natural and social sciences as prevalent emergent phenomena in complex systems, from shaken granular gases to traffic congestion. The understanding of the critical behaviour in these systems is currently a major research topic, in particular a mathematically rigorous treatment of the associated metastable dynamics. In this thesis we study these phenomena in the context of interacting particle systems. In particular we focus on two generic models for condensation, the zero-range process, and the recently introduced inclusion process. We firstly give a brief review of some relevant aspects of Markov processes, with a particular emphasis on interacting particle systems and a heuristic description of metastability. Subsequently, we present a general framework for studying the equivalence of ensembles, and the large deviations of the maximum site occupation, in interacting particle systems that exhibit product stationary measures. These original results are based on relative entropy methods and techniques from the theory of large deviations. They form the theoretical basis of this thesis, which enables us to find the relevant scales on which metastability is observed, and to derive refined results on the equivalence of ensembles. The general approach is to first derive a large deviation principle for the density and maximum site occupation under a reference measure. This gives rise to the large deviations of the maximum in the thermodynamic limit, or a more refined scaling limit, which describe the metastable behaviour. Also it gives rise to equivalence of ensembles results, which are used to find the limiting expectation of important observables (such as the stationary current). In the second part of the thesis we use these general results to give a detailed analysis of three cases. We derive the leading order finite size effects for a generic family of condensing zero-range process. At this scale we observe, and are able to characterise, metastable switching between uid and condensed states. Secondly, we study a zerorange process with size-dependent rates, for which metastable effects are stabilised in the thermodynamic limit. Here we are able to describe two distinct mechanisms of condensate motion. Finally, we study condensation of a different origin in the inclusion process. In this case our general results can no longer be applied directly, since some of the regularity assumptions do not hold, however the guiding principles still apply. Following these we make formal calculations that give rise to the relevant stationary properties. We give a heuristic analysis of the dynamics which turn out to be very different from those in the zero-range process. Throughout the thesis theoretical results are supported by Monte Carlo simulations and numerical calculations where appropriate. Our results contribute to a detailed understanding of the nature of finite size effects and metastable dynamics close to condensation and jamming transitions. This is vital in applications in complex systems such as granular media and traffic flow, which exhibit moderate system sizes and cannot be fully described by the usual thermodynamic limit.
|Item Type:||Thesis or Dissertation (PhD)|
|Subjects:||Q Science > QA Mathematics|
|Library of Congress Subject Headings (LCSH):||Dynamics of a particle, Markov processes, Condensation -- Mathematical models|
|Institution:||University of Warwick|
|Theses Department:||Mathematics Institute|
|Supervisor(s)/Advisor:||Grosskinsky, Stefan ; Somfai, Ellak|
|Sponsors:||Engineering and Physical Sciences Research Council (EPSRC)|
|Extent:||ix, 176 leaves : charts|
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