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
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|
|Official Date:||October 2011|
|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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