Rafferty, Thomas (2016) Dynamic properties of condensing particle systems. PhD thesis, University of Warwick.

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Abstract

Condensation transitions are observed in many physical and social systems, ranging from Bose-Einstein condensation to traffic jams on the motorway. The understanding of the critical phenomena prevalent in these systems presents many interesting mathematical challenges. We are interested in understanding the various definitions of condensation which are suitable in the field of stochastic particle systems and how they are related. Furthermore, we are also interested in dynamic properties of processes that undergo the condensation transition, such as typical convergence time scales and monotonicity properties.

Condensation can be defined in many different ways; considering the thermodynamic limit, a weak law of large numbers for the maximum occupation number, and an infinite particle limit on fixed finite lattices. For the latter definition, and processes that exhibit a family of stationary product measures, we prove an equivalent characterisation in terms of sub-exponential distributions generalising previous known results.

All known examples of condensing processes that exhibit homogeneous stationary product measures are non-monotone, i.e. the dynamics do not preserve a partial ordering of the state space. This non-monotonicity is typically characterised by an overshoot of the canonical current, which on a heuristic level is related to metastability. We prove that these processes with a finite critical density are necessarily non-monotone confirming a previous conjecture. If the critical density is infinite, condensation can still occur on finite lattices. We present partial evidence that there also exist monotone condensing processes.

We also study the typical convergence time scales of condensing inhomogeneous zero-range processes. Our results represent a first rigours calculation of the relaxation time of a condensing zero-range process, where we prove a dynamic transition in the order of the relaxation time as the density crosses a critical value. We also derive bounds for homogeneous condensing models and obtain results consistent with known metastable time scales.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Condensation -- Mathematical models, Dynamics of a particle -- Mathematical models, Stochastic systems -- Mathematical models
Official Date:	November 2016
Dates:	Date Event November 2016 Submitted
Institution:	University of Warwick
Theses Department:	Centre for Complexity Science
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Grosskinsky, Stefan ; Chleboun, P. I. (Paul I.)
Format of File:	pdf
Extent:	vi, 140 leaves : illustrations, charts
Language:	eng

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