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Title: The combinatorics of the cluster and virial expansions
Author: Tate, Stephen James
ISNI:       0000 0004 5349 3259
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2014
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The interpretation of cluster and virial expansions as weighted exponential generating functions of connected and two-connected graphs respectively was given by Mayer in the 1940s. Combinatorial approaches, either through the tree graph identities, introduced by Brydges, Battle and Federbush, or the fixed point equations of Kotecký-Preiss, generalised by Fernández-Procacci, have led to results pertaining to the convergence of these series, notably in the case of the cluster expansion. Recent interest in these expansions has been stimulated by the connection to Joyal's combinatorial species of structure, presented both in the work of Leroux and his collaborators and Faris. Virial expansions have also gained renewed interest through the Canonical Ensemble methods of Pulvirenti and Tsagkarogiannis, through which the convergence conditions of Lebowitz and Penrose are obtained. This thesis obtains combinatorial interpretations of the cancellations in the virial expansions for the one-particle hardcore gas and the Tonks gas. Improved bounds are obtained for the virial expansion, through an original approach, depending on cluster coefficient bounds. Separate bounds are also found by using the cluster coefficient bounds of Poghosyan and Ueltschi. Furthermore, a generalisation to multispecies expansions, through Lagrange-Good inversion, is given, providing Kotecký-Preiss type conditions of convergence for the multispecies virial expansion. The tree-graph expansions are analysed in the context of these results and are used to understand the key structure necessary to replicate such bounds for virial expansions.
Supervisor: Not available Sponsor: Engineering and Physical Sciences Research Council (EP/G056390/1)
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics