Jan Fiala

Date of Award

2004

Level of Access Assigned by Author

Open-Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Physics

Advisor

Peter Kleban

Second Committee Member

R. Dean Astumian

Third Committee Member

William Bray

Abstract

This thesis considers several statistical models defined on the Farey fractions. Two of these models, considered first, may be regarded as "spin chains", with long-range interactions, another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle-Perron-F'robenius operator), which is defined using the maps (presentation functions) generating the Farey "tree". The spectrum of this operator was completely determined by Prellberg. It follows that all these models have a second-order phase transition with a specific heat divergence of the form C - [c ln2 el-'. The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition. The second part of this work extends our model by introducing an external field h. From rigorous and more heuristic arguments, we determine the phase diagram and phase transition behavior of the extended model. Our results are fully consistent with scaling theory (for the case when a "marginal" field is present) despite the unusual nature of the transition for h = 0. The third part of this thesis introduces a new family of partition functions with the same free energy. These models generalize one of the spin chain by introducing a new real parameter x. The structure of the Farey fractions then leads to a recurrence formula which has a direct connection to the operator studied by Prellberg. This connection provides a new and simple relation of the Contucci and Knauf "canonical" and "grand canonical" partition functions for any length of spin chain to the function obtained by the action of the operator on a constant function. In addition, we use the new partition functions to calculate certain expectation values and correlation functions.

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