Geometric and Combinatorial Aspects of NonEquilibrium Statistical Mechanics
Contribuinte(s) |
Bazzani, Armando |
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Data(s) |
16/03/2012
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Resumo |
Non-Equilibrium Statistical Mechanics is a broad subject. Grossly speaking, it deals with systems which have not yet relaxed to an equilibrium state, or else with systems which are in a steady non-equilibrium state, or with more general situations. They are characterized by external forcing and internal fluxes, resulting in a net production of entropy which quantifies dissipation and the extent by which, by the Second Law of Thermodynamics, time-reversal invariance is broken. In this thesis we discuss some of the mathematical structures involved with generic discrete-state-space non-equilibrium systems, that we depict with networks in all analogous to electrical networks. We define suitable observables and derive their linear regime relationships, we discuss a duality between external and internal observables that reverses the role of the system and of the environment, we show that network observables serve as constraints for a derivation of the minimum entropy production principle. We dwell on deep combinatorial aspects regarding linear response determinants, which are related to spanning tree polynomials in graph theory, and we give a geometrical interpretation of observables in terms of Wilson loops of a connection and gauge degrees of freedom. We specialize the formalism to continuous-time Markov chains, we give a physical interpretation for observables in terms of locally detailed balanced rates, we prove many variants of the fluctuation theorem, and show that a well-known expression for the entropy production due to Schnakenberg descends from considerations of gauge invariance, where the gauge symmetry is related to the freedom in the choice of a prior probability distribution. As an additional topic of geometrical flavor related to continuous-time Markov chains, we discuss the Fisher-Rao geometry of nonequilibrium decay modes, showing that the Fisher matrix contains information about many aspects of non-equilibrium behavior, including non-equilibrium phase transitions and superposition of modes. We establish a sort of statistical equivalence principle and discuss the behavior of the Fisher matrix under time-reversal. To conclude, we propose that geometry and combinatorics might greatly increase our understanding of nonequilibrium phenomena. |
Formato |
application/pdf |
Identificador |
http://amsdottorato.unibo.it/4305/1/polettini_matteo_tesi.pdf urn:nbn:it:unibo-3021 Polettini, Matteo (2012) Geometric and Combinatorial Aspects of NonEquilibrium Statistical Mechanics, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in Fisica <http://amsdottorato.unibo.it/view/dottorati/DOT244/>, 24 Ciclo. DOI 10.6092/unibo/amsdottorato/4305. |
Idioma(s) |
en |
Publicador |
Alma Mater Studiorum - Università di Bologna |
Relação |
http://amsdottorato.unibo.it/4305/ |
Direitos |
info:eu-repo/semantics/openAccess |
Palavras-Chave | #FIS/07 Fisica applicata (a beni culturali, ambientali, biologia e medicina) |
Tipo |
Tesi di dottorato NonPeerReviewed |