A Categorical Approach to Statistical Mechanics

Abstract

In this document, accepted for the Twelfth Symposium on Compositional Structures (SYCO 12) [1], we aim to gather various results related to a compositional/categorical approach to rigorous Statistical Mechanics [2][3][4][5][6][7][8][9]. Rigorous Statistical Mechanics is centered on the mathematical study of statistical systems. Central concepts in this field have a natural expression in terms of diagrams in a category that couples measurable maps and Markov kernels [8]. We showed that statistical systems are particular representations of partially ordered sets (posets), that we call A -specifications, and expressed their phases, i.e., Gibbs measures, as invariants of these representations. It opens the way to the use of homological algebra to compute phases of statistical systems. Two central results of rigorous Statistical Mechanics are, firstly, the characterization of extreme Gibbs measure as it relates to the zero-one law for extreme Gibbs measures, and, secondly, their variational principle which states that for translation invariant Hamiltonians, Gibbs measures are the minima of the Gibbs free energy. We showed in [9] how the characterization of extreme Gibbs measures extends to A -specifications; we proposed in [10] an Entropy functional for A -specifications and gave a message-passing algorithm, that generalized the belief propagation algorithm of graphical models (see O. Peltre's SYCO 8 talk, [11] and [12]), to minimize variational free energy.