Compositional statistical mechanics, entropy and variational inference - Analytical Genomics Access content directly
Conference Papers Year : 2024

Compositional statistical mechanics, entropy and variational inference

Grégoire Sergeant-Perthuis
  • Function : Author
  • PersonId : 1215559

Abstract

In this document, we aim to gather various results related to a compositional/categorical approach to rigorous Statistical Mechanics. 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. 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 how the characterization of extreme Gibbs measures extends to A-specifications; we proposed an Entropy functional for A-specifications and gave a message-passing algorithm, that generalized the belief propagation algorithm of graphical models, to minimize variational free energy.
Fichier principal
Vignette du fichier
SYCO24_arxiv.pdf (318.74 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-04518736 , version 1 (24-03-2024)

Licence

Attribution - NonCommercial - NoDerivatives

Identifiers

  • HAL Id : hal-04518736 , version 1

Cite

Grégoire Sergeant-Perthuis. Compositional statistical mechanics, entropy and variational inference. Twelfth Symposium on Com- positional Structures (SYCO 12), Apr 2024, Birmingham (UK), United Kingdom. ⟨hal-04518736⟩
16 View
12 Download

Share

Gmail Facebook X LinkedIn More