A categorical approach to Statistical Mechanics

28.04.2023, 11:00  –  Golm campus, Building 9, Room 2.22 + Zoom
Arbeitsgruppenseminar Analysis

Grégoire Sergeant (INRIA and Sorbonne Université, Paris)

Statistical physics is a framework that focuses on the probabilistic description of complex systems: a collection of interacting 'particles' or components of a whole in most generality. In (rigorous) statistical mechanics, a statistical system is encoded by the notion of 'specification' [1]. We will show that a 'specification' can be extended into some representation of a partially ordered set (poset). In particular, the phases of a statistical system (Gibbs measures) are geometric invariants of the associated representation [2,3]. We propose that these poset representations serve as a novel framework for statistical mechanics; we call this framework 'generalized statistical mechanics'. In this framework, generalized statistical systems are described by collections of sub-systems together with the way these sub-systems interact with one another (compositionality). It allows for an increase in modeling power for probabilistic modeling. We will introduce the category of generalized statistical systems. We will give algebraic tools to compute their invariants. We introduce a variational principle for these systems. We will also discuss model selection and sampling for these generalizations and their application in molecular biology.

[1] 'Gibbs Measures and Phase Transitions', H.O. Georgii,  De Gruyter, Berlin, New York (2011)
[2] 'Intersection property, interaction decomposition, regionalized optimization and applications', G. Sergeant-Perthuis, PhD Thesis,
https://www.researchgate.net/publication/349732943_Intersection_property_interaction_decomposition_regionalized_optimization_and_applications
[3] 'A categorical approach to Statistical Mechanics', G. Sergeant-Perthuis,
https://www.researchgate.net/publication/369325487_A_categorical_approach_to_Statistical_Mechanics

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