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arXiv · 2212.13618

Functorial Statistical Physics: Feynman--Kac Formulae and Information Geometries

Dalton A R Sakthivadivel
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Abstract

The main results of this paper comprise proofs of the following two related facts: (i) the Feynman--Kac formula is a functor $F_*$, namely, between a stochastic differential equation and a dynamical system on a statistical manifold, and (ii) a statistical manifold is a sheaf generated by this functor with a canonical gluing condition. Using a particular locality property for $F_*$, recognised from functorial quantum field theory as a `sewing law,' we then extend our results to the Chapman--Kolmogorov equation {\it via} a time-dependent generalisation of the principle of maximum entropy. This yields a partial formalisation of a variational principle which takes us beyond Feynman--Kac measures driven by Wiener laws. Our construction offers a robust glimpse at a deeper theory which we argue re-imagines time-dependent statistical physics and information geometry alike.

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Dalton A R Sakthivadivel. 2022-12-27. Functorial Statistical Physics: Feynman--Kac Formulae and Information Geometries. https://arxiv.org/abs/2212.13618

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