Information geometry of perturbed gradient flow systems on hypergraphs: A perspective towards nonequilibrium physics

TL;DR

This paper reviews the connection between gradient flow systems on hypergraphs and information geometry, integrating modern nonequilibrium physics principles to reveal new geometric concepts and physical insights.

Contribution

It formulates modern nonequilibrium principles within the framework of perturbed gradient flows on hypergraphs, introducing geometric concepts like moduli spaces and thermodynamical area.

Findings

Geometrical perspective leads to new concepts such as moduli spaces.

Introduces thermodynamical area as key to understanding speed limits.

Connects information geometry with nonequilibrium physics principles.

Abstract

This article serves to concisely review the link between gradient flow systems on hypergraphs and information geometry which has been established within the last five years. Gradient flow systems describe a wealth of physical phenomena and provide powerful analytical technquies which are based on the variational energy-dissipation principle. Modern nonequilbrium physics has complemented this classical principle with thermodynamic uncertaintly relations, speed limits, entropy production rate decompositions, and many more. In this article, we formulate these modern principles within the framework of perturbed gradient flow systems on hypergraphs. In particular, we discuss the geometry induced by the Bregman divergence, the physical implications of dual foliations, as well as the corresponding infinitesimal Riemannian geometry for gradient flow systems. Through the geometrical perspective, we are naturally led to new concepts such as moduli spaces for perturbed gradient flow systems and thermodynamical area which is crucial for understanding speed limits. We hope to encourage the readers working in either of the two fields to further expand on and foster the interaction between the two fields.