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

TL;DR

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 are discussed.

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.

Figures (2)

• Figure 1: Illustration of the dual foliations of $\mathcal{F}_{\boldsymbol{x}}$ and $\mathcal{J}_{\boldsymbol{x}}$ resulting from the interplay between the hypergraph structure and information geometry. Each space $\mathcal{F}_{\boldsymbol{x}}^{\textrm{c}}(\boldsymbol{f}')$ acts as a base for a foliation of $\mathcal{F}_{\boldsymbol{x}}$ with leaves $\{\mathcal{F}_{\boldsymbol{x}}^{\textrm{b}}(\boldsymbol{f}) | \boldsymbol{f} \in \mathcal{F}_{\boldsymbol{x}}^{\textrm{c}}(\boldsymbol{f}')\}$, and dually, each space $\mathcal{F}_{\boldsymbol{x}}^{\textrm{b}}(\boldsymbol{f})$ is a base space for a foliation with leaves $\{\mathcal{F}_{\boldsymbol{x}}^{\textrm{c}}(\boldsymbol{f}') | \boldsymbol{f}' \in \mathcal{F}_{\boldsymbol{x}}^{\textrm{b}}(\boldsymbol{f}) \}$. The analogous situation holds for the space $\mathcal{J}_{\boldsymbol{x}}$. The spaces $\mathcal{F}_{\boldsymbol{x}}^{\textrm{b}}(\boldsymbol{f})$ and $\mathcal{J}_{\boldsymbol{x}}^{\textrm{c}}(\boldsymbol{j})$ are affine while $\mathcal{F}_{\boldsymbol{x}}^{\textrm{c}}(\boldsymbol{f})$ and $\mathcal{J}_{\boldsymbol{x}}^{\textrm{b}}(\boldsymbol{j})$ are curved. The leaves intersect the base orthogonally according to the Pythoagorean relation (\ref{['eq:Pythagoras']}), which is also illustrated in the figure.
• Figure 2: Illustration of the geometry underlying the dissipation rate decomposition (\ref{['eq:IG_decomposition']}). On $\mathcal{F}_{\boldsymbol{x}}$, the composition is achived via the cycle force $\boldsymbol{f}^{\textrm{c}} = \mathcal{F}_{\boldsymbol{x}}^{\textrm{b}}(\boldsymbol{f}) \cap \mathcal{F}_{\boldsymbol{x}}^{\textrm{c}}(\boldsymbol{0})$ and the correspoding orthogonal decomposition of $\Psi^*(\boldsymbol{x},\boldsymbol{f})$, and dually, on $\mathcal{J}_{\boldsymbol{x}}$ via the boundary flux $\boldsymbol{j}^{\textrm{b}} = \mathcal{J}_{\boldsymbol{x}}^{\textrm{b}}(\boldsymbol{0}) \cap \mathcal{J}_{\boldsymbol{x}}^{\textrm{c}}(\boldsymbol{j})$. Note that one can also define directly $\boldsymbol{f}^{\textrm{b}} := \mathcal{F}_{\boldsymbol{x}}^{\textrm{b}}(\boldsymbol{0}) \cap \mathcal{F}_{\boldsymbol{x}}^{\textrm{c}}(\boldsymbol{f})$ to get the Legendre dual of $\boldsymbol{j}^{\textrm{b}}$, and analogously for $\boldsymbol{j}^{\textrm{c}} := \mathcal{J}_{\boldsymbol{x}}^{\textrm{b}}(\boldsymbol{j}) \cap \mathcal{J}_{\boldsymbol{x}}^{\textrm{c}}(\boldsymbol{0})$.

Theorems & Definitions (7)

• Remark 2.1
• Definition 3.1
• Remark 3.1
• Definition 3.2
• Theorem 3.1
• Lemma 4.1
• Remark 1.1