Geometric decomposition of information flow: New insights into information thermodynamics

TL;DR

This work develops a geometric housekeeping–excess decomposition of information flow for bipartite Markov jump processes by projecting the thermodynamic force onto the conservative subspace and separating nonconservative (housekeeping) from conservative (excess) contributions. It generalizes the second law of information thermodynamics and the cyclic decomposition to nonsteady states, introduces a short-time TUR and an information-thermodynamic speed limit grounded in 2-Wasserstein geometry, and extends the framework to marginal distributions via contracted graphs. The approach yields a clear interpretation of information flow: housekeeping components sustain nonequilibrium steady states while excess components drive changes in mutual information between subsystems. The results illuminate how Maxwellian demons can be classified with respect to housekeeping and excess dissipation and offer practical inequalities and speed limits for information processing, validated on a simple four-state model. The framework bridges stochastic thermodynamics, optimal transport, and information theory, with potential extensions to non-bipartite, Langevin, chemical-network, and quantum systems.

Abstract

We propose a decomposition of information flow into housekeeping and excess components for autonomous bipartite systems subject to Markov jump processes. We introduce this decomposition by using the geometric structure of probability currents and the conjugate thermodynamic forces. The housekeeping component arises from the cyclic modes caused by the detailed balance violations and maintains the correlations between the two subsystems. In contrast, the excess component, is a contribution of conservative forces that alters the mutual information between the two subsystems. With this decomposition, we generalize previous results, such as the second law of information thermodynamics, the cyclic decomposition, and the information-thermodynamic extensions of thermodynamic trade-off relations.

Figures (8)

• Figure 1: (a) Schematic of a bipartite system. Two subsystems interact while in contact with (potentially, multiple) baths. (b) Example of a graph. The nodes represent the system's states, and the edges represent the transitions. (c) Information flow is decomposed into housekeeping and excess components. The housekeeping information flow arises from nonconservative cyclic modes, while the excess information flow arises from conservative nonstationary modes.
• Figure 2: A graph and its incidence matrix (transposed, giving the divergence). The graph $G$ consists of the nodes $\mathcal{Z} = \{1, 2, 3, 4\}$ and the edges $\mathcal{E} = \{e_1, e_2, e_3, e_4, e_5, e_6\}$. Each node represents a state of the system, and the edges indicate reversible transitions between two different states. We will refer to this graph repeatedly in figures.
• Figure 3: An example of cycles and their associated vectors in the graph $G$ presented in Fig. \ref{['fig:graph_incidencematrix']}. The graph $G$ has three fundamental cycles $C^1 = \{-e_3, e_5\}, C^2 = \{-e_4, e_6\}$ and $C^3 = \{e_1, -e_2, -e_3, e_4\}$, whose corresponding vectors are denoted by $\bm{\mathcal{S}}^1, \bm{\mathcal{S}}^2$, and $\bm{\mathcal{S}}^3$, respectively. Any cycle on $G$ can be expressed as a superposition of $C^1$, $C^2$ and $C^3$, and the kernel of $\nabla^{\top}$ is given by $\ker\nabla^{\top} = \{r_1\bm{\mathcal{S}}^1 + r_2\bm{\mathcal{S}}^2 + r_3\bm{\mathcal{S}}^3 \mid r_1, r_2, r_3\in\mathbb{R}\}$.
• Figure 4: (a) An example of a graph representing a bipartite system, which is structurally the same as the graph $G$ presented in Fig. \ref{['fig:graph_incidencematrix']}. Let $\mathcal{X}$ and $\mathcal{Y}$ be $\mathcal{X} = \{0, 1\}$ and $\mathcal{Y} = \{0, 1\}$, respectively. The original states $\{1,2,3,4\}$ are identified with $\{(0,0),(0,1),(1,0),(1,1)\}$. In this setting, the edges are split into $\mathcal{E}_X = \{e_1, e_2\}$ and $\mathcal{E}_Y = \{e_3, e_4, e_5, e_6\}$, and the original graph $G$ is decomposed into the subgraphs $G_X = (\mathcal{X}\times\mathcal{Y}, \mathcal{E}_X)$ and $G_Y = (\mathcal{X}\times\mathcal{Y}, \mathcal{E}_Y)$. (b) Corresponding to the decomposition of the graph, the incidence matrix $\nabla^{\top}$ is partitioned into $\nabla^{\top}_{G_{X}}$ and $\nabla^{\top}_{G_{Y}}$. We also show the marginalization matrix $\Pi_{\alpha}$, and the condition for bipartite systems $\Pi_{\alpha}\nabla^{\top}_{G_{\alpha^{\rm c}}} = O$ for $\alpha \in \{X, Y\}$.
• Figure 5: (a) Schematic of the model. The system $X$ exchanges particles with the particle bath that has the chemical potential $\mu_X$ and the inverse temperature $\beta_X$. The system $Y$ exchanges particles with the two particle baths that have different chemical potentials $\mu_{\mathrm{L}}$ and $\mu_{\mathrm{R}}\;(<\mu_{\mathrm{L}})$ and the same inverse temperature $\beta\;(<\beta_X)$. There is also an interaction between $X$ and $Y$. (b) The graph corresponding to the model. The presence or absence of a particle in each system represents the states of $X$ and $Y$, which are written as $x$ and $y$, respectively. All possible states of the system can be described as $(x, y) \in \{(0, 0), (0, 1), (1, 0), (1, 1)\}$ where $1$ represents an occupied state and $0$ an unoccupied one. The nodes are connected by edges $\{e_i\}_{i\in\{1,\ldots, 6\}}$.