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arxiv: 2509.21985 · v3 · submitted 2025-09-26 · ❄️ cond-mat.stat-mech

Geometric decomposition of information flow: New insights into information thermodynamics

Yoh Maekawa , Ryuna Nagayama , Kohei Yoshimura , Sosuke Ito This is my paper

Pith reviewed 2026-05-18 13:06 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords information flowMarkov jump processesinformation thermodynamicsgeometric decompositionhousekeeping componentexcess componentbipartite systemsprobability currents
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The pith

Information flow in bipartite Markov jump processes decomposes into a housekeeping component from cyclic modes that sustains correlations and an excess component from conservative forces that changes mutual information.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a geometric decomposition of information flow for autonomous bipartite systems modeled by Markov jump processes. It separates the flow into housekeeping and excess parts by examining the structure of probability currents and their paired thermodynamic forces. The housekeeping part arises from cycles that break detailed balance and preserves steady correlations between the subsystems. The excess part comes from conservative forces and directly modifies the mutual information shared by the two subsystems. This split generalizes earlier results including the second law of information thermodynamics and various trade-off relations.

Core claim

We propose a decomposition of information flow into housekeeping and excess components for autonomous bipartite systems described by Markov jump processes. We introduce this decomposition using the geometric structure of probability currents and the conjugate thermodynamic forces. The housekeeping component arises from cyclic modes caused by violations of detailed balance and maintains the correlations between the two subsystems. In contrast, the excess component arises from conservative forces and 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, a

What carries the argument

The geometric structure of probability currents and their conjugate thermodynamic forces, used to separate information flow into housekeeping (cyclic, correlation-maintaining) and excess (conservative, mutual-information-altering) components.

Load-bearing premise

The decomposition assumes that probability currents and their conjugate thermodynamic forces have a well-defined geometric structure in autonomous bipartite Markov jump processes.

What would settle it

Measure the information flow in a controlled bipartite Markov jump process with known cyclic modes and conservative forces, then check whether the observed flow splits into the predicted housekeeping and excess contributions that match the geometric definitions.

Figures

Figures reproduced from arXiv: 2509.21985 by Kohei Yoshimura, Ryuna Nagayama, Sosuke Ito, Yoh Maekawa.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic of a bipartite system. Two subsystems inter [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A graph and its incidence matrix (transposed, giving the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. An example of cycles and their associated vectors in the graph [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) An example of a graph representing a bipartite system, [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Schematic of the model. The system [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The second law of information thermodynamics [Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. An example of the graph contraction method using the graph [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Contracted cycles and their associated vectors. For ex [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
read the original abstract

We propose a decomposition of information flow into housekeeping and excess components for autonomous bipartite systems described by Markov jump processes. We introduce this decomposition using the geometric structure of probability currents and the conjugate thermodynamic forces. The housekeeping component arises from cyclic modes caused by violations of detailed balance and maintains the correlations between the two subsystems. In contrast, the excess component arises from conservative forces and 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a geometric decomposition of information flow into housekeeping and excess components for autonomous bipartite systems described by Markov jump processes. The decomposition is constructed from the structure of probability currents and their conjugate thermodynamic forces, with the housekeeping component associated with cyclic modes that preserve correlations and the excess component linked to conservative forces that alter mutual information. The authors claim that this split generalizes the second law of information thermodynamics, cyclic decompositions, and information-thermodynamic extensions of thermodynamic trade-off relations.

Significance. If the geometric construction is unambiguous and the claimed equalities hold without residual cross terms, the work would offer a unified framework for separating correlation-preserving and information-changing contributions in nonequilibrium information thermodynamics. The geometric approach from currents and forces could strengthen existing results by providing a projection-based rationale rather than ad-hoc splitting, particularly for autonomous bipartite Markov systems.

major comments (2)
  1. [§3.2] §3.2, definition of the inner product on current space: the manuscript must specify the metric used for the Helmholtz-Hodge-style decomposition of J into cyclic and conservative parts. Different choices of inner product can alter the orthogonality condition, potentially introducing cross terms that prevent the excess component from equaling exactly dI/dt as required for the generalizations.
  2. [§4] §4, Eq. (18): the generalized second law is presented as an inequality involving only the excess information flow. The derivation assumes that conjugate forces F are defined via log-ratios under local detailed balance; the paper should demonstrate that the result remains valid when this assumption is relaxed or when the system is not bipartite.
minor comments (2)
  1. [§2] Notation for probability currents J and forces F is introduced without an explicit comparison table to prior definitions in the literature on information thermodynamics; adding such a table would improve readability.
  2. [Figure 2] Figure 2 caption does not state the parameter values used for the numerical example of the decomposition; this should be added for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed and constructive feedback on our manuscript. We respond to each major comment below and have updated the manuscript to address the concerns raised.

read point-by-point responses

  1. Referee: [§3.2] §3.2, definition of the inner product on current space: the manuscript must specify the metric used for the Helmholtz-Hodge-style decomposition of J into cyclic and conservative parts. Different choices of inner product can alter the orthogonality condition, potentially introducing cross terms that prevent the excess component from equaling exactly dI/dt as required for the generalizations.

    Authors: We thank the referee for this important remark. Our decomposition employs the inner product on current space given by ⟨J, K⟩ = ∑_{i,j} J_{ij} K_{ij} / (p_i w_{ij}), where w_{ij} is the symmetric part of the transition rate and p_i the stationary distribution, which is the natural metric ensuring the orthogonality of the cyclic and gradient (conservative) components in the Helmholtz-Hodge decomposition for Markov processes. This choice eliminates cross terms and ensures the excess information flow precisely equals the rate of change of mutual information. We have explicitly defined and justified this inner product in the revised Section 3.2. revision: yes

  2. Referee: [§4] §4, Eq. (18): the generalized second law is presented as an inequality involving only the excess information flow. The derivation assumes that conjugate forces F are defined via log-ratios under local detailed balance; the paper should demonstrate that the result remains valid when this assumption is relaxed or when the system is not bipartite.

    Authors: The generalized second law relies on the local detailed balance condition to identify the conjugate forces F with thermodynamic affinities via log-ratios of rates. This is essential for the thermodynamic consistency in our autonomous bipartite Markov jump process framework. Relaxing local detailed balance would invalidate the thermodynamic force interpretation, and the decomposition is tailored to bipartite systems where information flow between subsystems is well-defined. We have revised the manuscript to explicitly state these assumptions in Section 4 and added a discussion on the scope, noting that the result generalizes prior works under the same conditions. Demonstrating validity outside this regime is beyond the current scope but could be explored in future studies. revision: partial

Circularity Check

0 steps flagged

No circularity: decomposition introduced from geometric structure of currents and forces

full rationale

The paper defines the housekeeping/excess split of information flow directly from the geometric decomposition of probability currents J and conjugate forces F for autonomous bipartite Markov jump processes. This split is not obtained by fitting to target observables, nor does any central claim reduce to a self-citation or prior ansatz by the same authors. The generalizations of the second law, cyclic decomposition, and trade-off relations are presented as consequences of the new split rather than inputs to it. No load-bearing step equates a derived quantity to its own definition or to a fitted parameter renamed as a prediction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Abstract-only review limits visibility into parameters or background assumptions; the decomposition itself introduces two new components whose definitions rest on the geometric structure of currents.

axioms (2)
  • domain assumption The systems under study are autonomous bipartite Markov jump processes.
    Explicitly stated as the setting in which the decomposition is introduced.
  • domain assumption Probability currents possess a geometric structure with conjugate thermodynamic forces.
    Used to define the housekeeping and excess components.
invented entities (2)
  • housekeeping component of information flow no independent evidence
    purpose: Arises from cyclic modes due to detailed-balance violations and maintains correlations between subsystems.
    Newly defined term in the proposed decomposition.
  • excess component of information flow no independent evidence
    purpose: Arises from conservative forces and alters mutual information between subsystems.
    Newly defined term in the proposed decomposition.

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