Geometric decomposition of information flow for overdamped Langevin systems and optimal transport in subsystems

Abstract: Information flow between subsystems is a central concept in information thermodynamics, which provides the second-law-like inequalities for subsystems. This paper discusses the geometric decomposition of information flow, which was introduced for Markov jump systems [Y Maekawa, R Nagayama, K Yoshimura and S Ito, arXiv:2509.21985 (2025)], and applies it to overdamped Langevin systems. For overdamped Langevin systems, the geometric decomposition of information flow into excess and housekeeping contributions is related to the conventional definition of the $2$-Wasserstein distance between marginal distributions in optimal transport theory. This formulation offers an optimal-transport interpretation of subsystem dynamics, and this optimal-transport formulation is simpler for overdamped Langevin systems than for general Markov jump systems. It is also possible to handle features that are specific to overdamped Langevin systems, such as representations based on the Koopman mode decomposition, as well as their relationship with the Fisher information matrix. As with the results for Markov jump systems, we generalize the second law of information thermodynamics using housekeeping and excess information flow, leading to the concept of excess and housekeeping demons. We also derive a thermodynamic uncertainty relation and an information-thermodynamic speed limit incorporating excess information flow. These results are illustrated for the Gaussian case, and we discuss the conditions under which the excess and housekeeping demons emerge.

• The paper introduces a geometric decomposition that splits information flow into excess and housekeeping components in overdamped Langevin systems.
• It applies optimal transport theory to link subsystem entropy production with the 2-Wasserstein distance, providing a metric-based thermodynamic bound.
• The study outlines subsystem-specific generalized second laws, modal decompositions, and conditions for demon-like negative dissipation phenomena.

Geometric Decomposition of Information Flow in Overdamped Langevin Systems

Introduction

The paper "Geometric decomposition of information flow for overdamped Langevin systems and optimal transport in subsystems" (2512.22890) offers a continuous-state, stochastic thermodynamics-based formulation for the geometric decomposition of information flow in bipartite overdamped Langevin systems. This work extends the discrete-state results of [maekawa2025geometric] to Langevin dynamics, enabling a transparent connection between excess/housekeeping splits of information flow and the optimal transport framework, specifically leveraging the $2$-Wasserstein metric.

The authors develop a differential geometric perspective on information thermodynamics and provide generalized second law inequalities, uncertainty relations, speed limits, and a modal decomposition for subsystem dynamics. They introduce the concepts of "housekeeping demon" and "excess demon"—subsystem-specific Maxwellian agents for negative apparent dissipation—whose operational regimes are delineated and illustrated for Gaussian linear processes.

Stochastic Thermodynamic Framework and Information Flow

The central thermodynamic object is a general bipartite overdamped Langevin system, described by

$$\dot{\boldsymbol{z}}(t) = \mathsf{\mu} \boldsymbol{F}_t (\boldsymbol{z}) + \sqrt{2 T} \mathsf{\mu}^{1/2} \boldsymbol{\xi}_t$$

with $\boldsymbol{z} = (\boldsymbol{x}, \boldsymbol{y})$, state decomposition, and block-diagonal mobility $\mathsf{\mu}$. This structure ensures uncorrelated noise and force partitions for subsystems X and Y.

Thermodynamic quantities (system entropy, entropy flow to bath, entropy production rate—EPR) decompose correspondingly. Crucially, information flows $\dot{I}^{\mathrm{X}}_t$, $\dot{I}^{\mathrm{Y}}_t$ between subsystems are defined via partial derivatives of mutual information and play a compensatory role in the second law for subsystems. These information flows generically do not vanish in steady states for coupled subsystems.

Geometric Decomposition: Excess and Housekeeping Components

A principal contribution is the geometric, metric-based splitting of the entropy production and information flow into "excess" (conservative/gradient) and "housekeeping" (nonconservative/cyclic) components:

• Excess terms capture redistribution associated with net time evolution (non-steady relaxation).
• Housekeeping terms encode persistent nonequilibrium flux maintained by nonconservative drives.

The decomposition is defined via projection of the thermodynamic force onto the gradient (conservative) manifold. The excess force $-\nabla\phi$ satisfies a Poisson equation tied to the instantaneous Fokker-Planck evolution. These components are $p_t \mathsf{D}$-orthogonal, yielding a Pythagorean decomposition of the entropy production.

Figure 1: Probability distributions and phase-space streamlines for (a) weakly nonconservative ($r=0.1$) and (b) strongly nonconservative ($r=-1$) driving. Left and center panels: two distinct anisotropic Gaussian initializations. Right: steady-state.

Excess and Housekeeping Information Flow

This structural split extends to information flow: for subsystem X, the rate decomposes as

$$\dot{I}^{\mathrm{X}}_t = \dot{I}^{\mathrm{ex;X}}_t + \dot{I}^{\mathrm{hk;X}}_t$$

with analogous expressions for Y. Excess information flow is directly tied to time variation of mutual information, while housekeeping flow is antisymmetric between subsystems at all times, not only steady-state ($$\dot{I}^{\mathrm{hk;X}}_t = -\dot{I}^{\mathrm{hk;Y}}_t$$).

Generalized Second Laws and Thermodynamic Bounds

The authors derive subsystem-specific, generalized second law inequalities, encapsulating both excess and housekeeping dissipation: $$\sigma^{\mathrm{ex;X}}_t \geq \dot{I}^{\mathrm{ex;X}}_t \qquad \sigma^{\mathrm{hk;X}}_t \geq \dot{I}^{\mathrm{hk;X}}_t$$ where the "apparent" rates incorporate only local entropic and heat flux contributions, exclusive of information flow.

Importantly, they establish that excess demon phenomena—negative $\sigma^{\mathrm{ex;X}}_t$ triggered by negative $\dot{I}^{\mathrm{ex;X}}_t$—arise only in transient, non-steady regimes; conversely, housekeeping demons (enabled by negative $\dot{I}^{\mathrm{hk;X}}_t$) persist in steady, driven nonequilibria if a nonconservative force is present.

Thermodynamic uncertainty relations (TURs) and information-thermodynamic speed limits for these decomposed EPRs are computed using $2$-Wasserstein geodesic distances for marginal system distributions, leveraging variational formulas for minimum action.

Optimal Transport, Information Flow, and Subsystem Geometry

A major outcome is the rigorous connection between excess (and local) entropy production rates and the $2$-Wasserstein (and generalized) distances between time-adjacent marginal distributions: $$\dot{\Sigma}_t^{\mathrm{local\ ex;X}} = \lim_{\Delta t \to 0} \frac{\mathcal{W}_2^2(p_t^{\mathrm{X}}, p_{t+\Delta t}^{\mathrm{X}})}{(\Delta t)^2}$$ This identifies subsystem transitions with geodesics in probability space under the appropriate kinetic metric, providing geometric lower bounds on dissipation. The generalized transport metric is shown to reduce to a standard $2$-Wasserstein under a diffusion rescaling.

Koopman Mode Decomposition

The nonconservative (housekeeping) entropy production is further decomposed into modal contributions using the spectral expansion of the Koopman operator associated with the stationary, divergence-free cyclic currents. Each mode's contribution is weighted by its oscillatory frequency and participation intensity, analogous to cycle decompositions in finite Markov settings.

Explicit Illustration: Gaussian Example

The theory is solved analytically for linear drift, Gaussian-initialized dynamics with coupling and rotational (antisymmetric) force parameters. The analytic accessibility of excess and housekeeping flows and dissipation rates allows nontrivial regimes to be explored, especially the emergence and temporal domain of excess and housekeeping demon behaviors.

Figure 2: Time evolution of excess demon signatures: $\sigma_t^{\mathrm{ex;X}}$ and $\dot{I}_t^{\mathrm{ex;X}}$ show transient negative dissipation rates indicating demon-like entropy reduction during relaxation for substantial nonconservative coupling.

Figure 3: Housekeeping demon dynamics: panel (a) shows transient demon activity; panel (b) longer time demon operation in steady drive; panel (c) landscape of minimal $\sigma_t^{\mathrm{hk;X}}$, identifying operational regimes.

Implications and Future Directions

This geometric, mode-resolved framework offers a precise information-geometric lens on subsystem thermodynamics and information transfer, embedding optimal transport as a structural component at the mesoscopic level. The theory enables:

• Quantitative predictions for entropy reduction (demon action) in coupled systems;
• Design of minimal-dissipation protocols for information erasure, control, and generative AI models governed by diffusion;
• Analytical assessment of speed limits and performance tradeoffs for subsystems under feedback or game-theoretic intervention.

The extension to time-inhomogeneous diffusions, multiple heat baths, and underdamped/stochastic Hamiltonian settings is mapped out, presenting significant opportunities for generalization. Modal decompositions suggest spectral approaches to identifying persistent nonequilibrium modes in high-dimensional systems, with potential for applications to neural, sensor-adaptive, and molecular networks.

Conclusion

The paper establishes a rigorous, extensible foundation for geometric decomposition of information thermodynamic flows in continuous systems, deeply linking nonequilibrium second law extensions, subsystem optimal transport, and modal geometry. The identification of excess and housekeeping demons introduces formal, operational distinctions in subsystem thermodynamics, with wide-ranging implications for nonequilibrium control and information-driven engines.

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Cited Paper: "Geometric decomposition of information flow for overdamped Langevin systems and optimal transport in subsystems" (2512.22890).