🔬 condensed matter

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

This paper extends the geometric decomposition of information flow from Markov jump systems to overdamped Langevin systems, linking excess and housekeeping contributions to optimal transport theory and Koopman mode decomposition to derive generalized thermodynamic laws, uncertainty relations, and speed limits.

Original authors: Sosuke Ito, Yoh Maekawa, Ryuna Nagayama, Andreas Dechant, Kohei Yoshimura

Published 2026-08-21

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Sosuke Ito, Yoh Maekawa, Ryuna Nagayama, Andreas Dechant, Kohei Yoshimura

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the study of how things change over time, physicists have long been fascinated by systems that are far from balance. Think of a cup of hot coffee cooling in a room, or a cell maintaining its internal order while surrounded by a chaotic environment. These are nonequilibrium systems, and understanding them requires more than just tracking energy; it demands a way to measure how different parts of a system talk to each other. A key concept in this field is "information flow," which describes how the state of one part of a system influences the future of another. This idea is crucial for understanding "Maxwell's demon," a famous thought experiment where a tiny, intelligent agent uses information about particles to seemingly break the laws of thermodynamics by sorting them without expending energy. While we know such demons cannot truly violate the second law of thermodynamics, they can create the appearance of doing so by trading information for order. The challenge for modern science has been to quantify exactly how this trade-off works in complex, continuous systems, distinguishing between temporary fluctuations and permanent, steady-state behaviors.

A team of researchers at the University of Tokyo, Kyoto University, and RIKEN has now provided a clearer map of this territory by applying a new geometric perspective to systems that move through a fluid-like environment, known as overdamped Langevin systems. These are systems where particles move slowly through a viscous medium, like pollen grains drifting in water, rather than bouncing around freely. The researchers took a complex mathematical framework they had previously developed for systems that jump between discrete states and adapted it for these continuous, fluid-like movements. Their work reveals that the flow of information between two interacting parts of a system is not a single, uniform stream. Instead, it can be cleanly split into two distinct types: one that is temporary and changing, and another that is constant and maintenance-oriented. This separation allows scientists to see exactly when and why a subsystem might appear to violate the second law of thermodynamics, and what kind of "demon" is responsible for the illusion.

The core of their discovery is a geometric decomposition that separates the total information flow into an "excess" component and a "housekeeping" component. The excess part represents the information flow that arises from the system changing its state, such as when two parts of a system are moving toward a new equilibrium or reacting to a sudden shift. This type of flow is conservative, meaning it is tied to the actual evolution of the system's correlations and can only exist while the system is in a state of flux. The housekeeping part, by contrast, represents the information flow required just to keep the system running in its current state, maintaining correlations against the natural tendency to drift apart. This flow is nonconservative and persists even when the system has settled into a steady rhythm. By separating these two, the researchers showed that the "demons" that appear to break thermodynamic laws are actually two different creatures with different rules.

The "excess demon" is a transient figure. It only appears when the system is changing, such as during the initial moments of a process. It uses the excess information flow to make the entropy, or disorder, of a subsystem appear to decrease temporarily. This is akin to a system borrowing order from the future to create a momentary pocket of calm. However, this effect is fleeting; once the system stops changing and settles down, the excess demon vanishes. On the other hand, the "housekeeping demon" is a permanent resident, but it has a strict requirement: it can only exist if the system is being driven by a force that does not conserve energy, such as a constant push or a chemical reaction that never reaches equilibrium. This demon uses the housekeeping information flow to maintain a steady, apparent violation of the second law, effectively acting as an autonomous agent that constantly pumps entropy out of one part of the system and into another. The researchers found that in a steady state, only the housekeeping demon can exist, and it can only operate on one side of the interaction at a time, ensuring that the overall laws of physics are never truly broken.

To make these abstract concepts concrete, the team applied their theory to a specific, mathematically tractable case involving Gaussian distributions, which describe systems where variables follow a familiar bell-curve pattern. In this scenario, they could calculate the exact moments when each type of demon emerges. They confirmed that the excess demon appears strictly during the transient phase of a process, while the housekeeping demon requires the presence of non-conservative forces to persist. This distinction is vital because it clarifies that the "violation" of thermodynamic laws is not a single phenomenon but depends entirely on the nature of the forces driving the system and the timing of the observation. The work also connects these findings to a branch of mathematics called optimal transport, which studies the most efficient way to move mass from one distribution to another. The researchers showed that the cost of moving the probability distribution of a subsystem is directly linked to the excess entropy production, providing a geometric way to measure the thermodynamic cost of information processing.

The implications of this work extend to understanding the fundamental limits of how fast a system can change and how much uncertainty is inherent in its behavior. The researchers derived new "uncertainty relations" that set a lower bound on the energy dissipation required for a subsystem to change its state, linking the speed of change to the amount of information flow and the system's sensitivity to fluctuations. They also established "speed limits" for these systems, showing that the time it takes for a subsystem to evolve is constrained by the geometric distance between its starting and ending states. These results suggest that there is a fundamental trade-off between how fast a system can process information, how much energy it must dissipate, and how much it can reduce its own uncertainty. By framing these relationships in terms of excess and housekeeping contributions, the study offers a more nuanced view of the thermodynamic costs of information, revealing that the price of maintaining order is different from the price of changing it.

Ultimately, this research provides a unified language for describing how information and thermodynamics interact in continuous systems. It moves beyond the simple question of whether information can lower entropy to a more sophisticated understanding of how and when this happens. The geometric decomposition allows scientists to identify the specific mechanisms driving apparent violations of thermodynamic laws, distinguishing between the temporary effects of a system in transition and the steady-state effects of a system driven by non-conservative forces. This clarity is essential for designing future technologies that rely on information processing, from biological motors to artificial nanomachines, ensuring that we understand the true thermodynamic cost of every bit of information we manipulate. The study does not claim to have solved all the mysteries of nonequilibrium physics, but it has drawn a much sharper map of the landscape, showing exactly where the excess demons and housekeeping demons live, and under what conditions they can be seen.

Technical Summary: Geometric Decomposition of Information Flow for Overdamped Langevin Systems and Optimal Transport in Subsystems

Problem Statement
Information flow between subsystems is a central concept in information thermodynamics, providing second-law-like inequalities that account for apparent violations of the second law of thermodynamics in interacting systems. While previous work established a geometric decomposition of information flow into "excess" and "housekeeping" components for discrete Markov jump systems, a corresponding formulation for continuous-state overdamped Langevin systems was lacking. This paper addresses the need to extend this geometric framework to overdamped Langevin dynamics, clarifying the physical interpretation of these components and their relationship to optimal transport theory, thermodynamic uncertainty relations, and the concepts of autonomous and transient "demons."

Methodology
The authors analyze bipartite overdamped Langevin systems where the total state $z = (x, y)$ is decomposed into two subsystems $X$ and $Y$ . The dynamics are governed by the Langevin equation with uncorrelated noise acting on each subsystem. The methodology proceeds through the following steps:

Stochastic Thermodynamic Framework: The authors review the standard definitions of entropy production rates, partial entropy production rates, and information flow for the total system and individual subsystems. They establish the second law of information thermodynamics, which relates the partial entropy production rate to the entropy change rate of a subsystem and the information flow from the other.
Geometric Decomposition of Forces: Building on the geometric decomposition of the total entropy production rate into excess (conservative) and housekeeping (nonconservative) contributions, the authors decompose the thermodynamic forces $f_t$ into excess forces $f^{ex}_t$ and housekeeping forces $f^{hk}_t$ . This decomposition is based on a projection onto the space of conservative forces, utilizing the continuity equation to define a potential $\phi_t$ such that $f^{ex}_t = -\nabla \phi_t$ .
Decomposition of Information Flow: By applying the decomposition of thermodynamic forces to the subsystems, the authors define excess information flow ( $\dot{I}^{ex}$ ) and housekeeping information flow ( $\dot{I}^{hk}$ ). They derive the mathematical properties of these flows, including their relationship to the time derivative of mutual information and their antisymmetric properties.
Optimal Transport and Variational Formulations: The paper connects the excess entropy production rate to the 2-Wasserstein distance (specifically a generalized version weighted by the diffusion matrix) via the Benamou-Brenier formula. This allows for the derivation of thermodynamic speed limits and uncertainty relations.
Koopman Mode Decomposition: For the housekeeping component, the authors utilize the Koopman generator to decompose the entropy production rate into intrinsic oscillatory modes, linking nonconservative forces to spectral properties.
Gaussian Analysis: The theoretical framework is illustrated using analytically tractable Gaussian systems with linear forces, allowing for explicit calculation of all derived quantities.

Key Contributions and Results

Geometric Decomposition of Information Flow: The paper introduces a rigorous decomposition of information flow in overdamped Langevin systems into excess and housekeeping components.
- Excess Information Flow ( $\dot{I}^{ex}$ ): This component is conservative and accounts for the time variation of mutual information. It vanishes in the steady state.
- Housekeeping Information Flow ( $\dot{I}^{hk}$ ): This component is nonconservative and maintains correlations between subsystems. It satisfies an antisymmetric relation ( $\dot{I}^{hk}_X = -\dot{I}^{hk}_Y$ ) even in non-steady states.
Generalized Second Laws: The authors derive generalized second laws of information thermodynamics for each component.
- $\sigma^{ex}_X \ge \dot{I}^{ex}_X$ and $\sigma^{hk}_X \ge \dot{I}^{hk}_X$ , where $\sigma$ represents the apparent entropy change rate.
- These inequalities imply that apparent violations of the second law (negative entropy change) can be compensated by negative information flow.
Excess and Housekeeping Demons: The paper formalizes two distinct types of "demons" based on the generalized second laws:
- Excess Demon: A subsystem acting as a demon via excess information flow. This phenomenon is transient, occurring only during non-steady dynamics, and is associated with conservative forces.
- Housekeeping Demon: A subsystem acting as a demon via housekeeping information flow. This can persist in the steady state and requires the presence of nonconservative forces. This corresponds to the concept of an autonomous demon.
Optimal Transport and Speed Limits: The local excess entropy production rate is shown to be the square of the generalized 2-Wasserstein speed of the marginal distribution. This leads to an information-thermodynamic speed limit for subsystems, bounding the time integral of excess entropy production by the squared generalized 2-Wasserstein distance between initial and final marginal distributions.
Uncertainty Relations and Fisher Information: The paper derives thermodynamic uncertainty relations for the partial excess and housekeeping entropy production rates. It establishes bounds on the apparent entropy change rates involving the conditional Fisher information matrix, showing that excessively large information flows can prohibit apparent second-law violations.
Koopman Mode Decomposition: The housekeeping entropy production rate is decomposed into contributions from Koopman eigenmodes, providing a spectral interpretation of nonconservative dissipation.

Significance
The paper claims to provide a unified geometric framework that connects information thermodynamics, stochastic thermodynamics, and optimal transport theory for continuous-state systems. By extending the excess/housekeeping decomposition to overdamped Langevin systems, the authors clarify the physical mechanisms behind information-driven entropy reduction. Specifically, the work distinguishes between transient phenomena (excess demons) and steady-state phenomena (housekeeping/autonomous demons), offering a more nuanced understanding of Maxwell's demon in different dynamical regimes. The formulation simplifies the optimal transport interpretation for Langevin systems compared to discrete Markov jump systems and establishes new trade-off relations (speed limits and uncertainty relations) that incorporate information flow. The results are illustrated analytically in the Gaussian case, demonstrating the feasibility of calculating these quantities and identifying the conditions under which the different types of demons emerge.

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