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

TL;DR

The paper addresses how information flow between interacting subsystems can be decomposed geometrically in nonequilibrium overdamped Langevin systems and connects this decomposition to optimal transport via the $2$-Wasserstein distance. It develops a comprehensive framework: a conservative excess information flow and a nonconservative housekeeping information flow, with generalized second laws, thermodynamic uncertainty relations, Koopman-mode analyses, variational (Benamou–Brenier) formulations, and an information-geometric speed limit, all tied to an optimal-transport interpretation for subsystems. The authors provide explicit results for Gaussian dynamics and illustrate transient and steady-state demon-like phenomena (excess and housekeeping demons) through both analytic and numerical examples. The framework offers a unified geometric view of subsystem thermodynamics and information exchange, with potential applications to nanoscale control, diffusion-based models, and diffusion-model training where optimal transport metrics naturally arise.

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.

Figures (4)

• Figure 1: (a) Schematic showing the geometric decomposition of information flow. Excess information flow represents a conservative contribution and can vary the correlation. Housekeeping information flow represents a nonconservative contribution and maintains the correlation. (b) Schematic showing the excess demon and the housekeeping demon. The excess demon in system $\rm Y$ uses excess information flow $\dot{I}^{\rm ex;X}_t$ to make the apparent excess entropy change rate $\sigma^{\rm ex;X}_t$ in system $\rm X$ negative. The housekeeping demon in system $\rm Y$ uses housekeeping information flow $\dot{I}^{\rm hk;X}_t$ to make the apparent housekeeping entropy change rate $\sigma^{\rm hk;X}_t$ in system $\rm X$ negative.
• Figure 2: The probability distributions and streamlines for (a) $r=0.1$ and (b) $r=-1$. (Left) The initial distribution for $(v_+, v_-) =(1, 1/2)$ and its streamlines. (Center) The initial distribution for $(v_+, v_-) =(1/2, 1)$ and its streamlines. (Right) The steady-state distribution and its streamlines.
• Figure 3: (a) Time evolution of $\sigma_t^{\mathrm{ex};\mathrm{X}}$ and $\dot{I}_t^{\mathrm{ex};\mathrm{X}}$ for $r=-1.0$. (b) Minimum value of $\sigma_t^{\mathrm{ex};\mathrm{X}}$ over time $t$.
• Figure 4: (a,b) Time evolution of $\sigma_t^{\mathrm{hk};\mathrm{X}}$ and $\dot{I}_t^{\mathrm{hk};\mathrm{X}}$ for (a) $r=-0.1$ and (b) $r=0.1$. (c) Minimum value of $\sigma_t^{\mathrm{hk};\mathrm{X}}$ over time $t$.