Geometric excess-housekeeping decomposition of entropy production beyond overdamped diffusions: underdamped Langevin dynamics and Markov jump processes

For overdamped Langevin diffusions with constant diffusion matrix, the entropy production (EP) admits a geometric decomposition into excess and housekeeping parts via an orthogonal decomposition of the probability current, with variational expressions and an optimal-transport interpretation: the minimum EP required to change the state in a given time is a Wasserstein length. Dechant, Sasa, and Ito posed the generalization of this framework to underdamped (Kramers) dynamics and Markov jump processes, asking whether the Wasserstein-type lower bounds known in those settings can be identified with an excess EP analogous to the Maes-Netočný decomposition, and whether an orthogonal decomposition of the relevant current fields exists. The jump-process half has since been substantially resolved: orthogonal information-geometric and Hessian-geometric decompositions exist, and the Wasserstein identification holds under fixed dynamical activity or Onsager coefficients but fails in general — EP-minimizing forces on discrete networks are typically nonconservative, and a renormalized EP has been introduced and shown to be uniquely minimized by conservative forces. The underdamped half remains open: no general geometric excess-housekeeping decomposition of the phase-space current with variational expressions is established, the naive Wasserstein identification is known to fail (the phase-space optimal-transport bound is generally not tight), and only partial results exist for linear systems and for Hamiltonian-augmented metrics. The surviving open problem is the construction, or a no-go theorem, for the underdamped case.

Background

Stochastic thermodynamics assigns thermodynamic quantities such as heat, work, and entropy production to the trajectories of small systems whose dynamics are modeled as Markovian stochastic processes. The simplest setting is overdamped Langevin dynamics: the state $x_t$ of a system in a thermal bath at temperature $T$ obeys $\dot{x}=\mu F_t(x)+\sqrt{2\mu T}\,\xi_t$ with mobility $\mu$, force $F_t$, and Gaussian white noise $\xi_t$, and its probability density $p_t$ evolves by a Fokker-Planck equation. The entropy production (EP) rate quantifies the time-irreversibility of the dynamics, vanishing identically only when the dynamics are reversible at equilibrium; for this dynamics it is a quadratic form in the local mean velocity $\nu_t(x)=\mu(F_t(x)-T\nabla\ln p_t(x))$, the conditional deterministic drift at $x$. A system can be out of equilibrium in two qualitatively different ways: through time-dependent driving (relaxation) and through nonconservative forces that maintain steady circulating currents. Hatano and Sasa (2001), in steady-state thermodynamics, split the EP into an excess part (work stored in time-dependent driving) and a housekeeping part (dissipation sustaining a nonequilibrium steady state); Maes and Netočný gave a closely related decomposition in which the excess EP is the minimum EP compatible with the observed evolution of the state. Dechant, Sasa, and Ito (2022) gave these decompositions a geometric formulation for overdamped diffusions: the local mean velocity is split by orthogonal projection (with respect to a density-weighted inner product) into a gradient part and a solenoidal (divergence-free) part, giving the excess and housekeeping EPs, each with a variational (minimum/maximum) expression computable from trajectory data. In the same setting, optimal transport theory supplies a complementary statement: by the Benamou-Brenier formula, which recasts optimal transport as a dynamical mass-flow problem, the minimum total EP needed to carry $p_0$ to $p_\tau$ in time $\tau$ is proportional to the squared $L^2$-Wasserstein distance — the minimal quadratic cost of transporting one probability distribution onto another — between the two distributions, divided by the transition time, which is why the EP is bounded below by a Wasserstein length. The framework, however, is restricted to overdamped Fokker-Planck equations with constant diffusion matrix and natural or periodic boundary conditions. Two other workhorse model classes of stochastic thermodynamics lie outside it: underdamped Langevin (Kramers) dynamics, where the state includes velocities and the phase-space probability current splits into a reversible (Hamiltonian-like) part and an irreversible part; and Markov jump processes on discrete state spaces, where the current is a set of transition fluxes. For both classes, Wasserstein-type lower bounds on the total EP had been established (Dechant and Sakurai 2019 for diffusions; Van Vu and Hasegawa 2021 and Dechant 2022 for jump processes, where the relevant distance is built on graph distances and the dynamical activity, i.e., the overall rate of state transitions). The question posed by Dechant, Sasa, and Ito was whether the full geometric picture — orthogonal decomposition of the current fields, variational expressions for both EP parts, and an identification of the Wasserstein-type lower bound with the excess EP — extends to these classes; for jump processes it has since been answered in the affirmative in qualified, constraint-dependent form (summarized in the research objective above), while for underdamped Langevin dynamics it remains open.

Problem Statement

Establish, or refute, a geometric formulation of the excess-housekeeping decomposition of entropy production, with the associated variational expressions, for underdamped Langevin (Kramers) dynamics in phase space — the class of stochastic dynamics for which this framework is still missing — deciding whether an orthogonal decomposition of the phase-space probability current exists and whether a Wasserstein-type lower bound on the total entropy production can be identified with an excess entropy production analogous to the Maes-Netočný decomposition, as the source speculates. Concretely, for general underdamped Langevin dynamics with nonconservative forcing, construct an orthogonal decomposition of the phase-space probability current into excess and housekeeping parts together with variational expressions for each part, or show that no such decomposition exists in general. Because the non-tightness of the naive phase-space optimal-transport bound is established background (Sabbagh, Movilla Miangolarra, and Georgiou 2024) rather than a result to be reproduced, the identification question is not discharged by restating it: the open problem is whether any Wasserstein-type geometric structure — that is, any metric on phase-space probability densities induced by an optimal-transport cost (a transport metric rather than a divergence between densities), of the kind proposed for the overdamped case or augmented to incorporate Hamiltonian dynamics — can be identified with an excess entropy production for general underdamped dynamics with nonconservative forcing. This branch must be settled over the whole class of Wasserstein-type distances so delimited, in one of three explicit ways: exhibit at least one transport metric whose Wasserstein-type lower bound coincides with an excess entropy production; show that no metric in the class admits such an identification; or, when the identification holds for some transport metrics but not for others, delimit exactly which metrics of the class admit it and which do not. Reduction to the known overdamped decomposition in the large-friction limit and consistency with the known results for linear Gaussian (Ornstein-Uhlenbeck) dynamics are physically desirable checks on a candidate construction, not mandatory acceptance conditions. The corresponding questions for Markov jump processes have received substantive answers and are summarized as resolved background in the research objective above; they are not part of the completion requirement here. The objective is to determine whether the optimal-transport geometry of the second law is generic across stochastic model classes or a peculiarity of overdamped diffusions.

References

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