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arxiv: 2510.01787 · v2 · submitted 2025-10-02 · ❄️ cond-mat.stat-mech · math-ph· math.MP

Variational formulation of stochastic thermodynamics: Finite-dimensional systems

H\'ector Vaquero del Pino , Fran\c{c}ois Gay-Balmaz , Hiroaki Yoshimura , Lock Yue Chew This is my paper

Pith reviewed 2026-05-18 10:56 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MP
keywords stochastic thermodynamicsvariational principlesfluctuation-dissipation relationsnonholonomic constraintsentropy productionlocal detailed balanceLagrange-d'Alembert principle
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The pith

Requiring the second law yields a consistent variational structure for stochastic thermodynamics with natural fluctuation-dissipation relations

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

The paper develops a variational foundation for stochastic thermodynamics of finite-dimensional continuous-time systems. By requiring non-negative average total entropy production, it systematically constructs a thermodynamic structure using a generalized Lagrange-d'Alembert principle. Irreversible and stochastic forces are incorporated as nonlinear nonholonomic constraints, with entropy as an independent dynamical variable. This leads to novel generalized fluctuation-dissipation relations that ensure local detailed balance and extends to open and closed systems with state-dependent parameters and cross-correlated noise. A sympathetic reader would care because it offers a geometrically consistent way to model stochastic processes while respecting thermodynamic laws at the trajectory level.

Core claim

Requiring the second law (non-negative average total entropy production) systematically yields a consistent thermodynamic structure, from which novel generalized fluctuation-dissipation relations emerge naturally, ensuring local detailed balance. This principle extends key results of stochastic thermodynamics including an individual trajectory level description of both configurational and thermal variables and fluctuation theorems in an extended thermodynamic phase space. It applies to both closed and open systems, while accommodating state-dependent parameters, nonlinear couplings between configurational and thermal degrees of freedom, and cross-correlated noise consistent with Onsager symm

What carries the argument

Generalized Lagrange-d'Alembert principle incorporating irreversible and stochastic forces as nonlinear nonholonomic constraints, with entropy treated as an independent dynamical variable

If this is right

  • The framework applies to both closed and open systems with state-dependent parameters.
  • It accommodates nonlinear couplings between configurational and thermal degrees of freedom.
  • Cross-correlated noise is handled consistently with Onsager symmetry.
  • Fluctuation theorems hold in an extended thermodynamic phase space.
  • Individual trajectories for configurational and thermal variables receive a variational description.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same variational construction may supply a route to thermodynamically consistent models of active and complex fluids once extended beyond finite dimensions.
  • Preserving the underlying geometric structure could simplify the derivation of fluctuation relations in systems with strong nonlinear noise correlations.
  • Treating entropy as a dynamical variable from the start may generalize to other variational formulations of nonequilibrium processes.

Load-bearing premise

Irreversible and stochastic forces can be consistently incorporated as nonlinear nonholonomic constraints within the generalized Lagrange-d'Alembert principle while preserving the variational structure without inconsistencies in the entropy dynamics or fluctuation relations

What would settle it

A numerical simulation of a specific finite system with cross-correlated noise in which the derived generalized fluctuation-dissipation relations fail to hold or average entropy production becomes negative would falsify the central claim

Figures

Figures reproduced from arXiv: 2510.01787 by Fran\c{c}ois Gay-Balmaz, H\'ector Vaquero del Pino, Hiroaki Yoshimura, Lock Yue Chew.

Figure 1
Figure 1. Figure 1: FIG. 1. The system Ω := [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic of an interconnected system with two [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematic of a thermomechanical open system. (1) [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
read the original abstract

In this paper, we develop a variational foundation for stochastic thermodynamics of finite-dimensional, continuous-time systems. Requiring the second law (non-negative average total entropy production) systematically yields a consistent thermodynamic structure, from which novel generalized fluctuation-dissipation relations emerge naturally, ensuring local detailed balance. This principle extends key results of stochastic thermodynamics including an individual trajectory level description of both configurational and thermal variables and fluctuation theorems in an extended thermodynamic phase space. It applies to both closed and open systems, while accommodating state-dependent parameters, nonlinear couplings between configurational and thermal degrees of freedom, and cross-correlated noise consistent with Onsager symmetry. This is achieved by establishing a unified geometric framework in which stochastic thermodynamics emerges from a generalized Lagrange-d'Alembert principle, building on the variational structure introduced by Gay-Balmaz and Yoshimura [Phil. Trans. R. Soc. A 381, 2256 (2023)]. Irreversible and stochastic forces are incorporated through nonlinear nonholonomic constraints, with entropy treated as an independent dynamical variable. This work provides a novel approach for thermodynamically consistent modeling of stochastic systems, and paves the way to applications in continuum systems such as active and complex fluids.

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 paper develops a variational foundation for stochastic thermodynamics of finite-dimensional continuous-time systems. It imposes the second law (non-negative average total entropy production) through a generalized Lagrange-d'Alembert principle that extends the authors' 2023 variational structure, treating irreversible and stochastic forces as nonlinear nonholonomic constraints while keeping entropy as an independent dynamical variable. From this, the authors derive a consistent thermodynamic structure, local detailed balance, generalized fluctuation-dissipation relations, and fluctuation theorems in an extended thermodynamic phase space, applicable to closed and open systems with state-dependent parameters, nonlinear couplings, and cross-correlated noise.

Significance. If the central derivations hold without hidden assumptions on the stochastic calculus or noise, the framework offers a geometric route to thermodynamically consistent modeling of stochastic systems and could facilitate extensions to continuum active matter. The explicit emergence of generalized FDRs and trajectory-level descriptions from the variational principle would be a notable contribution to stochastic thermodynamics.

major comments (2)
  1. [Introduction and stochastic derivation section] The abstract and introduction assert that requiring non-negative average total entropy production 'systematically yields' the thermodynamic structure and local detailed balance, but the manuscript does not provide an explicit step-by-step verification that the nonlinear nonholonomic constraint forces alone enforce 〈dS_tot〉 ≥ 0 for arbitrary state-dependent diffusion coefficients or cross-correlated noise (see the stochastic extension in the main derivation section).
  2. [Section on generalized Lagrange-d'Alembert principle with stochastic constraints] The construction relies on the 2023 Gay-Balmaz-Yoshimura variational structure; the extension to stochastic forces must demonstrate that the chosen stochastic calculus convention (Itô/Stratonovich) does not introduce cross terms that could violate the second-law constraint for nonlinear couplings between configurational and thermal variables.
minor comments (2)
  1. [Notation and preliminaries] Notation for the extended thermodynamic phase space and the precise definition of the total entropy production functional should be introduced earlier and used consistently.
  2. [Results on fluctuation-dissipation relations] The manuscript would benefit from a short table comparing the new generalized FDRs with standard Onsager relations to highlight the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments, which help clarify the presentation of the variational derivation. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

  1. Referee: [Introduction and stochastic derivation section] The abstract and introduction assert that requiring non-negative average total entropy production 'systematically yields' the thermodynamic structure and local detailed balance, but the manuscript does not provide an explicit step-by-step verification that the nonlinear nonholonomic constraint forces alone enforce 〈dS_tot〉 ≥ 0 for arbitrary state-dependent diffusion coefficients or cross-correlated noise (see the stochastic extension in the main derivation section).

    Authors: We agree that an explicit verification would improve the clarity of the argument. Although the main derivation shows that the generalized Lagrange-d'Alembert principle with the nonlinear nonholonomic constraints enforces the second law, we will add a dedicated paragraph in the stochastic extension section that computes the average total entropy production explicitly. This calculation will treat general state-dependent diffusion matrices and cross-correlated noise terms (consistent with Onsager symmetry) and confirm that ⟨dS_tot⟩ ≥ 0 follows directly from the constraint forces without further assumptions. revision: yes

  2. Referee: [Section on generalized Lagrange-d'Alembert principle with stochastic constraints] The construction relies on the 2023 Gay-Balmaz-Yoshimura variational structure; the extension to stochastic forces must demonstrate that the chosen stochastic calculus convention (Itô/Stratonovich) does not introduce cross terms that could violate the second-law constraint for nonlinear couplings between configurational and thermal variables.

    Authors: We appreciate this observation on the stochastic interpretation. The framework is formulated in the Stratonovich sense to preserve the geometric structure and the chain rule, which is essential for treating entropy as an independent dynamical variable. In the revised manuscript we will insert a short explicit calculation (either in the main text or as a brief appendix) showing that, under this convention, no spurious cross terms arise from nonlinear couplings that would violate the non-negativity of average entropy production. We will also note the equivalent Itô representation and the associated correction terms for completeness. revision: yes

Circularity Check

1 steps flagged

Moderate circularity from self-cited 2023 variational framework as load-bearing foundation

specific steps
  1. self citation load bearing [Abstract]
    "This is achieved by establishing a unified geometric framework in which stochastic thermodynamics emerges from a generalized Lagrange-d'Alembert principle, building on the variational structure introduced by Gay-Balmaz and Yoshimura [Phil. Trans. R. Soc. A 381, 2256 (2023)]. Irreversible and stochastic forces are incorporated through nonlinear nonholonomic constraints, with entropy treated as an independent dynamical variable."

    The requirement of the second law is said to yield the full thermodynamic structure, but the mechanism for incorporating irreversible/stochastic forces as nonlinear nonholonomic constraints and preserving the variational structure is imported directly from the overlapping authors' 2023 paper rather than re-derived or independently justified here.

full rationale

The paper's core claim—that requiring non-negative average total entropy production via the generalized Lagrange-d'Alembert principle systematically derives the thermodynamic structure, local detailed balance, and generalized FDRs—explicitly builds on the authors' prior 2023 work for the underlying variational structure and constraint incorporation. This self-citation is load-bearing for the geometric framework but the extension to stochastic forces, entropy dynamics, and fluctuation relations adds independent content. No reduction of the final results to a pure fit or definition occurs; the derivation remains partially self-contained against external benchmarks once the 2023 foundation is granted. No other circular patterns (self-definitional, fitted predictions, or ansatz smuggling) are exhibited in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the geometric variational structure from the 2023 reference, the interpretation of the second law as a nonholonomic constraint, and standard assumptions of stochastic calculus for continuous-time finite-dimensional systems.

axioms (2)
  • domain assumption The second law requires non-negative average total entropy production
    Invoked to systematically generate the thermodynamic structure and local detailed balance.
  • ad hoc to paper Irreversible and stochastic forces can be represented as nonlinear nonholonomic constraints
    Central modeling choice that allows incorporation into the Lagrange-d'Alembert principle.

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