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arxiv: 2604.22371 · v1 · submitted 2026-04-24 · ❄️ cond-mat.soft

Covariant Onsager and Onsager-Machlup principles for active and inertial dynamics

Kento Yasuda , Bin Zheng , Zhongqiang Xiong , Zhanglin Hou , Kenta Ishimoto , Xinpeng Xu , David Andelman , Shigeyuki Komura This is my paper

Pith reviewed 2026-05-08 09:29 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords Onsager principleOnsager-Machlup functionalactive matterstochastic thermodynamicscovariant formulationinertial dynamicsdetailed fluctuation theoremvariational principles
0
0 comments X p. Extension

The pith

Covariant Onsager-Machlup principle ensures thermodynamic consistency for active inertial dynamics.

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

The paper develops a geometrically invariant version of the Onsager variational principle for dissipative dynamics in active systems. It extends the approach to stochastic trajectories through the Onsager-Machlup functional and shows that enforcing the detailed fluctuation theorem on path probabilities produces results consistent with stochastic thermodynamics. The framework adds inertia by performing the variation while holding covariant acceleration fixed, which generates the appropriate covariant equations of motion. A sympathetic reader would care because this supplies a single variational route to phenomenological equations that remain valid under coordinate changes and respect nonequilibrium thermodynamic relations.

Core claim

Requiring that the path probability obeys the detailed fluctuation theorem, we show that the extended Onsager-Machlup theory is consistent with stochastic thermodynamics. Moreover, we incorporate inertia into the variational framework and show that the proper covariant equations follow when the covariant acceleration is held fixed during the variation.

What carries the argument

The covariant Onsager-Machlup functional, whose minimization subject to the detailed fluctuation theorem and fixed covariant acceleration produces the governing equations.

If this is right

  • Phenomenological equations for active dissipative systems are obtained by minimizing a covariant Rayleighian.
  • Fluctuating path probabilities automatically satisfy thermodynamic relations such as the detailed fluctuation theorem.
  • Inertial terms enter the dynamics while geometric covariance under coordinate transformations is preserved.
  • The same variational structure unifies deterministic active dynamics, fluctuations, and inertia.

Where Pith is reading between the lines

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

  • The same covariance requirement could be applied to derive consistent equations in other dissipative systems that live on manifolds.
  • Path-sampling algorithms built on this functional might generate trajectories that automatically obey both activity and inertia constraints.
  • The approach suggests a route to variational formulations for active particles in curved geometries or under external flows.

Load-bearing premise

The path probability for stochastic trajectories is required to obey the detailed fluctuation theorem in order to establish consistency with stochastic thermodynamics.

What would settle it

Direct numerical sampling of trajectory probabilities from the underlying active inertial stochastic differential equations and explicit verification that they satisfy the detailed fluctuation theorem predicted by the variational principle.

Figures

Figures reproduced from arXiv: 2604.22371 by Bin Zheng, David Andelman, Kenta Ishimoto, Kento Yasuda, Shigeyuki Komura, Xinpeng Xu, Zhanglin Hou, Zhongqiang Xiong.

Figure 1
Figure 1. Figure 1: FIG. 1. Entropy balance for a system coupled to a heat bath view at source ↗
read the original abstract

The Onsager principle provides a variational route to the phenomenological equations of dissipative dynamics through the minimization of the Rayleighian. We develop a covariant formulation of the Onsager principle for active systems, ensuring geometric consistency under coordinate transformations. To further incorporate thermal fluctuations, we formulate the Onsager-Machlup principle for active systems by considering the Onsager-Machlup functional and the corresponding path probability for stochastic trajectories. Requiring that the path probability obeys the detailed fluctuation theorem, we show that the extended Onsager-Machlup theory is consistent with stochastic thermodynamics. Moreover, we incorporate inertia into the variational framework and show that the proper covariant equations follow when the covariant acceleration is held fixed during the variation.

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 / 1 minor

Summary. The manuscript develops a covariant formulation of the Onsager principle for active systems via minimization of a Rayleighian, extends this to an Onsager-Machlup functional whose associated path measure encodes thermal fluctuations, and asserts consistency with stochastic thermodynamics by requiring that the path probability satisfy the detailed fluctuation theorem. It further incorporates inertia by performing variations while holding the covariant acceleration fixed, claiming that this yields the correct covariant inertial equations.

Significance. A self-consistent covariant variational framework that unifies active, fluctuating, and inertial dynamics would be valuable for geometric treatments of stochastic thermodynamics in soft matter. The work's potential impact is reduced by the fact that consistency is obtained through an imposed external constraint (the fluctuation theorem) rather than derived from the variational principle itself; if this imposition is not tautological, the result could provide a useful route to fluctuation theorems in active systems.

major comments (2)
  1. [Abstract / Onsager-Machlup derivation] Abstract and the section deriving consistency with stochastic thermodynamics: the central claim that 'requiring that the path probability obeys the detailed fluctuation theorem' establishes consistency is load-bearing, yet the manuscript does not demonstrate that the Onsager-Machlup functional is selected independently of this requirement. If the functional is constructed so that P[forward]/P[reverse] = exp(ΔS) holds identically, the consistency follows by construction rather than from covariance or the variational principle; an explicit check that the functional emerges first and then satisfies the theorem (or a counter-example where it fails) is needed.
  2. [Inertial dynamics section] The inertial extension (section on incorporation of inertia): holding the covariant acceleration fixed during the variation is presented as yielding the 'proper covariant equations,' but no explicit comparison is given to the standard covariant Langevin equation or to the known form of inertial active Brownian motion. Without this verification, it is unclear whether additional geometric terms are omitted or introduced by the fixed-acceleration constraint.
minor comments (1)
  1. [Notation and definitions] Notation for the covariant acceleration and the precise definition of the Rayleighian under coordinate transformations should be stated explicitly in the main text rather than deferred to appendices, to allow immediate verification of covariance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comments point by point below and outline the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: [Abstract / Onsager-Machlup derivation] Abstract and the section deriving consistency with stochastic thermodynamics: the central claim that 'requiring that the path probability obeys the detailed fluctuation theorem' establishes consistency is load-bearing, yet the manuscript does not demonstrate that the Onsager-Machlup functional is selected independently of this requirement. If the functional is constructed so that P[forward]/P[reverse] = exp(ΔS) holds identically, the consistency follows by construction rather than from covariance or the variational principle; an explicit check that the functional emerges first and then satisfies the theorem (or a counter-example where it fails) is needed.

    Authors: We agree that the independence of the functional derivation from the fluctuation theorem requirement needs to be clarified. The Onsager-Machlup functional is obtained by extending the covariant Rayleighian minimization to include stochastic paths, based solely on the geometric and variational principles for active dynamics. The detailed fluctuation theorem is then imposed as a consistency condition with stochastic thermodynamics, which selects the appropriate noise correlator or confirms the form. This is not by construction in the sense that the variational principle could in principle yield other functionals, but the theorem ensures thermodynamic consistency. We will revise the manuscript to include an explicit step-by-step derivation showing the functional prior to the theorem application, and provide a brief discussion of what would happen if the theorem were not satisfied. This addresses the concern directly. revision: yes

  2. Referee: [Inertial dynamics section] The inertial extension (section on incorporation of inertia): holding the covariant acceleration fixed during the variation is presented as yielding the 'proper covariant equations,' but no explicit comparison is given to the standard covariant Langevin equation or to the known form of inertial active Brownian motion. Without this verification, it is unclear whether additional geometric terms are omitted or introduced by the fixed-acceleration constraint.

    Authors: The referee correctly identifies that an explicit verification against known equations is necessary. In our approach, fixing the covariant acceleration during the variation ensures that the inertial terms transform correctly as a tensor, avoiding non-covariant artifacts. To demonstrate this, we will add a new subsection comparing our derived equations to the standard covariant inertial Langevin equation for active particles (e.g., the one with covariant derivatives and appropriate noise terms for active Brownian motion with inertia). We show that they coincide, with the fixed-acceleration constraint preventing the introduction of spurious geometric terms that would arise from naive variations. This verification will be included in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: external fluctuation theorem used to validate consistency, independent of variational construction.

full rationale

The derivation begins from the standard Onsager variational principle, extends it covariantly for active systems, and formulates the Onsager-Machlup functional for path probabilities. Consistency with stochastic thermodynamics is established by imposing the detailed fluctuation theorem as an external requirement on the path probability, rather than deriving the theorem from the functional or fitting parameters to force it. The inertial extension follows from holding covariant acceleration fixed in the variation. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the chain; the DFT is a standard external benchmark from stochastic thermodynamics, not constructed within the paper's ansatz.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Based solely on the abstract, the central claims rest on the standard variational structure of the Onsager principle, the existence of a Rayleighian for active systems, the validity of the detailed fluctuation theorem as an external constraint, and the definition of covariant acceleration. No explicit free parameters or new invented entities are stated.

axioms (3)
  • domain assumption The Onsager principle applies to active dissipative systems via minimization of a suitably defined Rayleighian.
    Invoked as the starting point for the covariant extension.
  • domain assumption Path probabilities for stochastic trajectories must obey the detailed fluctuation theorem.
    Used to enforce consistency with stochastic thermodynamics.
  • domain assumption Covariant acceleration is the appropriate quantity to hold fixed during variation when including inertia.
    Stated as the condition that yields the proper covariant equations.

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