arxiv: 2605.07949 · v1 · submitted 2026-05-08 · ❄️ cond-mat.stat-mech

Recognition: no theorem link

Mutual Linearity in Nonequilibrium Langevin Dynamics

Jiming Zheng, Zhiyue Lu

Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:55 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords nonequilibrium dynamicsLangevin dynamicsstationary statesmutual linearityoverdamped systemsresponse theoryF1-ATPase

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The pith

In nonequilibrium overdamped Langevin dynamics, locally perturbing one dynamical parameter linearly relates stationary densities at any two positions.

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

This paper establishes that in continuous nonequilibrium systems governed by overdamped Langevin dynamics, a local perturbation to a dynamical parameter at one position produces a linear relationship between the stationary densities at any pair of positions. The same structure generates mutual linearity among different stationary state-current observables and carries over to relaxation dynamics when analyzed in the Laplace domain. The result traces to a shared one-dimensional response structure that also appears in discrete systems and stays intact for perturbations of finite but localized width, as shown by explicit calculation in the F1-ATPase rotary motor.

Core claim

When a dynamical parameter is locally perturbed at a single position in nonequilibrium overdamped Langevin systems, the stationary densities at any two positions are linearly related. This leads to mutual linearity among different stationary state-current observables. The theory extends the mutual linearity to non-stationary relaxation processes in the Laplace domain, revealing that it originates from the same one-dimensional response structure as in discrete systems, and demonstrates robustness under finite-width perturbations with an application to the F1-ATPase rotary motor model.

What carries the argument

The mutual linearity relation, which states that stationary densities at distinct positions remain proportional under a localized parameter perturbation and thereby connects multiple current observables through the underlying one-dimensional response structure.

If this is right

The framework supplies a direct rule for controlling and designing nonequilibrium responses in continuous systems.
Mutual linearity connects any pair of stationary state-current observables without additional calculation.
The linear relation extends from stationary states to relaxation processes when examined in the Laplace domain.
The relation remains valid for perturbations that have finite but still localized width.
The same linearity appears in the F1-ATPase rotary motor model and can be used to predict its response.

Where Pith is reading between the lines

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

If the relation is general, it may let researchers predict responses in complex continuous systems by measuring densities or currents at only two locations rather than solving the full equation everywhere.
The one-dimensional structure identified here could be checked for signatures in higher-dimensional or multi-particle nonequilibrium models.
Because the linearity survives finite-width perturbations, it may apply to experimental setups where perfect point perturbations are impossible.

Load-bearing premise

The dynamics must be overdamped Langevin with a unique stationary state, and any perturbation must be applied locally at one position.

What would settle it

Solve the Fokker-Planck equation for a one-dimensional overdamped Langevin system, apply a local perturbation of varying strength at position x, and test whether the ratio of stationary densities at positions y and z stays exactly constant.

Figures

Figures reproduced from arXiv: 2605.07949 by Jiming Zheng, Zhiyue Lu.

**Figure 2.** Figure 2: Non-stationary mutual linearity in the Laplace do [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Complete steady-state mutual-linearity tests for the F [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

read the original abstract

Understanding how nonequilibrium systems respond to perturbations is a central challenge in physics. In this work, we establish mutual linearity in nonequilibrium overdamped Langevin systems. This theory provides a framework for controlling and designing nonequilibrium responses in continuous systems. When a dynamical parameter is locally perturbed at a single position, the stationary densities at any two positions are linearly related. It further leads to mutual linearity among different stationary state-current observables. We also extend the mutual linearity to non-stationary relaxation processes in the Laplace domain. Our theory reveals that mutual linearity in both discrete and continuous systems originates from the same one-dimensional response structure. We further show that mutual linearity is robust under finite-width perturbations. As an application, we demonstrate the mutual linearity and its finite-width robustness in the F$_1$-ATPase rotary motor model.

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.

Desk Editor's Note private letter to a colleague

The paper shows mutual linearity of stationary densities under local perturbations in 1D overdamped Langevin systems, derived from the constant-current Fokker-Planck equation.

read the letter

The key point is that local perturbations in one-dimensional overdamped Langevin dynamics produce linear relations between stationary densities at different locations because the probability current is constant in the stationary state. This leads to mutual linearity among observables and carries over to the Laplace domain for time-dependent relaxation. The paper does a good job laying out how this comes from the one-dimensional response structure, which is the same origin as in discrete Markov chains. They derive the explicit linear map by integrating the Fokker-Planck equation across the perturbation region and show robustness when the perturbation has finite width rather than being a delta function. The numerical demonstration on the F1-ATPase rotary motor model serves as a practical check that the relations hold in a realistic biophysical setting. On the downside, everything is confined to one dimension. In higher dimensions the current is no longer a scalar that can be integrated so simply, so the result does not generalize immediately. The work assumes a well-defined stationary state exists, which is standard but worth noting for systems that might not relax. No obvious circularity or fitting parameters appear in the construction. This is aimed at people in statistical mechanics and biophysics who deal with continuous nonequilibrium models, particularly molecular machines. A reader looking for practical ways to predict responses to localized changes will get value from the explicit formulas. It is solid enough on its own terms to merit peer review, even if the scope is narrow. I would send it out for refereeing.

Referee Report

0 major / 2 minor

Summary. The manuscript establishes mutual linearity in nonequilibrium overdamped Langevin dynamics. When a dynamical parameter is locally perturbed at a single position, the stationary densities at any two positions are linearly related; this follows from integrating the first-order stationary Fokker-Planck equation (constant current J) under a localized drift perturbation. The result extends to mutual linearity among different stationary state-current observables, to non-stationary relaxation processes in the Laplace domain, and remains valid for finite-width perturbations by continuity. The same one-dimensional response structure is shown to underlie mutual linearity in both discrete and continuous systems. The theory is illustrated by a direct numerical check in the F1-ATPase rotary motor model.

Significance. If the central derivation holds, the work supplies a simple, algebraically explicit framework for controlling and designing nonequilibrium responses in continuous systems. The unification of discrete and continuous cases through the shared one-dimensional structure, the parameter-free character obtained by direct integration of the Fokker-Planck ODE, and the explicit robustness proof for finite-width perturbations are notable strengths. The numerical demonstration on the biologically relevant F1-ATPase model adds practical value and falsifiability.

minor comments (2)

[Abstract] The abstract states the main results but does not indicate the explicit integration step that yields the linear relation; a single sentence referencing the constant-current Fokker-Planck structure would improve immediate clarity.
[Application section] In the F1-ATPase application, the specific dynamical parameter that is perturbed and the precise definition of the state-current observables should be stated explicitly in the text (rather than only in a figure caption) to make the numerical check fully reproducible from the manuscript alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript and for recommending acceptance. We are pleased that the referee recognized the value of the one-dimensional response structure underlying mutual linearity, its extension to discrete and continuous systems, the robustness to finite-width perturbations, and the numerical illustration with the F1-ATPase model.

Circularity Check

0 steps flagged

No circularity: derivation is direct integration of 1D stationary Fokker-Planck under localized perturbation

full rationale

The paper derives mutual linearity by integrating the first-order ODE obtained from the stationary Fokker-Planck equation (constant probability current J) when the drift perturbation is confined to a single point or finite interval. This algebraic step produces an explicit linear relation between stationary densities at arbitrary positions without invoking fitted parameters, self-definitions, or prior results from the same authors. The extension to Laplace-domain relaxation and finite-width cases follows by continuity of the integrating factor. The F1-ATPase example is a numerical verification of the same 1D construction. No load-bearing step reduces to a tautology or self-citation chain; the result is a straightforward consequence of the 1D continuity equation and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the mathematical structure of the overdamped Langevin equation and the assumption of a stationary state under local perturbations; no free parameters, invented entities, or additional axioms are visible in the abstract.

axioms (2)

domain assumption The system obeys overdamped Langevin dynamics
Explicitly stated as the setting for the theory in the abstract.
domain assumption A stationary state exists
Required for the definition of stationary densities and currents.

pith-pipeline@v0.9.0 · 5428 in / 1294 out tokens · 40408 ms · 2026-05-11T02:55:35.972042+00:00 · methodology

discussion (0)

Reference graph

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