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arxiv: 2601.16114 · v2 · pith:ZR6PCDQKnew · submitted 2026-01-22 · ❄️ cond-mat.stat-mech

Langevin equations with non-Gaussian thermal noise: Valid but superfluous

Alex V. Plyukhin This is my paper

Pith reviewed 2026-05-21 15:03 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords generalized Langevin equationnon-Gaussian noiseJarzynski equalityBrownian oscillatoradditive noisefinite-time thermodynamicsperturbative expansion
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The pith

The generalized Langevin equation with non-Gaussian noise satisfies the Jarzynski equality only up to seventh order in the pulse duration τ.

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

The paper checks the consistency of the generalized Langevin equation with linear dissipation and additive noise against the Jarzynski equality at finite times for a driven Brownian oscillator, without assuming ergodicity. A perturbative expansion in the duration τ of a rectangular pulse perturbation shows that the equality holds for any noise distribution through order seven. Beyond that order the equality requires Gaussian noise. A reader would care because the result limits the practical scope of such equations to quantities that are linear or quadratic in the noise, which do not depend on its higher statistics.

Core claim

The author considers a classical Brownian oscillator whose initial stiffness is perturbed by a rectangular pulse of duration τ and derives the statistics of work from the generalized Langevin equation. The key result is that the Jarzynski equality is satisfied unconditionally only up to the seventh order in τ; in higher orders the equality holds if and only if the noise is Gaussian. These results imply that, unless it is exact, the Langevin equation can only be used to evaluate properties that are linear or quadratic in noise and its derivatives, properties that are insensitive to the noise statistics.

What carries the argument

perturbative expansion in powers of the rectangular-pulse duration τ applied to the Jarzynski equality for solutions of the generalized Langevin equation with additive noise

If this is right

  • The equation with non-Gaussian noise remains consistent with the Jarzynski equality for any quantity linear or quadratic in the noise.
  • Properties insensitive to noise statistics can be computed without regard to whether the noise is Gaussian.
  • The equation becomes superfluous for general nonequilibrium calculations unless the noise distribution is exactly Gaussian.
  • Long-time equilibrium behavior alone does not guarantee finite-time consistency with fluctuation theorems.

Where Pith is reading between the lines

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

  • The same order limitation may appear in other nonequilibrium equalities that involve higher moments of work or heat.
  • Exact non-Markovian models with controlled non-Gaussian noise could be constructed to capture effects beyond seventh order.
  • Direct trajectory simulations with Poisson or Lévy noise would locate the precise τ value at which deviations from the Jarzynski equality first appear.

Load-bearing premise

The system is a classical Brownian oscillator obeying the generalized Langevin equation with linear dissipation and additive noise driven by a rectangular pulse of duration τ, allowing a power-series expansion without assuming ergodicity.

What would settle it

A numerical simulation of the oscillator trajectories with a chosen non-Gaussian noise distribution at a pulse duration where eighth-order terms become appreciable, checking whether the ensemble average of exp(−βW) still equals the free-energy difference.

read the original abstract

We discuss the statistics of additive thermal (internal) noise in systems governed by the generalized Langevin equation with linear dissipation. To assess the equation's validity, it is common to assume that the system is ergodic and to verify that solutions approach correct equilibrium values at asymptotically long times. In this paper, we instead consider the consistency of the generalized Langevin equation with the Jarzynski equality at finite times and do not assume the system's ergodicity. Specifically, we consider a classical Brownian oscillator whose initial stiffness, or frequency, is perturbed by a rectangular pulse of duration $\tau$. We find that the Jarzynski equality is satisfied unconditionally only up to the seventh order in $\tau$; in higher orders, the Jarzynski equality holds if and only if the noise is Gaussian. These results imply that, unless it is exact, the Langevin equation can only be used to evaluate properties that are linear or quadratic in noise and its derivatives. Such properties are insensitive to the noise statistics, so the Langevin equation with linear dissipation and non-Gaussian noise (though not inconsistent by itself) is superfluous.

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 manuscript considers a classical Brownian oscillator obeying the generalized Langevin equation with linear dissipation and additive (possibly non-Gaussian) noise. It perturbs the oscillator stiffness with a rectangular pulse of duration τ and expands the Jarzynski average ⟨exp(−βW)⟩ in powers of τ without assuming ergodicity. The central claim is that this average equals the equilibrium value unconditionally only through O(τ^7); at O(τ^8) and higher the equality holds if and only if all cumulants beyond the second vanish, i.e., the noise is Gaussian. The authors conclude that the generalized Langevin equation with non-Gaussian noise is formally valid but superfluous for any observable linear or quadratic in the noise.

Significance. If the perturbative cancellations are correct, the result supplies a concrete, finite-time, non-ergodic consistency check between the generalized Langevin equation and the Jarzynski equality. It isolates the precise order at which higher cumulants first contribute, thereby furnishing a parameter-free derivation that non-Gaussian statistics are invisible to all work-related quantities up to seventh order in the pulse duration. This strengthens the case for using Gaussian noise in linear stochastic-thermodynamic models while clarifying the domain of applicability of the equation itself.

major comments (2)
  1. [perturbative expansion of the Jarzynski average (following the model definition)] The identification of unconditional validity through O(τ^7) and the first appearance of fourth-and-higher cumulants at O(τ^8) is the load-bearing step for the entire claim. The expansion of ⟨exp(−βW)⟩ involves multi-time integrals over the noise cumulants, the memory kernel, and the Green's function of the linear oscillator; the manuscript provides no intermediate expressions or auxiliary verification that all non-Gaussian contributions cancel through seventh order.
  2. [derivation of the order-by-order contributions] The rectangular-pulse protocol and the quadratic form of the work W are used to generate the time integrals; any algebraic or combinatorial error in those integrals would shift the reported threshold order. Because the central conclusion (“valid but superfluous”) rests on the exact location of this threshold, the absence of explicit cancellation bookkeeping constitutes a verification gap.
minor comments (2)
  1. Notation for the cumulant-generating function and the precise definition of the work functional W could be stated once in a single equation block to reduce cross-referencing.
  2. [abstract] The abstract states the result for “higher orders” without naming the eighth-order threshold; adding the explicit order would improve immediate clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their detailed and constructive feedback, which highlights important aspects of our perturbative analysis. We respond to the major comments point by point below and commit to revisions that enhance the transparency of our derivations.

read point-by-point responses

  1. Referee: The identification of unconditional validity through O(τ^7) and the first appearance of fourth-and-higher cumulants at O(τ^8) is the load-bearing step for the entire claim. The expansion of ⟨exp(−βW)⟩ involves multi-time integrals over the noise cumulants, the memory kernel, and the Green's function of the linear oscillator; the manuscript provides no intermediate expressions or auxiliary verification that all non-Gaussian contributions cancel through seventh order.

    Authors: The referee is correct that the manuscript would benefit from more explicit intermediate steps to facilitate verification of the cancellations. Our calculation relies on the fact that the work is a quadratic functional of the trajectory, which is itself a linear functional of the noise. Consequently, the Jarzynski average is the moment-generating function evaluated at the appropriate kernel. For the rectangular pulse, symmetry properties of the Green's function ensure that all contributions involving cumulants of order greater than 2 vanish identically for total orders less than 8 in τ. We will add an appendix detailing the leading terms at each order in the revised manuscript to address this concern. revision: yes

  2. Referee: The rectangular-pulse protocol and the quadratic form of the work W are used to generate the time integrals; any algebraic or combinatorial error in those integrals would shift the reported threshold order. Because the central conclusion (“valid but superfluous”) rests on the exact location of this threshold, the absence of explicit cancellation bookkeeping constitutes a verification gap.

    Authors: We acknowledge the potential for algebraic errors in the multi-time integrals and agree that explicit bookkeeping is desirable. In the revision, we will include a table or systematic listing of the integral contributions order by order, showing how non-Gaussian terms cancel up to O(τ^7) and appear at O(τ^8). This will allow independent checking of the threshold. The central claim remains unchanged as it follows directly from the structure of the linear system. revision: yes

Circularity Check

0 steps flagged

No circularity: direct perturbative expansion of Jarzynski equality

full rationale

The paper derives its central result—that the Jarzynski equality holds unconditionally only through O(τ^7) and requires Gaussian noise thereafter—via an explicit power-series expansion in the rectangular-pulse duration τ applied to the generalized Langevin equation for a classical Brownian oscillator. The expansion of ⟨exp(−βW)⟩ produces multi-time integrals over the noise cumulants; the claimed cancellations of fourth-and-higher cumulants through seventh order follow directly from the linearity of the solution x(t) in the additive noise and the quadratic form of the work W, without any fitted parameters, self-definitional relations, or load-bearing self-citations. The analysis is self-contained against the stated model assumptions and does not invoke prior author results or uniqueness theorems to force the order threshold.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions of classical statistical mechanics together with the specific modeling choice of a rectangular pulse; no new physical entities are postulated and no parameters are fitted to data.

axioms (2)
  • domain assumption The system is governed by the generalized Langevin equation with linear dissipation and additive thermal noise.
    This is the governing dynamical equation stated at the outset of the analysis.
  • domain assumption The Jarzynski equality provides a valid finite-time consistency condition independent of ergodicity.
    Invoked explicitly as the test that replaces the usual long-time equilibrium check.

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