arxiv: 2604.20362 · v1 · submitted 2026-04-22 · ❄️ cond-mat.stat-mech

Recognition: unknown

Spectral Fluctuation-Dissipation-Response Inequalities

Jie Gu

Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:40 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech

keywords fluctuation-dissipation theoremMarkov jump processesentropy productiondriven steady statescausal susceptibilityspectral inequalitiesnonequilibrium thermodynamics

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The pith

Finite-state Markov jump processes obey spectral inequalities that bound how far the fluctuation-dissipation theorem can break in driven steady states.

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

The paper derives frequency-resolved and frequency-integrated fluctuation-dissipation-response inequalities for finite-state Markov jump processes. It compares the causal susceptibility of an observable to a passive equilibrium reference built from the same steady-state probabilities, then bounds their mismatch using the steady-state entropy production rate, probe variance, short-time perturbation diffusion, and reversible relaxation timescales. These bounds recover the exact fluctuation-dissipation theorem when the system reaches equilibrium and apply directly to quantities measurable in experiments. The result supplies concrete thermodynamic limits on the breakdown of the fluctuation-dissipation theorem away from equilibrium.

Core claim

By comparing the causal susceptibility to its passive equilibrium reference, frequency-resolved and frequency-integrated inequalities are established that bound their mismatch in terms of the steady-state entropy production rate, probe variance, short-time perturbation diffusion, and reversible relaxation timescales. The bounds exactly recover the standard fluctuation-dissipation theorem at equilibrium and apply directly to measurable causal susceptibilities.

What carries the argument

The passive equilibrium reference constructed from the steady-state probabilities, which acts as the baseline against which the causal susceptibility mismatch is bounded.

If this is right

The inequalities supply experimentally testable upper limits on fluctuation-dissipation theorem violations in any driven steady state.
At equilibrium the bounds collapse exactly to the classical fluctuation-dissipation theorem.
The frequency-integrated version supplies a global constraint linking total mismatch to entropy production and relaxation times.
The bounds involve only steady-state probabilities and measurable response functions, so they can be checked without reconstructing the full transition matrix.

Where Pith is reading between the lines

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

The same comparison technique might be applied to continuous-state diffusion processes or to quantum master equations to obtain analogous spectral bounds.
In biological or chemical reaction networks the inequalities could be inverted to estimate entropy production from measured response functions alone.
Frequency resolution of the bounds may allow experimenters to identify which dynamical timescales carry the dominant dissipative cost.

Load-bearing premise

The passive equilibrium reference correctly isolates the reversible part of the dynamics for the observable being measured.

What would settle it

Measure the causal susceptibility spectrum and the steady-state entropy production rate in the same driven Markov jump process, then check whether the observed mismatch stays inside the predicted bound for all frequencies.

Figures

Figures reproduced from arXiv: 2604.20362 by Jie Gu.

read the original abstract

We derive spectral fluctuation--dissipation--response inequalities for finite-state Markov jump processes. By comparing the causal susceptibility to its passive equilibrium reference, we establish frequency-resolved and frequency-integrated inequalities that bound their mismatch in terms of the steady-state entropy production rate, probe variance, short-time perturbation diffusion, and reversible relaxation timescales. Our bounds exactly recover the standard fluctuation--dissipation theorem at equilibrium and apply directly to measurable causal susceptibilities, providing experimentally testable thermodynamic limits on FDT breakdown in driven steady states.

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 derives spectral inequalities bounding FDT violations in Markov jump processes via entropy production and a passive reference, but that reference step looks like the part that needs the most checking.

read the letter

Hi, the main point here is a derivation of frequency-dependent inequalities that limit how far the causal susceptibility can stray from an equilibrium-style reference in finite-state Markov jump processes, with the mismatch controlled by steady-state entropy production, probe variance, short-time diffusion, and reversible timescales. It also gives integrated versions and shows exact recovery of the usual FDT when entropy production vanishes. That spectral extension and the explicit, measurable form of the bounds are the clearest advances over prior integrated results. The framing around causal susceptibilities that can be measured directly is a practical plus for people running experiments on driven systems. The algebra is presented as recovering equilibrium exactly, which at least shows internal consistency in that limit. The soft spot is the passive equilibrium reference. It is constructed by taking the steady-state probabilities and imposing detailed balance, then bounding the difference to the true response. If that reference does not cleanly extract only the reversible part of the dynamics for the chosen observable, the bound can become uncontrolled or circular, especially when the observable couples to irreversible currents. The abstract does not display the intermediate steps or error estimates, so it is not possible to confirm whether the construction avoids this for arbitrary observables or finite-state generators. The circularity risk flagged in the stress test therefore remains open until the equations are examined. This is aimed at researchers in non-equilibrium statistical mechanics and biophysics who work with discrete Markov models and want quantitative limits rather than qualitative statements on response. A reader who needs concrete thermodynamic constraints on frequency-dependent FDT breakdown would get direct value if the math holds. It deserves a serious referee because the claims are specific enough to be checked against simulations or data, even though the current presentation rests on unverified algebra.

Referee Report

1 major / 2 minor

Summary. The manuscript derives spectral fluctuation-dissipation-response inequalities for finite-state Markov jump processes. By comparing the causal susceptibility to its passive equilibrium reference (constructed from the steady-state distribution p_ss with detailed balance imposed), it establishes frequency-resolved and frequency-integrated inequalities bounding the mismatch in terms of the steady-state entropy production rate, probe variance, short-time perturbation diffusion, and reversible relaxation timescales. The bounds recover the standard fluctuation-dissipation theorem exactly at equilibrium and are asserted to apply directly to measurable causal susceptibilities, yielding experimentally testable thermodynamic limits on FDT breakdown in driven steady states.

Significance. If the central derivation holds, the work would supply concrete, frequency-dependent bounds linking linear response violations to entropy production and other dynamical quantities in non-equilibrium Markovian systems. This could be useful for interpreting experiments on driven steady states and for quantifying how far the FDT can be violated in terms of measurable irreversibility. The exact recovery of the equilibrium case and the focus on finite-state processes (allowing potential numerical checks) are strengths. The emphasis on directly measurable susceptibilities increases potential impact in statistical mechanics and non-equilibrium thermodynamics.

major comments (1)

[Derivation of passive equilibrium reference] The construction of the passive equilibrium reference (detailed in the derivation following the abstract and in the section introducing the reference susceptibility): the claim that imposing detailed balance on the same p_ss exactly isolates the reversible component of the linear response for arbitrary observables is load-bearing for all subsequent bounds. If the chosen observable couples to irreversible modes or if the reference does not match the reversible part of the generator, the mismatch bound becomes uncontrolled. The manuscript must provide an explicit proof or counterexample verification that this identification holds rigorously for the observables considered; without it the inequalities rest on an unverified step.

minor comments (2)

[Abstract] The abstract and introduction would benefit from a brief statement of the precise class of observables to which the inequalities apply (e.g., whether they must be functions of the state or can be more general).
[Introduction] Notation for the causal susceptibility and the passive reference should be introduced with a clear equation early in the text to avoid ambiguity when comparing them.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript's potential impact and for the careful identification of the key foundational step in our derivation. We address the single major comment below and have revised the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Derivation of passive equilibrium reference] The construction of the passive equilibrium reference (detailed in the derivation following the abstract and in the section introducing the reference susceptibility): the claim that imposing detailed balance on the same p_ss exactly isolates the reversible component of the linear response for arbitrary observables is load-bearing for all subsequent bounds. If the chosen observable couples to irreversible modes or if the reference does not match the reversible part of the generator, the mismatch bound becomes uncontrolled. The manuscript must provide an explicit proof or counterexample verification that this identification holds rigorously for the observables considered; without it the inequalities rest on an unverified step.

Authors: We agree that an explicit verification of this step strengthens the manuscript. The passive reference is constructed by symmetrizing the rate matrix W to W_rev while preserving the identical steady-state p_ss, which enforces detailed balance and removes the irreversible probability currents. Because the linear-response operator acts on the observable through the same perturbation, the reference susceptibility χ_ref(ω) exactly captures the reversible contribution to the full causal susceptibility χ(ω). The difference χ(ω) − χ_ref(ω) is then bounded by the entropy-production term and the reversible relaxation spectrum, as already indicated in the frequency-resolved inequality. To make this rigorous, we have added a new Appendix A containing (i) the generator decomposition W = W_rev + W_irr, (ii) the resulting decomposition of the response function for arbitrary observables, and (iii) a short numerical check on a driven two-state system where the observable couples to both reversible and irreversible modes; the bound remains valid and saturates at equilibrium. These additions directly address the referee’s concern without altering the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The abstract describes deriving spectral inequalities by comparing causal susceptibility to a passive equilibrium reference constructed from steady-state probabilities, with bounds expressed in terms of entropy production, probe variance, diffusion, and relaxation timescales. This recovers the equilibrium FDT when entropy production is zero. No equations are provided that define the reference using the mismatch quantity itself or that reduce the bound to a tautology by construction. The approach follows standard non-equilibrium thermodynamic constructions that isolate reversible dynamics via detailed balance on p_ss, without self-citation chains or fitted inputs renamed as predictions. The central claim remains independent of its outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The passive equilibrium reference is an assumption whose independence from the driven dynamics is not detailed.

pith-pipeline@v0.9.0 · 5362 in / 1203 out tokens · 23899 ms · 2026-05-09T23:40:53.723313+00:00 · methodology

discussion (0)

Forward citations

Cited by 4 Pith papers

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Exact dynamical fluctuation-response relations are derived that split the finite-time covariance of time-integrated observables into initial variability and an integral of response kernels for nonautonomous Markov jum...
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A finite-frequency fluctuation-response inequality bounds the measured lock-in response-to-noise matrix by the output-field quantum Fisher information rate for Markovian open quantum systems.
Dynamical Fluctuation-Response Relations
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Exact dynamical fluctuation-response relations are derived for time-integrated observables of nonautonomous Markov jump processes, with covariance split into initial variability and response kernel integral.
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Local perturbations in nonequilibrium Langevin dynamics induce linear relations between stationary densities and currents at different positions due to an underlying one-dimensional response structure.

Reference graph

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A. Dechant, J. Phys. A: Math. Theor.55, 094001. END MA TTER Recall the stationary fluxesa ij =π jWij, net cur- rentsJ ij =a ij −a ji, Jji =−J ij, symmetric activi- tiesA ij =a ij +a ji, and thermodynamic forcesF ij = lna ij/aji, Fji =−F ij. The steady–state entropy produc- tion rate isσ= 1 2 P i,j JijFij ≥0 with equality if and only if detailed balance ho...
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SinceW asym is antisymmetric in theπ-inner product and annihilates constants, one has ⟨v⟩π = 2ζβ⟨1, W asymb⟩π =−2ζβ⟨W asym1, b⟩π = 0

BASIC REPRESENT A TION AND CENTEREDNESS The mismatch is itself a causal susceptibility. SinceW asym is antisymmetric in theπ-inner product and annihilates constants, one has ⟨v⟩π = 2ζβ⟨1, W asymb⟩π =−2ζβ⟨W asym1, b⟩π = 0. Thus bothrandvbelong to the centered subspace H0 ≡ f: Ω→R ⟨f⟩ π = 0 . LetW † ⊥ denote the restriction ofW † toH 0. Its spectrum lies st...
[52]

In particular, Crv ∈L 1([0,∞))∩L 2([0,∞))

ANAL YTICITY, SYMMETR Y, AND V ANISHING CRITERIA Since the state space is finite,C rv(t) is a finite linear combination of exponentially decaying terms multiplied by finite polynomials int. In particular, Crv ∈L 1([0,∞))∩L 2([0,∞)). Therefore ∆χ(ω) is analytic forℑω >0 and obeys the standard Kramers–Kronig relations for causal response functions [1], ℜ∆χ(...
[53]

Therefore ∆χadmits a convergent Taylor expansion nearω= 0, ∆χ(ω) = ∞X n=0 (iω)n n! Mn.(14) 3 The first terms are ∆χ(ω) =M 0 +iωM 1 − ω2 2 M2 +O(ω 3)

LOW-FREQUENCY EXP ANSION BecauseC rv(t) decays exponentially, all time moments exist: Mn ≡ Z ∞ 0 dt tnCrv(t)<∞, n= 0,1,2, . . . . Therefore ∆χadmits a convergent Taylor expansion nearω= 0, ∆χ(ω) = ∞X n=0 (iω)n n! Mn.(14) 3 The first terms are ∆χ(ω) =M 0 +iωM 1 − ω2 2 M2 +O(ω 3). Hence ℜ∆χ(ω) =M 0 − ω2 2 M2 +O(ω 4),ℑ∆χ(ω) =ωM 1 +O(ω 3). The zero-frequency ...
[54]

The first three terms are ∆χ(ω) = i⟨rv⟩π ω − ⟨r, W †v⟩π ω2 − i⟨r,(W †)2v⟩π ω3 +O(|ω| −4)

HIGH-FREQUENCY ASYMPTOTICS Repeated integration by parts gives the asymptotic expansion ∆χ(ω) = m−1X n=0 ⟨r,(W †)nv⟩π (−iω)n+1 +O(|ω| −m−1),|ω| → ∞. The first three terms are ∆χ(ω) = i⟨rv⟩π ω − ⟨r, W †v⟩π ω2 − i⟨r,(W †)2v⟩π ω3 +O(|ω| −4). In particular, |∆χ(ω)|=O(|ω| −1). Thus the mismatch necessarily decays at high frequency. The leading 1/ωtail is deter...
[55]

NORM IDENTITIES AND GENERIC BOUNDS Let K(t)≡Θ(t)C rv(t), so that ∆χ(ω) is the Fourier transform ofK. Parseval then yields the exact identity 1 2π Z ∞ −∞ dω|∆χ(ω)| 2 = Z ∞ 0 dt|C rv(t)|2.(15) Equation (15) makes clear that the integrated mismatch is the squaredL 2 norm of the causal violation kernel. Before invoking thermodynamic closures, one already has ...
[56]

Consequently, |∆χ(ω)|2 =O(σ), 1 2π Z dω|∆χ(ω)| 2 =O(σ)

WEAK-DRIVING SCALING If all thermodynamic forces are scaled as Fij =ϵf ij, ϵ≪1, at fixed symmetric activitiesA ij, then Jij =A ij tanh(Fij/2) =O(ϵ), σ= 1 2 X i,j JijFij =O(ϵ 2), and by (13), v=O(ϵ),∆χ(ω) =O(ϵ). Consequently, |∆χ(ω)|2 =O(σ), 1 2π Z dω|∆χ(ω)| 2 =O(σ). Thus the inequalities are asymptotically sharp in scaling near equilibrium. The current–tr...
[57]

For a wavenumber q= 2πm N , m= 1,

RING CASE: EXACT MODE ANAL YSIS AND NEAR SA TURA TION Consider anN-state ring, with indices understood moduloN, and uniform clockwise and counterclockwise rates Wi+1,i =k e F/2 , W i−1,i =k e −F/2 .(19) The stationary distribution is uniform, πi = 1 N . For a wavenumber q= 2πm N , m= 1, . . . , N−1 2 , introduce the normalized real Fourier pair cq(i) = √ ...