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arxiv: 2605.05038 · v1 · submitted 2026-05-06 · ❄️ cond-mat.stat-mech

Recognition: unknown

Nonequilibrium Fluctuation-Response Theory in the Frequency Domain

Euijoon Kwon , Hyun-Myung Chun , Hyunggyu Park , Jae Sung Lee
Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:18 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords nonequilibrium steady statesfluctuation-response relationsfrequency domainpower spectrumuncertainty relationsLangevin dynamicsMarkov jump processes
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The pith

The power spectrum of observables in nonequilibrium steady states equals a quadratic form of local responses at the same frequency.

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

The paper establishes a single identity that writes the power spectral density of any observable exactly as a quadratic combination of local response functions evaluated at the identical frequency. This identity holds for overdamped Langevin dynamics and for Markov jump processes in nonequilibrium steady states, and it applies to state-dependent observables, current observables, and mixtures of the two. A reader would care because the same relation immediately yields frequency-domain uncertainty relations, recovers the equilibrium fluctuation-dissipation theorem, and produces Harada-Sasa-type equalities without additional assumptions.

Core claim

For systems in nonequilibrium steady states governed by overdamped Langevin dynamics or Markov jump processes, the power spectrum of a general observable is expressed exactly as a quadratic form of the local responses measured at the same frequency; the decomposition is spatial for Langevin systems and edge-resolved for jump processes.

What carries the argument

The frequency-domain fluctuation-response identity, which decomposes the power spectrum into a quadratic form of frequency-matched local responses.

If this is right

  • Response uncertainty relations hold at each frequency separately.
  • Kinetic and thermodynamic uncertainty relations follow directly from the same identity.
  • The equilibrium fluctuation-dissipation theorem is recovered as a special case.
  • Harada-Sasa-type relations appear as immediate corollaries.
  • Fluctuation spectra in stochastic networks or driven diffusive systems resolve into explicit edge-wise or spatial contributions.

Where Pith is reading between the lines

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

  • The relation suggests that frequency-dependent trade-offs between fluctuations, response speed, and dissipation can be read directly from measurable spectra without full knowledge of the underlying rates or forces.
  • It may allow experimental estimation of hidden dissipation rates in biological or colloidal systems by combining fluctuation spectra with local response measurements at selected frequencies.
  • Similar decompositions, if they exist, could extend the framework to underdamped or non-Markovian dynamics.

Load-bearing premise

The systems must obey overdamped Langevin dynamics or Markov jump processes and sit in a nonequilibrium steady state.

What would settle it

A direct measurement, in an overdamped Langevin or Markov-jump system, showing that the observed power spectrum at a chosen frequency deviates from the quadratic combination of the independently measured local responses at that same frequency.

Figures

Figures reproduced from arXiv: 2605.05038 by Euijoon Kwon, Hyunggyu Park, Hyun-Myung Chun, Jae Sung Lee.

Figure 1
Figure 1. Figure 1: FIG. 1. A hierarchy of the relations derived in this work. The frequency-domain FRR sits at the top of the hierarchy as the view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustration of the frequency-domain FRR for a unicyclic network [Eq. ( view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. A numerical illustration of the frequency-domain view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Illustration of the frequency-domain FRR in the view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. A numerical illustration of Eq. ( view at source ↗
read the original abstract

We develop a unified fluctuation-response theory in the frequency domain for nonequilibrium steady states governed by overdamped Langevin dynamics and Markov jump processes. The relation expresses the power spectrum of general observables exactly as a quadratic form of local responses measured at the same frequency, thereby extending static nonequilibrium fluctuation-response relations to finite frequencies. The decomposition is spatial for Langevin systems and edge-resolved for Markov jump processes, and applies uniformly to state-dependent observables, current-like observables, and their combinations. As consequences of the same identity, we derive frequency-domain response uncertainty relations, kinetic and thermodynamic uncertainty relations, the equilibrium fluctuation-dissipation theorem, and Harada-Sasa-type relations. Applications to stochastic networks and driven diffusive systems illustrate how the theory resolves fluctuation spectra into edge-wise contributions and reveals frequency-dependent tradeoffs between fluctuations, response, and dissipation.

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

0 major / 3 minor

Summary. The manuscript develops a unified fluctuation-response theory in the frequency domain for nonequilibrium steady states (NESS) governed by overdamped Langevin dynamics and Markov jump processes. The central claim is an exact identity expressing the power spectrum of general observables (state-dependent, current-like, or combinations) as a quadratic form of local responses measured at the same frequency. This extends static nonequilibrium fluctuation-response relations to finite frequencies, with a spatial decomposition for Langevin systems and an edge-resolved decomposition for jump processes. Consequences include derivations of frequency-domain response uncertainty relations, kinetic and thermodynamic uncertainty relations, recovery of the equilibrium fluctuation-dissipation theorem, and Harada-Sasa relations. Applications to stochastic networks and driven diffusive systems illustrate resolution of fluctuation spectra into local contributions.

Significance. If the central identity holds, the work provides a significant exact framework for frequency-domain analysis in nonequilibrium stochastic thermodynamics without additional approximations for the stated dynamical classes. The uniform treatment of different observable types and the local decompositions enable new insights into how specific edges or spatial regions contribute to global spectra and frequency-dependent tradeoffs between fluctuations, response, and dissipation. Recovery of known equilibrium and nonequilibrium limits serves as a strong internal consistency check, and the derived uncertainty relations may find applications in analyzing biological and driven systems.

minor comments (3)
  1. [§3.2] §3.2, Eq. (18): the definition of the local response operator could include an explicit statement of its action on the probability current to clarify the edge-resolved decomposition for Markov processes.
  2. [Figure 2] Figure 2: the color scale for the edge contributions in the driven diffusive system example is not labeled with units, making quantitative comparison to the total spectrum difficult.
  3. [Introduction] The abstract states the relation is 'exact' for the two dynamical classes; a brief remark in the introduction on why the derivation does not extend immediately to underdamped Langevin dynamics would help scope the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript on the frequency-domain fluctuation-response theory for nonequilibrium steady states. We appreciate the recognition of the exact identity's scope, its uniform treatment of observables, the local decompositions, and the recovery of known limits as consistency checks. Since the report lists no specific major comments, we have no revisions to propose at this time but remain available to address any minor points the editor or referee may identify.

Circularity Check

0 steps flagged

No significant circularity; derivation follows directly from stochastic generator

full rationale

The central identity expresses the power spectrum exactly as a quadratic form in same-frequency local responses, obtained by direct algebraic manipulation of the Fokker-Planck or master-equation generator for overdamped Langevin dynamics and Markov jump processes in NESS. No step reduces a fitted parameter to a prediction, renames a known result as new unification, or relies on a load-bearing self-citation whose content is itself unverified. Recovery of the equilibrium FDT and Harada-Sasa relations functions only as an internal consistency check. The derivation remains self-contained against the stated dynamical assumptions and does not import uniqueness theorems or ansatzes from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are introduced or mentioned; the theory rests on standard overdamped Langevin and Markov jump process assumptions for steady states.

pith-pipeline@v0.9.0 · 5446 in / 938 out tokens · 34101 ms · 2026-05-08T16:18:26.807222+00:00 · methodology

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Reference graph

Works this paper leans on

69 extracted references · 5 canonical work pages · 1 internal anchor

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    Perturbed Fokker–Planck generator and local response formulas In this subsection, we derive the local response for- mulas stated in Sec. III. We consider the overdamped Langevin dynamics ˙x(t) =M(x(t))F(x(t)) + √ 2B(x(t))⊛ξ(t),(A1) where⊛denotes the anti-Itˆ o product, and ˆLx is the Fokker–Planck generator given by ˆLx =−∇ T x ˆJx, ˆJx =M(x) F(x)−T(x)∇ x...

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    Properties of the excess propagator In this subsection, we derive the identities for the ex- cess propagator used in Sec. III. We recall its definition, H(x,z;ω) = Z ∞ 0 P(x, t|z,0)−π(x) eiωt dt.(A24) The propagator satisfies both the forward and backward Fokker–Planck equations, ∂tP(x, t|z,0) = ˆLxP(x, t|z,0),(A25) ∂tP(x, t|z,0) = ˆL† zP(x, t|z,0),(A26) ...

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    Covariances of empirical density and current We now derive the covariance formulas for the empiri- cal density and current. We begin with ρ(x, t) =δ(x−x(t)).(A34) Fort >0, the connected two-time correlation is given by ⟨ρ(x, t)ρ(y,0)⟩ −π(x)π(y) = P(x, t|y,0)−π(x) π(y), (A35) whereas fort <0, ⟨ρ(x, t)ρ(y,0)⟩ −π(x)π(y) = P(y,−t|x,0)−π(y) π(x), (A36) which f...

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    Proof of the quadratic representation for the density–density covariance In this subsection, we prove the quadratic representa- tion (A38) of the density–density covariance. We first rewriteH(x,y;ω)π(y) in terms of a delta function to exploit the resolvent identity: H(x,y;ω)π(y) = Z dzπ(z)H(x,z;ω)δ(y−z).(A50) Using the identity (A33) we rewrite the delta ...

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    Explicit forms of the perturbation weight matricesA ϕ(z) In this subsection, we collect the explicit forms of the perturbation weight matricesA ϕ(z) that enter the FRR. We recall the definition Aϕ(z) = 1 2π(z) N ϕ(z) T D(z) −1N ϕ(z).(A55) For perturbationsϕ∈ {F,lnM, T}, using (A20)–(A22), we obtain the following explicit forms ofA ϕ(z): [AF (z)]kl =δ kl π...

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    Perturbed master equation and local response formulas In this subsection, we derive the local response for- mulas stated in Sec. IV. The structure closely parallels Appendix A 1. We consider a local perturbation applied at timet= 0 on the edgek↔l(withk > l). The unperturbed dynamics is governed by ∂tp(t) =Wp(t),(B1) with stationary distributionπsatisfying...

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    Properties of the discrete excess propagator We now derive several identities for the discrete excess propagator defined in (B15). To this end, we use the following limiting behaviors: lim t→0+ [eWt]nk =δ nk,lim t→∞ [eWt]nk =π n.(B28) First, we show that X m Hnm(ω)πm = 0.(B29) Multiplying (B15) byπ m and summing overm, we obtain X m Hnm(ω)πm = Z ∞ 0 dt ei...

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    Covariances of empirical state indicators and edge currents We now derive the covariance formulas for the empir- ical state indicators and empirical edge currents. We begin with ηn(t) =δ X(t),n .(B37) 24 Fort >0, ⟨ηn(t)ηm(0)⟩ −π nπm = [eWt]nm −π n πm,(B38) whereas fort <0, by time-translation invariance, ⟨ηn(t)ηm(0)⟩ −π nπm = [eW(−t)]mn −π m πn.(B39) Taki...

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    Proof of the discrete quadratic identity for the state–state covariance In this subsection, we prove the quadratic representa- tion (B41). To this end, we first rewriteH nm(ω)πm in terms of a Kronecker-delta function as Hnm(ω)πm = X k Hnk(ω)πkδkm.(B55) Using the identity (B36) at frequency−ω, we rewrite the Kronecker-delta as δkm =− X l −iωδlk +W lk Hml(−...

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    Derivation of the frequency-domain response uncertainty relation In this subsection, we derive the frequency-domain RUR stated in Sec. V A. This follows directly from the Cauchy–Schwarz inequality applied to the quadratic re- sponse representation of the covariance matrix. We begin with the overdamped Langevin case. For an arbitrary vectoru∈C NO, multiply...

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    We begin with the overdamped Langevin case

    Derivation of the frequency-domain thermodynamic uncertainty relations In this subsection, we derive the frequency-domain TURs, (97) and (105). We begin with the overdamped Langevin case. Con- sider the homogeneous perturbationψ k(x, ω) = 1 (for all k,x, andω). For a current-like observable θ(t) = Z dxL(x)ȷ(x, t),(C18) the corresponding global response is...

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    Hence, for this choice of perturbation, the response equals the mean value of the current-like observable: Rθ B7→B+ϵ(ω) =⟨θ⟩ ss. This relation extends componen- twise to a vector of current-like observables Rθ B7→B+ϵ(ω) =⟨θ⟩ ss.(C26) Substituting this into (104) yields ⟨θ⟩T ssCθ,θ T(ω)−1⟨θ⟩ss ≤ σps 2 ,(C27) which is the frequency-domain TUR for Markov jum...

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    Proof of the equilibrium reciprocity identities In this subsection, we prove the reciprocity identities used in Sec. V C. We consider the overdamped Langevin and Markov jump cases separately. We begin with the overdamped Langevin case. At equilibrium, we assume spatially homogeneous temper- atureT k(x) =Tand conservative forceF k(x) = −∂xk U(x), so that t...

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