Dynamical Fluctuation-Response Relations

Abstract: We derive exact dynamical fluctuation-response relations (FRRs) for time-integrated observables of any nonautonomous Markov jump process. The finite-time covariance splits into an initial variability and an integral of response kernels along the driven dynamics. The identity sharpens the dynamical response thermodynamic and kinetic uncertainty relations and fluctuation-response inequalities (FRIs). It also recovers steady-state FRRs, fluctuation-dissipation theorem and Onsager reciprocity, identifies known autonomous FRIs as the zero-frequency mode.

• The paper derives an exact finite-time covariance identity for arbitrary integrated state and current observables in nonautonomous Markov jump processes, separating initial variability from an integral of squared dynamical response kernels.
• The framework recovers and sharpens dynamical TURs, kinetic uncertainty relations, fluctuation-response inequalities, the fluctuation-dissipation theorem, Onsager reciprocity, and steady-state relations, with tighter finite-time bounds when initial-state variability is known.
• The analysis shows that periodically driven no-pumping states can have zero net current while still violating FDT and Onsager reciprocity, demonstrating that equilibrium-like response requires slow driving rather than merely vanishing transport.

This paper derives exact fluctuation-response relations (FRRs) for time-integrated observables of nonautonomous Markov jump processes, extending a framework previously restricted to nonequilibrium steady states (2604.24626). The central result expresses the finite-time covariance between arbitrary integrated state and current observables as a sum of an initial-variability term and an integral of squared dynamical response kernels evaluated along the unperturbed driven evolution. From this single identity the authors recover and sharpen a broad set of results in stochastic thermodynamics: dynamical response thermodynamic and kinetic uncertainty relations (R-TURs, R-KURs), the Dechant–Sasa fluctuation-response inequality, the fluctuation-dissipation theorem (FDT) and Onsager reciprocity in the detailed-balance autonomous limit, and the steady-state FRRs as the long-time limit.

Setup and dynamical response theory

The system is a Markov jump process on $N$ states with a time-dependent rate matrix $W(t) = \Delta \cdot \Gamma(t)$, decomposed via the incidence matrix $\Delta$ and weighted incidence matrix $\Gamma$. The observables of interest are integrated quantities of the form $\mathcal{Q} = \mathcal{O} + \mathcal{J}$, combining a state contribution $o_{n_\tau}(\tau)$ and an edge-current contribution $x_e(\tau)\,dk_e(\tau)$.

The key structural object is the conditional mean $Q_n(\tau, T)$, the expected value of $\mathcal{Q}$ over trajectories starting in state $n$ at time $W(t) = \Delta \cdot \Gamma(t)$0. It obeys a backward Kolmogorov equation, a dynamical generalization of the steady-state Poisson equation. The response kernel to a perturbation of a rate-controlling parameter $W(t) = \Delta \cdot \Gamma(t)$1 takes the compact form

$W(t) = \Delta \cdot \Gamma(t)$2

where $W(t) = \Delta \cdot \Gamma(t)$3 is the mean edge current. Only derivatives of the rates appear; the kernel is an exact functional derivative along the unperturbed driven dynamics, not a linearization of the dynamics itself. A notable side result is a dynamical extension of mutual linearity: the ratio of responses to forward and backward rate perturbations on the same edge equals $W(t) = \Delta \cdot \Gamma(t)$4 at any time along the trajectory, generalizing the steady-state corollary of Bebon and Speck.

The central covariance identity

The main result states that the scaled covariance between two integrated observables splits as

$W(t) = \Delta \cdot \Gamma(t)$5

where $W(t) = \Delta \cdot \Gamma(t)$6 is the edge activity and $W(t) = \Delta \cdot \Gamma(t)$7 is the initial-variability term, built from excess observables $W(t) = \Delta \cdot \Gamma(t)$8 measuring how the conditional mean depends on the initial state. The derivation uses finite-time cumulant-generating functions and their backward evolution.

Three properties of the initial-variability term determine the structure of the theory. For variances it is nonnegative; it vanishes for deterministic initial preparation; and it decays to zero once the dynamics loses memory of the initial ensemble, including in the periodic (Floquet) state of a periodically driven system. Consequently, the response-integral term alone governs the asymptotic regime, while the full identity is strictly sharper at finite time. This is the mechanism by which the paper tightens existing finite-time bounds: every inequality derived below by discarding $W(t) = \Delta \cdot \Gamma(t)$9 recovers a previously known, looser bound.

Specific perturbations and uncertainty relations

With the rate parameterization $\Delta$0, where $\Delta$1 are kinetic barriers and $\Delta$2 nonconservative affinities, the response kernels to barrier and affinity perturbations satisfy the exact ratio $\Delta$3, extending known steady-state relations to nonautonomous dynamics. Substituting these kernels into the FRR yields an exact dynamical identity involving only $\Delta$4 and activities. Applying Sedrakyan's integral inequality then gives local dynamical R-TURs and R-KURs bounding the edge-resolved pseudo-entropy production rate and activity by response-to-fluctuation ratios, corrected by the initial-variability term.

For a global kinetic perturbation (a "virtual" rescaling of all barriers), the paper derives a dynamical R-TUR that, specialized to current observables, becomes

$\Delta$5

where $\Delta$6 is the driving speed. Dropping the variability term reproduces exactly the time-dependent-driving TUR of Koyuk and Seifert and, for autonomous relaxation, the bound of Liu, Gong, and Ueda; the presence of $\Delta$7 makes the new bound uniformly tighter at finite time.

For thermodynamic perturbations, with forces $\Delta$8 decomposed via the edge–force incidence $\Delta$9, the paper derives a dynamical R-KUR involving the weighted activity matrix $\Gamma$0. Optimizing over the perturbation direction $\Gamma$1 yields an optimal R-KUR with the inverse activity matrix, in which the exact FRR identity degrades to an inequality. Combining the R-KUR with the quadratic expansion of the Kullback–Leibler divergence between perturbed and unperturbed path probabilities gives a sharpened linear fluctuation-response inequality,

$\Gamma$2

which reduces to the Dechant–Sasa FRI when the variability term is discarded. The paper notes that only the linear-response, Markov-jump version of the Dechant–Sasa FRI is derived here.

Recovery of FDT, Onsager reciprocity, and the steady-state hierarchy

For autonomous detailed-balance dynamics prepared in equilibrium, the framework yields the standard FDT $\Gamma$3 and Onsager reciprocity $\Gamma$4 at finite time. For driven detailed-balance dynamics, only the symmetrized identity $\Gamma$5 survives; Onsager symmetry is not recovered under driving, consistent with the no-pumping analysis below. The paper also derives a covariance representation of responses in the spirit of auxiliary-observable FDTs: choosing the observable as the integrated discrepancy between stochastic current and drift, the covariance with any observable equals the mean response, for arbitrary driving and distance from equilibrium.

In the autonomous steady-state limit, a Fourier resolution of the FRR via Parseval's theorem shows that the variance decomposes into a sum over frequencies of squared Fourier coefficients of the response kernel. Retaining only the zero-frequency mode (the scaled step response) reproduces the autonomous FRI of Kwon et al. as a strictly looser bound, identifying previously known autonomous fluctuation-response inequalities as the zero-frequency sector of the exact theory. The long-time limit recovers the steady-state FRRs, the optimal TUR-type inequality $\Gamma$6, and, via variance minimization over mixed current-state observables, the improved bound of Dechant and Sasa using correlations.

No-pumping theorem and FDT violation at zero current

As an application, the paper gives a compact proof of the no-pumping theorem: for instantaneously detailed-balance rates $\Gamma$7 with symmetric barriers under a cyclic protocol, the integrated current over one period vanishes in the Floquet state. The proof solves the backward Kolmogorov equation explicitly, writing the current as $\Gamma$8, which vanishes by periodicity.

The more substantive point is a negative result: despite zero net pumped current, the no-pumping state does not satisfy the FDT or Onsager reciprocity at finite driving frequency. Numerical results for a periodically driven two-cycle network (with $\Gamma$9, period $\mathcal{Q} = \mathcal{O} + \mathcal{J}$0) show that the FDT violation norm $\mathcal{Q} = \mathcal{O} + \mathcal{J}$1 and nonreciprocity $\mathcal{Q} = \mathcal{O} + \mathcal{J}$2 remain nonzero in the Floquet state and are restored only as $\mathcal{Q} = \mathcal{O} + \mathcal{J}$3. This contradicts the intuition that zero current, as near equilibrium, implies equilibrium-like response behavior, and demonstrates that the FDT is a property of slow driving, not of vanishing transport.

Limitations and open questions

Several restrictions are stated explicitly. The framework is confined to Markov jump processes on discrete state spaces; extensions to diffusive or Langevin dynamics are not derived here, although the authors cite parallel developments. The derivation assumes constant observable weights $\mathcal{Q} = \mathcal{O} + \mathcal{J}$4; the general case requires a corrected response kernel, which is provided but complicates the formulas. No exact FRR analog exists for vertex (energy) perturbations $\mathcal{Q} = \mathcal{O} + \mathcal{J}$5, in contrast to the barrier and affinity cases, and the reason for this asymmetry is not resolved. The Dechant–Sasa FRI is recovered only in its linear-response Markov-jump version, leaving the general nonlinear FRI outside the present identity. Finally, the sharpened bounds involve $\mathcal{Q} = \mathcal{O} + \mathcal{J}$6, which requires knowledge of the initial ensemble; for experiments with unknown preparation, the practical advantage over the looser steady-state-style bounds depends on how quickly memory of the initial state decays, a system-dependent question the paper does not quantify in general.

Conclusion

The paper establishes an exact, finite-time fluctuation-response identity for arbitrary nonautonomous Markov jump processes and shows that a substantial body of stochastic-thermodynamic results—dynamical TURs, R-KURs, FRIs, the FDT, Onsager reciprocity, steady-state FRRs, and the no-pumping theorem—emerges as special cases or loosened versions of it. The initial-variability term is the structural element responsible for the uniform tightening of finite-time bounds, and the demonstration that no-pumping Floquet states violate the FDT clarifies that zero current and equilibrium response are logically distinct. The framework leaves open the extension to non-Markovian observables, to vertex-perturbation identities, and to quantifying the memory-time scale over which the sharpened bounds retain their advantage.