Thermodynamic and Kinetic Bounds for Finite-frequency Fluctuation-Response

Abstract

Fluctuation-response relations encode fundamental constraints on nonequilibrium systems. While time-domain static response is bounded by activity and entropy production, finite-frequency extensions for time-dependent perturbations remain largely unexplored. Here, we derive frequency-domain fluctuation-response inequalities for steady-state Markov processes with time-dependent perturbations. For barrier and entropic perturbations, the spectral signal-to-noise ratio (SNR) is universally bounded by dynamical activity. Furthermore, for state-current observables, the SNR is bounded by the entropy production rate (EPR). We illustrate our results using the F1-ATPase model to infer EPR. These finite-frequency inequalities provide a practical route to infer dissipation from power spectra measurements.

Figures (2)

• Figure 1: Numerical verification of \ref{['eq: FRIs 1a', 'eq: FRIs 1b', 'eq: FRIs 2a', 'eq: FRIs 2b']}. The transition rates are randomly generated from the exponential distribution $r_{ij} \sim \operatorname{Exp}(1)$. The observable coefficients are randomly generated from the uniform distribution $U(0, 1)$. Each data point is averaged over $4\times 10^5$ trajectories.
• Figure 2: (a). Minimal model for F1-ATPase rotatory motor. $x_0 \to x_1$: ATP binding step; $x_1 \to x_2$: hydrolysis with $\Delta\theta = 120^\circ$ counterclockwise rotation; $x_2 \to x_0$: ADP unbinding step. (b). The finite-frequency thermodynamic bound for the F1-ATPase model with different types of perturbations. The black dashed line represents the thermodynamic upper bound $\max_{(i, j)}\{(\partial_\zeta b_{ij})^2\}\cdot\frac{\dot{\sigma}}{2}$. The average is taken over $4 \times 10^5$ trajectories.