Fluctuation-Response Theory for Nonequilibrium Langevin Dynamics

TL;DR

We generalize the fluctuation-dissipation theorem to nonequilibrium Langevin dynamics by deriving a unified fluctuation-response relation (FRR) that links long-time fluctuations $C_{\Theta_1,\Theta_2}$ to local perturbations of $\phi$ in {\mu, F, T} via $C_{\Theta_1,\Theta_2} = \int 2\pi(z) D(z) A_{\Theta_1}(z) A_{\Theta_2}(z) dz$, with $A_{\Theta}(z) = \delta \langle \Theta \rangle_{ss} / \delta \phi(z)$. It reduces to the equilibrium FDT through the local Onsager reciprocal relation. A finite-time Cramér–Rao bound yields the fluctuation-response inequality: $\mathrm{Var}[\Theta(\tau)] \ge \int dx \int_0^{\tau} ds \; \frac{2 T(x)}{\mu(x) p(x,s)} \left( \frac{\delta \langle \Theta(\tau) \rangle}{\delta F(x,s)} \right)^2$. From the FRI we derive practical R-URs, including a Langevin analogue of the R-TUR with perturbation $\phi = \ln \mu$: $\frac{[\delta_{\ln \mu} \langle \Theta(\tau) \rangle]^2}{\mathrm{Var}[\Theta(\tau)]} \le \frac{\psi_{\max}^2 \Sigma_{\tau}}{2}$, where $\Sigma_{\tau}$ is the total entropy production. The framework is illustrated by applying these bounds to the ${F}_{1}$-ATPase motor to bound the long-time diffusion coefficient $D_{\infty}$.

Abstract

We establish a unified fluctuation-response relation for Langevin dynamics. By exploiting the common mathematical structures underlying fluctuations and responses of empirical density and current, we derive a unified identity that generalizes the fluctuation-dissipation theorem from equilibrium to nonequilibrium settings. This relation connects global fluctuations of observables with their local responses to perturbations in force, mobility, and temperature. We further derive finite-time fluctuation-response inequalities, leading to response uncertainty relations that complement the identity by providing more practical bounds. These derivations establish a unified theoretical framework linking the fluctuation-dissipation theorem and thermodynamic uncertainty relations. Using the $F_1$-ATPase molecular motor model, we illustrate how these response-based bounds constrain the long-time diffusion coefficient.

Figures (2)

• Figure 1: Hierarchy among the FRI, FRR, FDT, R-TUR, and TUR.
• Figure 2: Application of the response uncertainty relations to F$_1$-ATPase. (a) Comparison of $D_\infty$ with the three response-uncertainty bounds associated with perturbations in $F$, $\ln\mu$, and $T$. Inset: Schematic of the experimental setup for F$_1$-ATPase attached to a duplex of polystyrene beads Hayashi2015giant. (b) Temperature dependence of $D_\infty /D_0$ at $F \approx 120\,\mathrm{pN\cdot nm}$, compared with the three bounds. $T_0=298\text{K}$ denotes the room temperature. In (a) and (b), solid, dotted, dashed, and dashdot lines represent the analytical results of $D_\infty$, $\frac{T}{\mu} (\partial_F \omega)^2$, $(\partial_{\ln \mu} \omega)^2 / \sigma$, and $\frac{T}{\mu}(\partial_T \omega)^2/I$, respectively, while markers with error bars (indicating the standard deviation) denote numerical simulation results obtained from Eq. \ref{['eq:F1ATPase']}.