Finite-frequency fluctuation-response inequality

Abstract

We derive an inequality relating the finite-frequency linear response and fluctuations of an observable in a physical system. The relation holds for arbitrary observables and perturbations in general Markovian dynamics, including over- and underdamped Langevin systems and jump processes, both in and out of equilibrium. As a consequence, we obtain a universal upper bound on the broad-band signal-to-noise ratio of noisy dynamics, which only depends on the damping constant and temperature. We further show that the inequality reduces to an equality for appropriately chosen observables or perturbations in linear systems, both overdamped and underdamped and both in and out of equilibrium.

Figures (4)

• Figure 1: Power spectral density (dashed lines) and magnitude of the response (solid lines) of the velocity of an underdampled particle in a periodic potential, Eq. (\ref{['langevin-perpot']}). The black lines are for vanishing tilt $f_0 = 0$, where the system is in equilibrium. The orange lines are for a non-equilibrium steady state with $f_0 = 0.175 \cdot 2 \pi$. Other parameter values are $m = 1, \gamma = 1/3, T = 1/2, U_0 = 1$ and $L = 1$. Both in and out of equilibrium, the power spectral density upper bounds the response, as predicted by Eq. (\ref{['fffri-1D']}).
• Figure 2: A network consisting of 21 oscillators (disks), coupled to each other by springs with spring constant $\kappa = 0.75$; the outermost ring is attached to the black points by springs with spring constant $k = 2$. The mass of each oscillator is $m = 1$, the damping constant is $\gamma = 0.01$ and the temperature $T = 1$. The perturbation is applied to the red oscillator.
• Figure 3: Response efficiency of a linear network. The response efficiency Eq. (\ref{['response-efficiency']}) of the perturbed oscillator (red line) approaches $1$ in the high-frequency limit, where the perturbation is localized. At intermediate frequencies, the response efficiency in other parts of the network (blue) can be larger. Also measuring the response of the neighbors (light blue) yields additional information about the perturbation and increases the response efficiency. In particular, measuring the perturbed oscillator and its neighbors (light red) results in a response efficiency of unity at all frequencies.
• Figure 4: The structure of the coupling matrix $\bm{A}$ (left) and diffusion matrix $\bm{B}$ (right). The perturbed degrees of freedom $a^*$ (red) and their neighbors $N(a^*)$ (yellow) are the measured degrees of freedom $a$ (green). Since all other degrees of freedom $p$ (white) are not connected to $a^*$, the corresponding elements of the coupling matrix $\bm{A}$ are zero. For the diffusion matrix, we assume a block-diagonal structure between the perturbed degrees of freedom $a^*$, their neighbors $N(a^*)$ and the remaining degrees of freedom $p$.