Nonequilibrium fluctuation-response relations for state observables

TL;DR

This work addresses fluctuations of time-integrated state observables in nonequilibrium steady states of Markov jump processes. It develops exact Fluctuation-Response Relations (FRRs) that connect covariances of state observables to their static responses under perturbations of transition rates, with a concrete edge-transport interpretation. The authors derive a first-known upper bound on state-observable fluctuations, relate FRRs to fluctuation-response inequalities, and demonstrate, via a quantum-dot example, that covariance signs can reflect the underlying network topology. The results provide a topology-aware framework for inferring network structure from data and extend the theorization of universal response-fluctuation relations to state observables in nonequilibrium systems.

Abstract

Time-integrated state observables, which quantify the fraction of time spent by the system in a specific pool of states, are important in many fields, such as chemical sensing or the theory of fluorescence spectroscopy. We derive exact identities, called Fluctuation-Response Relations (FRRs), that connect the fluctuations of such observables to their response to external perturbations in nonequilibrium steady state of Markov jump processes. Using these results, we derive a first known upper bound on fluctuations of state observables, as well as some new lower bounds. We further demonstrate how our identities provide a deeper understanding of the mechanistic origin of fluctuations and reveal their properties dependent only on system topology, which may be relevant for model inference using measured data.

Figures (1)

• Figure 1: (a) Scheme of a quantum dot with two spin levels whose energies are Zeeman-splitted by the magnetic field. $\varepsilon$ denotes the average level energy, and $\Delta$ denotes the Zeeman splitting. The dot is coupled to two reservoirs $L$ and $R$ with chemical potentials $\mu_L=V/2$, $\mu_R=-V/2$, and temperature $T$. (b) The system dynamics is described by four-state Markov network with an empty state $0$, states occupied by single electron with spin $\uparrow$ or $\downarrow$, and doubly occupied state $2$. The transition rates read as $W_{\pm e}=\sum_{r \in \{L,R \}} \Gamma_r f[(E_{t(\pm e)}-E_{s(\pm e)} \mp \mu_r)/k_B T]$, where $\Gamma_r$ are tunnel couplings to reservoirs, $f(x) \equiv 1/[1+\exp(x)]$ is the Fermi-Dirac distribution, and the state energies read $E_0=0$, $E_\uparrow=\varepsilon+\Delta/2$, $E_\downarrow=\varepsilon-\Delta/2$, $E_2=U+2\varepsilon$, $U \geq 0$ being the Coulomb coupling. (c) For $\Delta=0$, the electron-tunneling is spin independent: $W_{\pm 1a}=W_{\pm 1b}$, $W_{\pm 2a}=W_{\pm 2b}$. As a result, the system dynamics can be described using coarse-grained one-dimensional Markov network, where $1$ is the union of states $\uparrow$ and $\downarrow$, $W_{+1}=2W_{+1a}$, $W_{-1}=W_{-1a}$, $W_{+2}=W_{+2a}$, $W_{-2}=2W_{-2a}$. (d) The covariance $C_{02}$ and its components $C_{02}^{(e)} \equiv 4 d_{S_e} \pi_0 d_{S_e} \pi_2/\tau_e$ as a function of $\Delta$. Parameters: $\Gamma_R=0.2\Gamma_L$, $k_B T=0.02U$, $\varepsilon=-0.3U$, $V=1.6U$. The analytic expressions for $C_{02}^{(e)}$ are presented in Appendix \ref{['app:c02e']}.