Nonequilibrium fluctuation-response relations for state-current correlations

TL;DR

The paper introduces exact Fluctuation-Response Relations (FRRs) that connect covariances of nonequilibrium state and current observables to their static responses in Markov jump processes, and extends these to mixed state-current covariances. It presents inverse FRRs, expressing individual state or current responses in terms of covariances, and generalizes FRRs to correlations with generic flux observables. The authors show that Onsager symmetry breaking requires nonzero state-current covariances and demonstrate practical utility by analyzing quantum-dot transport and enzymatic inhibition networks, including edge-resolved decompositions and channel-specific contributions. This framework provides a principled method to infer fluctuations from responses and to diagnose nonequilibrium behavior in nanoscale and biochemical systems, with potential extensions to continuous Langevin dynamics.

Abstract

Recently, novel exact identities known as Fluctuation-Response Relations (FRRs) have been derived for nonequilibrium steady states of Markov jump processes. These identities link the fluctuations of state or current observables to a combination of responses of these observables to perturbations of transition rates. Here, we complement these results by deriving analogous FRRs applicable to mixed covariances of one state and one current observable. We further derive novel Inverse FRRs expressing individual state or current response in terms of a combination of covariances rather than vice versa. Using these relations, we demonstrate that the breaking of the Onsager symmetry requires the presence of state-current correlations. On the practical side, we demonstrate the applicability of FRRs for explaining the behavior of fluctuations in quantum dot devices or enzymatic reaction networks.

Figures (6)

• Figure 1: Scheme of a unicyclic network with each state $n \in \{1,N-1\}$ being a source of a single edge $+n$ pointing to the state $n+1$, and the edge $+N$ pointing from $N$ to $1$. The transtion $+N$ is considered unidirectional ($W_{-N}=0$).
• Figure 2: Scheme of one-dimensional Markov network with each state $n \in \{0,N\}$ apart from $n=N$ being a source of several edges $+(n+1)\nu$ (here $\nu \in \{a,b\}$) pointing to a tip $n+1$. The index $\nu$ denotes the channel of transition.
• Figure 3: (a) Scheme of a quantum dot connected to two reservoirs $1$ and $2$. The applied voltage induces transitions between states occupied by $N$ and $N+1$ electrons. The voltage is much larger than the thermal energy $k_B T$, so that the electrons transitions can be considered unidirectional. (b) A Markov network describing the system. Here, labels $\{1,2\}$ denote the states with $N$ and $N+1$ electrons.
• Figure 4: The covariance $C^\mathfrak{m}_{11}$ and its individual components $C_{11}^{\mathfrak{m}(e)}$ as a function of the asymmetry coefficient $a$.
• Figure 5: Scheme of a Markov model corresponding to reaction \ref{['eq:inh']}. The system can reside in three states $n \in \{\text{ES},\text{E},\text{EI} \}$, corresponding to enzyme-substrate complex, unbound enzyme, and enzyme-inhibitor complex. The transition $+1b$ is associated with the release of the product P.