Integrated covariances as excess observables weighted by currents and activities

Abstract

Near equilibrium, the symmetric part of the time-integrated steady-state covariance, i.e., the time integral of correlation functions, is governed by the fluctuation-dissipation theorem, while the antisymmetric part vanishes due to Onsager reciprocity. Far from equilibrium, where these principles no longer apply, we develop a unified formalism for both symmetric and antisymmetric components of integrated covariances. We derive exact, computationally tractable expressions for these quantities, valid in arbitrary nonequilibrium steady states of Markov jump processes. Both components are expressed in terms of excess observables, a notion central to both statistical physics and reinforcement learning. Furthermore, we establish thermodynamic upper bounds for antisymmetric covariances in terms of (pseudo-)entropy production and cycle affinities. Finally, we show that the speed up of self-averaging induced by nonequilibrium drivings which preserve kinetics (activity) is bounded by the cycle affinities (thermodynamic forces).

Figures (2)

• Figure 1: (a): Sketch of a generic excess observable showing that its value $X_n$ corresponds to the net area of the highlighted region (the area above is positive and the one below negative). (b): Molecular motor with three states, $n\in\{1,2,3\}$, driven by fuel $F$ and waste $W$ generating a nonequilibrium drive $\Delta\mu$ on the transition from $1$ to $2$. (c): Excess observables $\boldsymbol{X}=(X_1,X_2,X_3)^\intercal$ and $\boldsymbol{Y} = (Y_1,Y_2,Y_3)^\intercal$ associated to observables $x=(1,0,0)^\intercal$ and $y = (0,1,0)^\intercal$ for various chemical potentials $\Delta \mu$. (d): AICov (purple curve) [\ref{['eq:asymmetry-result-J', 'eq:AICov-def']}], geometric bound (green) [\ref{['eq:bound-geometric']}] and excess bound (blue) [\ref{['eq:bound-excess']}]. (e): Purple curve is the relative speed up $(\tau_x^\text{eq}-\tau_x)/\tau_x^\text{eq}$ as a function of thermodynamic force $\Delta\mu$ calculated from \ref{['eq:speed-up-exact']}, where $\tau_x^\text{eq}$ corresponds to the equilibrium partner with the same probability and activity. Green curve is the goemetric bound \ref{['eq:bound-tau_x']} with $\mathcal{M}=\tanh(\Delta\mu/3)/\tan(\pi/6)$. Parameters: Matrix $\mathbb{W}$ has off-diagonal elements $W_{21}=e^{\Delta\mu/2}$, $W_{12}=e^{-\Delta\mu/2}$, $W_{31}=W_{13}=W_{23}=W_{32}=1$.
• Figure 2: Speed-up of self-averaging as a function of the cycle affinity generated by adding a clockwise cycle current $\mathcal{J}$ using the rotor model described in the main text. Inset: Equilibrium distribution $p^{\mathrm{eq}}(\theta_n)$. Simulations use $W^{\mathrm{eq}}_{n\pm1,n} = \exp[\beta V_0 \cos(2\theta_n)]$, $N=50$, $V_0 = 1$ and $\beta = 1$ with periodic boundary conditions $W^{\mathrm{eq}}_{1,N}=W^{\mathrm{eq}}_{N,1}$.