Thermodynamic Bound on the Asymmetry of Cross-Correlations

The principle of microscopic reversibility says that, in equilibrium, two-time cross-correlations are symmetric under the exchange of observables. Thus, the asymmetry of cross-correlations is a fundamental, measurable, and often-used statistical signature of deviation from equilibrium. Here we find a simple and universal inequality that bounds the magnitude of asymmetry by the cycle affinity, i.e., the strength of thermodynamic driving. Our result applies to a large class of systems and all state observables, and it suggests a fundamental thermodynamic cost for various nonequilibrium functions quantified by the asymmetry. It also provides a powerful tool to infer affinity from measured cross-correlations, in a different and complementary way to the thermodynamic uncertainty relations. As an application, we prove a thermodynamic bound on the coherence of noisy oscillations, which was previously conjectured by Barato and Seifert [Phys. Rev. E 95, 062409 (2017)]. We also derive a thermodynamic bound on directed information flow in a biochemical signal transduction model.

Figure 1

(a) A pair of observables $a(t)$ and $b(t)$ is measured in a stochastic system in steady state, from which the two-time correlations are calculated. (b) We define a normalized measure of asymmetry of cross-correlation ${\chi }_{ba}$ in Eq. (2) based on the short-time behavior of the correlation functions (black). (c) Our main result is a thermodynamic relation between ${\chi }_{ba}$ and cycle affinity, Eq. (3). (d) The result is derived in part by considering ${\chi }_{ba}$ as the ratio between the area and perimeter of the polygon formed by the values $(a,b)$ over a cycle and then using the isoperimetric inequality.

Figure 2

(a) Simple model of biological signal transduction [80]. (b) Formulation as a Markov jump system with a nonequilibrium cycle. (c) Validation of our bounds. Orange indicates ${\chi }_{ba}$ for varying chemical force $\mathrm{\Delta }\mu$, which determines the rate of the transition $k_b^{+,\mathrm{ON}}$ (other kinetic rates set to random but fixed values). ${\chi }_{ba}$ is nonnegative for $\mathrm{\Delta }\mu \ge 0$ in this model. Blue indicates the three upper bounds from Eq. (12). (d) The ratio between ${\chi }_{ba}$ and the tightest bound $\tanh (\mathrm{\Delta }\mu /8k_{B}T)$ in Eq. (12).