Estimating entropy production with odd-parity state variables via machine learning

Entropy production (EP) is a central measure in nonequilibrium thermodynamics, as it can quantify the irreversibility of a process as well as its energy dissipation in special cases. Using the time-reversal asymmetry in a system's path probability distribution, many methods have been developed to estimate EP from only trajectory data. However, for systems with odd-parity variables that prevail in nonequilibrium systems, EP estimation via machine learning has not been covered. In this study, we develop a machine-learning method for estimating the EP in a stochastic system with odd-parity variables through multiple neural networks, which enables us to measure EP with only trajectory data and parity information. We demonstrate our method with two systems, an underdamped bead-spring model and a one-particle odd-parity Markov jump process.

Figure 1

EP rate as a function of $T_1/T_2$ in the underdamped bead-spring model for $m=0.01$ (left), 0.1 (middle), and 1 (right). The solid lines (blue dots) represent the analytical EP rate $\dot{\sigma }$ (estimated EP rate $\widehat{\dot{\sigma }}$). The insets are scatter plots of the analytical $\mathrm{\Delta }{S}_{\mathrm{tot}}$ and the estimation $\mathrm{\Delta }{\widehat{S}}_{\mathrm{tot}}$ at $T_1/T_2=0.1$, where the red lines denote linear regression. Error bars represent the standard deviation of $\widehat{\dot{\sigma }}$ from five independently trained estimators.

Figure 2

EP per step over $\langle \mathrm{\Delta }{S}_{\mathrm{as}}\rangle$ as a function of mass $m$ at $T_1/T_2=0.1$.

Figure 3

EP rate as a function of $c$ in the one-particle odd-parity Markov jump process. The inset is a scatter plot of $\mathrm{\Delta }{S}_{\mathrm{tot}}$ and $\mathrm{\Delta }{\widehat{S}}_{\mathrm{tot}}$ at $c=10$ where the red line denotes the linear regression. The standard deviation (error bar) of the EP rate estimation $\widehat{\dot{\sigma }}$ is much smaller than the symbol size.