Stochastic approach to entropy production in chemical chaos

TL;DR

This work develops a stochastic thermodynamics framework for entropy production in chemical reaction networks that exhibit chaos, aligning stochastic and deterministic descriptions. It derives a path-based entropy production measure and a cycle decomposition tied to the stoichiometric matrix, showing that the macroscopic EPR can be written as a sum of cycle affinities and fluxes in the mass action regime. The authors apply the theory to a reversible GN83 like network that displays stationary, periodic, and chaotic dynamics, demonstrating that cycle contributions can be negative and that the direction of certain conversions can be tuned by parameter changes. The results validate the consistency of stochastic and deterministic approaches, enable energy conversion analysis in chaotic chemical networks, and provide tools for quantifying entropy production via cycle counting or path probabilities.

Abstract

Methods are presented to evaluate the entropy production rate in stochastic reactive systems. These methods are shown to be consistent with known results from nonequilibrium chemical thermodynamics. Moreover, it is proved that the time average of the entropy production rate can be decomposed into the contributions of the cycles obtained from the stoichiometric matrix in both stochastic processes and deterministic systems. These methods are applied to a complex reaction network constructed on the basis of Roessler's reinjection principle and featuring chemical chaos.

Figures (5)

• Figure 1: Bifurcation diagram of the dynamical system of Eqs. (\ref{['dot-x']})-(\ref{['dot-z']}) with the reaction rates (\ref{['w1']})-(\ref{['w10']}) and the parameter values (\ref{['parameters']}). The axis $k_{+2}$ is scanned every step $\Delta k_{+2}= 10^{-3}$. The dots depict every extremum where $\dot{z}(t)=0$ in the time interval $500<t<600$, after transients over the time interval $0<t<500$.
• Figure 2: (a) The concentrations $(x,y,z)$ versus time $t$ for the dynamical system of Eqs. (\ref{['dot-x']})-(\ref{['dot-z']}) with the reaction rates (\ref{['w1']})-(\ref{['w10']}) and the parameter values (\ref{['parameters']}) with $k_{+2}=0.55$. (b) The concentrations $(x,y,z)$ versus time $t$ for the corresponding stochastic process with the extensivity parameter $V=250$.
• Figure 3: (a) The chaotic attractor formed by the trajectory of Fig. \ref{['fig2']}(a) for $100<t<1100$. (b) The corresponding noisy attractor formed by the trajectory of Fig. \ref{['fig2']}(b) for $0<t<1000$ with the extensivity parameter $V=250$.
• Figure 4: Entropy production rate (EPR) of the reaction network (\ref{['GN83-CRN']}) for the parameter values (\ref{['parameters']}) versus $k_{+2}$, as computed with different methods in the deterministic and stochastic approaches. The deterministic EPR (open squares) is obtained with the time average of the entropy production rate (\ref{['sigma-det']}). The deterministic EPR$_{\rm c}$ (filled circles) is calculated by the sum (\ref{['av-sigma-det-3']}) over the $o=7$ cycles of the reaction network. The stochastic EPR (filled squares) is computed by extrapolation of the values $R/V$ given by Eq. (\ref{['R']}) using Gillespie's algorithm with $V=100$, $150$, $200$, and $250$. The stochastic EPR$_{\rm c}$ (open circles) is similarly obtained from Eq. (\ref{['Sigma-tilde-Jc']}) in the same simulations using Gillespie's algorithm. The averages are carried out over the time interval $t=1000$.
• Figure 5: Efficiency (\ref{['eta']}) of the opposite chemical conversion ${\rm A}_2\to{\rm A}_1$ in the reaction network (\ref{['GN83-CRN']}) for the parameter values (\ref{['parameters']}) versus $k_{+2}$, as computed in the deterministic and stochastic approaches. The deterministic values of the efficiency $\eta$ (open squares) are obtained using the time averages of the cycles (\ref{['cycle1']})-(\ref{['cycle7']}). The stochastic values of $\eta$ (filled squares) are computed by extrapolating the values obtained using Gillespie's algorithm with $V=100$, $150$, $200$, and $250$. The averages are carried out over the time interval $t=1000$. The statistical errors on the stochastic values are here smaller than or the same size as the filled squared.