Stochastic thermodynamics of reactive systems: An extended local equilibrium approach

The recently developed extended local equilibrium approach to stochastic thermodynamics is applied to reactive systems. The properties of the fluctuating entropy and entropy production are analyzed for general linear and for prototypical nonlinear kinetic processes. It is shown that nonlinear kinetics typically induces deviations of the mean entropy production from its value in the deterministic (mean-field) limit. The probability distributions around the mean are derived and shown to qualitatively differ in thermodynamic equilibrium, under nonequilibrium conditions and in the vicinity of criticalities associated to the onset of multistability. In each case large deviation-type properties are shown to hold. The results are compared with those of alternative approaches developed in the literature.

Figure 1

Mean normalized entropy production of fluctuations $\overline{\mathrm{\Delta }{\sigma }^{*}}$ (see text) for the model (24) and (25), as a function of the control parameter $c_B$. The other kinetic parameters are all set to 1 and $\mathrm{\Omega }=1000$. The dots correspond to the outcome of numerical integrations (total time = 10, the Stratonovich interpretation was used) averaged over $10000$ realizations of the corresponding stochastic differential equation, with $dt=1.0\times {10}^{-5}$. The plain curve is the analytical prediction as obtained from (30). The dashed line is used to highlight states of zero entropy production of fluctuations.

Figure 2

Probability distribution of the cumulative entropy production ${\mathrm{\Delta }}_{i}S$ for the Schlögl model (24) and (25). All constants are set to 1 as before, except for $c_B=1.1$. The size of the system is $\mathrm{\Omega }=50$ and the total integration time is 10. The plain curve is a Gaussian fitting, while the dots mark the statistics extracted from $50000$ numerical realizations of the stochastic model.

Figure 3

Verification of the fluctuation theorem (44) for the Schlögl model (24) and (25). All constants and parameters are the same as in Fig. 2. The plain straight line is the analytical prediction and the dots represent numerical estimates.

Figure 4

Equilibrium probability distribution $P\left({\mathrm{\Delta }}_{i}S\right)$ for the model (24) and (25), with all constants set to 1. The plain curves correspond to the asymptotic analytical expression (56). The different marks stand for the distributions of ${\mathrm{\Delta }}_{i}S$ obtained from $10000$ realizations of the stochastic differential equation (26) (the numerical method and the time step being the same as in Fig. 2). The circles correspond to a system of size $\mathrm{\Omega }=1000$ and a total integration time $t=50$, the squares are for $\mathrm{\Omega }=10000$ and $t=50$, while for triangles, $\mathrm{\Omega }=1000$ and $t=10$.

Figure 5

Dependence of the variance of the (rescaled) excess entropy production $\delta {\mathrm{\Sigma }}_t{k}_B$ on the size of the system for the rescaled Schlögl model (67) at criticality. The dots correspond to results obtained by numerical integrations of the corresponding stochastic evolution law. The line is a linear fitting done on the last four points.

Figure 6

Steady-state distribution of ${\mathrm{\Sigma }}_t$ for the Brusselator model with $k_1{c}_A=0.5,k_{-4}{c}_E=0.25,k_{-1}=k_4=0.25,k_{-2}{c}_D=k_{-3}=0.25,k_2{c}_B=6,k_3=1$. The dots correspond to histograms extracted from $10000$ realizations of a system of size $\mathrm{\Omega }=1\times {10}^3$ and a sampling time $t=200$. The curve is a Gaussian distribution whose mean and variance are those of the numerically obtained data. Note that we use nondimensional time units.

Figure 7

Equilibrium distribution of $\delta {\mathrm{\Sigma }}_t$ for the Brusselator model. The parameter values are the same as in Fig. 6, except for $k_2{c}_B=0.0625$. The marks were obtained from 5000 realizations of a system of size $\mathrm{\Omega }=1\times {10}^3$ and a sampling time $t=100$. The plain curve corresponds to Eq. (109).