Exponential volume dependence of entropy-current fluctuations at first-order phase transitions in chemical reaction networks

In chemical reaction networks, bistability can only occur far from equilibrium. It is associated with a first-order phase transition where the control parameter is the thermodynamic force. At the bistable point, the entropy production is known to be discontinuous with respect to the thermodynamic force. We show that the fluctuations of the entropy production have an exponential volume-dependence when the system is bistable. At the phase transition, the exponential prefactor is the height of the effective potential barrier between the two fixed-points. Our results obtained for Schlögl's model can be extended to any chemical network.

Figure 1

Phase transition in the Schlögl model. (a) Stationary distribution of chemical species $X$ for $\mathrm{\Omega }=100$ and different values of $\mathrm{\Delta }\mu$. (b) Mean entropy production rate $\sigma$ as a function of $\mathrm{\Delta }\mu$ for different system sizes $\mathrm{\Omega }$. (c) Maximum of the first derivative of $\sigma$ as a function of the system size $\mathrm{\Omega }$. Parameters are given in the main text.

Figure 2

Behavior of the diffusion coefficient $D_{\sigma }$ close to the bistable point. (a) Diffusion coefficient from simulations using Gillespie's algorithm for ${10}^4$ trajectories with a sampling time of $T={10}^4,{10}^5,{10}^6$. For the CME, we solve the linear system given by Eq. (34) numerically for $\mathrm{\Omega }=100$. (b) Diffusion coefficient for the two-state model obtained by evaluating Eq. (43). (c) Finite-size scaling of the maximum of the diffusion coefficients $D_{\sigma }^{\text{bi}}$. The corresponding slopes are given in Table 1.