Dynamic phase transitions in simple driven kinetic networks

We analyze the probability distribution for entropy production rates of trajectories evolving on a class of out-of-equilibrium kinetic networks. These networks can serve as simple models for driven dynamical systems, where energy fluxes typically result in nonequilibrium dynamics. By analyzing the fluctuations in the entropy production, we demonstrate the emergence, in a large system size limit, of a dynamic phase transition between two distinct dynamical regimes.

Figure 1

Diagrams of simple driven kinetic networks studied here. Arrows connecting a pair of vertices indicate Poisson-distributed transitions from one state to the other, labeled by the corresponding rate constants.

Figure 2

Entropy production distribution for the translationally symmetric triangle networks (red circles) and triangle networks with the heterogeneous link [see Fig. 1] (blue triangles). For each network, ${10}^7$ independent trajectories of length $\tau =250$ were generated with $x=20$, $y=1$, $b=0.1$, $h=0.025$, and $N=400$. The inset shows a trajectory illustrating the two dynamical regimes. Sites are numbered clockwise in a zig-zag fashion with the $h$ link connecting sites 1 and 800.

Figure 3

Scaled cumulant generating functions and average entropy production rates for the triangle networks, depicted in Fig. 1. All results are shown for rates $x=20,y=1,$ and $b=0.1$. Black solid curves show the exact behavior of the symmetric variant for comparison. Plots (a) and (b) show results for a variety of network sizes with $h=0.1$ for the symmetry-breaking link, suggesting a singularity in the large-$N$ limit. Plots (c) and (d) show results for $N=200$ and several values of $h$.

Figure 4

Large deviation function $I\left(\sigma \right)$ for the entropy production rate of the network in Fig. 1 with $x=20$, $y=1$, $b=0.1$, and several different values of $h$. The rate function envelope was determined by the LF transform of ${\psi }_{\omega }\left(\lambda \right)$, whose two singularities require the construction of tie lines (dashed curve). Kinetic Monte Carlo simulation results (${10}^7$ trajectories with $h=0.025,N=400,$ and $\tau =250$) are shown as red circles in the inset.

Figure 5

(a) Scaled cumulant generating function for the ring network with varying $h$. The values of ${\lambda }^{*}$ predicted by our theory are plotted as large blue points in plot, which coincide with the large-$N$ limit of the numerical calculations. (b) Singularity in the scaled cumulant generating function for the triangle network with rate constants $x=20,y=1,b=0.1,$ and $h=0.1$.