Nonequilibrium Lyapunov function and a fluctuation relation for stochastic systems: Poisson-representation approach

We present a statistical physics framework for the description of nonlinear nonequilibrium stochastic processes, modeled via a chemical master equation, in the weak-noise limit. Using the Poisson-representation approach and applying the large-deviation principle, we first solve the master equation. Then we use the notion of the nonequilibrium free energy to derive an integral fluctuation relation for nonlinear nonequilibrium systems under feedback control. We point out that the free energy as well as some functionals can serve as a nonequilibrium Lyapunov function which has an important property to decay monotonously to its minimal value at all times. The Poisson-representation technique is illustrated via exact stochastic treatment of biophysical processes, such as bacterial chemosensing and molecular evolution.

Figure 1

Temporal evolution of the probability distribution $P(n,t)$ as a function of the number of particles $n$ and time $t$ for the process of bacterial chemosensing with the following set of parameters: $N=6,n_0=1,\gamma =0.01,k_r=0.06{\mathrm{s}}^{-1},k_b=0.12{\mathrm{s}}^{-1},L=210\mu \mathrm{M},K_{\mathrm{on}}=3000\mu \mathrm{M}$, and $K_{\mathrm{off}}=18.2\mu \mathrm{M}$.

Figure 2

Probability distribution function $P\left(n\right)\equiv P(n,t_i)$ at $t_i=0$, 25, 50, 75, and 100 for the process of bacterial chemosensing with the same set of parameters as in Fig. 1.

Figure 3

Temporal evolution of the mean number of molecules $\overline{n}\left(t\right)$ [dashed (red) line], the Fano factor [the lower solid (dark) line], and the $\alpha$ variable (the solution of Eq. (27), [the upper solid (blue) line] for the process of bacterial chemosensing with the same set of parameters as in Fig. 1).