Path Integral in Irreversible Statistical Dynamics

Extending the path probability method for irreversible statistical dynamics proposed previously, the two-gate probability distribution for a stochastic Markoffian system is written in the form of a path integral. The most probable path is derived as a solution of the variational Euler-Lagrange equation. Introducing the concept of the anticausal path, a method to calculate the one-gate probability distribution function (for fluctuations away from the nonequilibrium steady state) is explained. The distribution function derived recently by Mathews, Shapiro, and Falkoff for a system of many levels is shown to follow. The concept of the pseudo-entropy $\tilde{S}$ is introduced. $\tilde{S}$ always increases in time as the system approaches its steady state. It is shown that the rate $\frac{d\tilde{S}}{\mathrm{dt}}$, rather than $\frac{\mathrm{dS}}{\mathrm{dt}}$ of Prigogine, is minimum in the steady state, although the latter can serve as an approximation when the steady state is near equilibrium.