Path Distribution for Irreversible Processes

An expression is derived which gives the probability of any specified response of a driven linear dissipative system. The probability that a system driven by a force $V_i\mathrm{}\left(t\right)\mathrm{}$ will follow an irreversible path taking it through the neighborhood of ${\tilde{Q}}_k$ at time $t$ is $P({\tilde{Q}}_k, t)=P_0\left({\tilde{Q}}_k\right)\exp \left(\beta \int {-\infty }^{t}d{t}^{\prime }{V}_i\left(t^{\prime }\right){〈〉\left(\frac{d}{\mathrm{dt}}{\tilde{Q}}_i(t-t^{\prime })\right)}_{{\tilde{Q}}_k\mathrm{}\left(0\right)\mathrm{}}\right),$ where $P_0\left({\tilde{Q}}_k\right)$ is the equilibrium distribution function, and the last factor is the mean value in the equilibrium ensemble of $\frac{d{\tilde{Q}}_i}{\mathrm{dt}}$ at time $t-t^{\prime }$, conditional on ${\tilde{Q}}_k$ at time zero. The path distribution function above is the classical ($\hslash \to 0$) approximation to the quantum-mechanical distribution, in which $\beta \equiv \frac{1}{\mathrm{kT}}$ is replaced by a somewhat more complicated form. The mean square path deviation is found.