Survival paths for reaction dynamics in fluctuating environments

We study rate processes in general Gaussian fluctuating environments using a path integral formalism. We derive a variational equation for the dominant survival path when the fluctuations relax exponentially or according to a stretched exponential law. In the case of a slowly varying barrier, the equilibrium regression approximation which is used by Frauenfelder and coworkers emerges. In this approximation, the survival path follows the ordinary law of relaxation to equilibrium. If the rate coefficients vary rapidly with environmental variables, however, the dominant survival paths exhibit more complex behaviour. Many phenomena analogous to geometrical optics occur. These include reflection off of rapid variations in rate constant, as well as refraction, giving paths very different from the equilibrium relaxation properties. A model with a piece-wise linear rate exhibits the basic phenomena, and the survival path equation is exactly solved for the general stretched exponential relaxing environment.