Frequency Factors in the Thermally Activated Process

Abstract

Discussions of the rate $\ensuremath{\nu}{e}^{\ensuremath{-}\frac{U}{\mathrm{kT}}}$ for thermally activated processes are usually based on the phase-space distribution function for thermal equilibrium. Kramers has gone beyond this and for the particle in a bistable one-dimensional well has treated the relaxation to equilibrium as a Brownian motion problem in which the one-dimensional motion is coupled to a reservoir through a viscosity. Kramers' arguments are readily extendable to many dimensions. In the overdamped case the reaction rate is reduced below the value derived from thermal equilibrium theory by the factor $\frac{{\ensuremath{\omega}}_{s}}{\ensuremath{\eta}}$, where ${\ensuremath{\omega}}_{s}$ is the angular frequency associated with the direction of steepest descent at the saddle point and $\ensuremath{\eta}$ the viscosity. In the underdamped case equilibrium theory is valid for many-dimensional systems, except for extreme degrees of underdamping.