Frequency Factors in the Thermally Activated Process

Discussions of the rate νe^-U/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 ωₛη, where ωₛ is the angular frequency associated with the direction of steepest descent at the saddle point and η the viscosity. In the underdamped case equilibrium theory is valid for many-dimensional systems, except for extreme degrees of underdamping.