Eigenvalues of the Fokker–Planck Operator and the Approach to Equilibrium for Diffusions in Potential Fields

Abstract

We consider the motion of a Brownian particle in an infinite potential field. The rate of approach to equilibrium is determined by the second eigenvalue of the stationary Fokker–Planck operator. The inverse of this eigenvalue is the expected time for the particle to overcome the potential barriers on its way to the deepest potential well. The height of the largest potential barrier is termed the activation energy, and the eigenvalue is computed asymptotically for large activation energies. Applications to the calculation of chemical reaction rates and ionic conductance in crystals are given.