Large barrier behavior of the rate constant from the diffusion equation

Many processes in chemistry, physics, and biology depend on thermally activated events in which the system changes its state by surmounting an activation barrier. Examples range from chemical reactions to protein folding and nucleation events. Parameterized forms of the mean field potential are often employed in the stochastic modeling of activated processes. In this contribution, we explore the alternative of employing parameterized forms of the equilibrium distribution by means of symmetric linear combination of two Gaussian functions. Such a procedure leads to flexible and convenient models for the landscape and the energy barrier whose features are controlled by the second moments of these Gaussian functions. The rate constants are examined through the solution of the corresponding diffusion problem, that is, the Fokker-Planck-Smoluchowski equation specified according to the parameterized equilibrium distribution. Numerical calculations clearly show that the asymptotic limit of large barriers does not agree with the results of the Kramers theory. The underlying reason is that the linear scaling of the potential, the procedure justifying the Kramers theory, cannot be applied when dealing with parameterized forms of the equilibrium distribution. A different kind of asymptotic analysis is then required and we introduce the appropriate theory when the equilibrium distribution is represented as a symmetric linear combination of two Gaussian functions: first in the one-dimensional case and afterward in the multidimensional diffusion model.