Activated rate processes in a multidimensional case. A new solution of the Kramers problem

Abstract

The problem of the escape of a classical particle from a multidimensional potential well due to the influence of a random force is studied. It is shown that for potentials of a certain type and a strong enough friction anisotropy, the well-known solution of the multidimensional Kramers problem results in an absurd dependence. A new solution of the Kramers problem free from this shortcoming has been obtained. In finding this solution, we used friction anisotropy and reduced the initial multidimensional Fokker-Planck equation to an effective one-dimensional equation by eliminating fast relaxing modes. It is shown that the solution of this equation depending on the friction anisotropy contains both the well-known solution of the multidimensional Kramers problem and a new solution which corresponds to a qualitative process picture appreciably different from the traditional one. In this anomalous decay regime, the kinetics still retains a simple one-exponential nature, P( t) = exp(− Γt), where P( t) is the probability to avoid the decay during the time t; however, the rate constant Γ is substantially lower than that predicted by the traditional formula.