Evaluation of the smallest nonvanishing eigenvalue of the fokker-planck equation for brownian motion in a potential: the continued fraction approach

Abstract

An equation for the smallest nonvanishing eigenvalue lambda(1) of the Fokker-Planck equation (FPE) for the Brownian motion of a particle in a potential is derived in terms of continued fractions. This equation is directly applicable to the calculation of lambda(1) if the solution of the FPE can be reduced to the solution of a scalar three-term recurrence relation for the moments (the expectation values of the dynamic quantities of interest) describing the dynamical behavior of the system under consideration. In contrast to the previously available continued fraction solution for lambda(1) [for example, H. Risken, The Fokker-Planck Equation, 2nd ed. (Springer, Berlin, 1989)], this equation does not require one to solve numerically a high order polynomial equation, as it is shown that lambda(1) may be represented as a sum of products of infinite continued fractions. Besides its advantage for the numerical calculation, the equation so obtained is also very useful for analytical purposes, e.g., for certain problems it may be expressed in terms of known mathematical (special) functions. Another advantage of such an approach is that it can now be applied to systems whose relaxation dynamics is governed by divergent three-term recurrence equations. To test the theory, the smallest eigenvalue lambda(1) is evaluated for several double-well potentials, which appear in various applications of the theory of rotational and translational Brownian motion. It is shown that for all ranges of the barrier height parameters the results predicted by the analytical equation so obtained are in agreement with those obtained by independent numerical methods. Moreover, the asymptotic results for lambda(1) previously derived for these particular problems by solving the FPE in the high barrier limit are readily recovered from the analytical equations.