Evaluation of the smallest nonvanishing eigenvalue of the fokker-planck equation for the brownian motion in a potential. II. The matrix 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 matrix-continued fractions. This equation is applicable to the calculation lambda(1) if the solution of the FPE can be reduced (by expanding the probability distribution function in terms of a complete set of appropriate functions) to the solution of a multiterm recurrence relation for the moments describing the dynamics of the Brownian particle. In contrast to the available continued fraction solution for lambda(1) [H. Risken, The Fokker-Planck Equation (Springer, Berlin, 1989)], this equation does not require one to solve numerically a high order polynomial equation. To test the theory, the smallest eigenvalue lambda(1) is evaluated for the FPE, which appears in the theory of magnetic relaxation of single domain (superparamagnetic) particles. Various regimes of relaxation of the magnetization in superparamagnetic particles are governed by a damping parameter alpha, the limiting values of which correspond to the high damping (alpha-->infinity) and the low damping (alpha<<1) limits in the theory of the escape rate over potential barriers. It is shown that for all ranges of the barrier height and damping parameters the smallest eigenvalue lambda(1) predicted by the continued fraction equation is in agreement with those gained by independent numerical methods and the asymptotic estimates for lambda(1) (in the high barrier limit) and, moreover, the matrix continued fraction approach may be successfully applied to the evaluation of lambda(1) in those ranges of parameters where traditional methods fail or are not applicable.