EXACT ANALYTIC SOLUTION FOR THE CORRELATION TIME OF A BROWNIAN PARTICLE IN A DOUBLE-WELL POTENTIAL FROM THE LANGEVIN EQUATION

The correlation time of the positional autocorrelation function is cal culated exactly for one-dimensional translational Brownian motion of a particle in a 2-4 double-well potential in the noninertial limit. The calculations are carried out using the method of direct conversion (b y averaging) of the Langevin equation for a nonlinear stochastic syste m to a set of differential-recurrence relations. These, in the present problem, reduce on taking the Laplace transform, to a three-term recu rrence relation. Thus the correlation time T-c of the positional autoc orrelation function may be formally expressed as a sum of products of infinite continued fractions which may be represented in series form a s a sum of two term products of Whittaker's parabolic cylinder functio ns. The sum of this series may be expressed as an integral using the i ntegral representation of the parabolic cylinder functions and subsequ ently the Taylor expansion of the error function, thus yielding the ex act solution for T-c. This solution is in numerical agreement with tha t obtained by Perico et al. [J. Chem. Phys. 98, 564 (1993)] using the first passage time approach while previous asymptotic results obtained by solving the underlying Smoluchowski equation an recovered in the l imit of high barrier heights. A simple empirical formula which provide s a close approximation to the exact solution for all barrier heights is also given. (C) 1996 American Institute of Physics.