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首页> 外文期刊>The Journal of Chemical Physics >EXACT ANALYTIC SOLUTION FOR THE CORRELATION TIME OF A BROWNIAN PARTICLE IN A DOUBLE-WELL POTENTIAL FROM THE LANGEVIN EQUATION
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EXACT ANALYTIC SOLUTION FOR THE CORRELATION TIME OF A BROWNIAN PARTICLE IN A DOUBLE-WELL POTENTIAL FROM THE LANGEVIN EQUATION

机译：朗格文方程双势势中布朗粒子相关时间的精确解析解

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摘要

The correlation time of the positional autocorrelation function is calculated 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 (by averaging) of the Langevin equation for a nonlinear stochastic system to a set of differential-recurrence relations. These, in the present problem, reduce on taking the Laplace transform, to a three-term recurrence relation. Thus the correlation time T-c of the positional autocorrelation function may be formally expressed as a sum of products of infinite continued fractions which may be represented in series form as a sum of two term products of Whittaker's parabolic cylinder functions. The sum of this series may be expressed as an integral using the integral representation of the parabolic cylinder functions and subsequently the Taylor expansion of the error function, thus yielding the exact solution for T-c. This solution is in numerical agreement with that 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 limit of high barrier heights. A simple empirical formula which provides a close approximation to the exact solution for all barrier heights is also given. (C) 1996 American Institute of Physics. [References: 24]
机译：位置自相关函数的相关时间是针对非惯性极限中2-4势阱中粒子的一维平移布朗运动精确计算的。使用非线性随机系统的Langevin方程直接转换（通过平均）为一组微分递归关系的方法进行计算。在当前问题中，这些将拉普拉斯变换简化为三项递归关系。因此，位置自相关函数的相关时间T-c可以形式表示为无限连续分数的乘积之和，其可以以惠特克抛物柱面函数的两个项乘积的总和形式表示。该系列的总和可以使用抛物柱面函数的积分表示以及随后的误差函数的泰勒展开表示为积分，从而得出T-c的精确解。该解决方案与Perico等人获得的解决方案在数值上一致。 [J.化学物理98，564（1993）]使用第一次通过时间方法，同时通过求解基础的Smoluchowski方程获得了先前的渐近结果，并在高势垒高度的极限中得到了恢复。还给出了一个简单的经验公式，它为所有势垒高度提供了与精确解的近似值。 （C）1996年美国物理研究所。 [参考：24]

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