Kalmykov YP
Centre d'Etudes Fondamentales, Universite de Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan Cedex, France.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Jun;61(6 Pt A):6320-9. doi: 10.1103/physreve.61.6320.
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.
利用连分数推导出了粒子在势场中做布朗运动的福克 - 普朗克方程（FPE）最小非零本征值(\lambda(1))的一个方程。如果FPE的解可以简化为描述所考虑系统动力学行为的矩（感兴趣的动态量的期望值）的标量三项递推关系的解，那么这个方程可直接用于计算(\lambda(1))。与之前可得的(\lambda(1))的连分数解[例如，H. Risken，《福克 - 普朗克方程》，第2版（施普林格出版社，柏林，1989年）]不同，这个方程不需要求解高阶多项式方程，因为已表明(\lambda(1))可以表示为无限连分数乘积的和。除了在数值计算方面的优势外，所得到的这个方程对于解析目的也非常有用，例如，对于某些问题它可以用已知的数学（特殊）函数表示。这种方法的另一个优点是它现在可以应用于弛豫动力学由发散三项递推方程支配的系统。为了检验该理论，对几个双阱势计算了最小本征值(\lambda(1))，这些双阱势出现在旋转和平移布朗运动理论的各种应用中。结果表明，对于势垒高度参数的所有取值范围，由此得到的解析方程所预测的结果与通过独立数值方法得到的结果一致。此外，之前通过在高势垒极限下求解FPE为这些特定问题推导的(\lambda(1))的渐近结果可以很容易地从解析方程中得到。
