Arnold P
Department of Physics, University of Virginia, Charlottesville, Virginia 22901, USA.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Jun;61(6 Pt A):6099-102. doi: 10.1103/physreve.61.6099.
A Langevin equation with multiplicative noise is an equation schematically of the form dq/dt=-F(q)+e(q)xi, where e(q)xi is Gaussian white noise whose amplitude e(q) depends on q itself. I show how to convert such equations into path integrals. The definition of the path integral depends crucially on the convention used for discretizing time, and I specifically derive the correct path integral when the convention used is the natural, time-symmetric one whose time derivatives are (q(t)-q(t-Deltat))/Deltat and coordinates are (q(t)+q(t-Deltat))/2. (This is the convention that permits standard manipulations of calculus on the action, like naive integration by parts.) It has sometimes been assumed in the literature that a Stratonovich Langevin equation can be quickly converted to a path integral by treating time as continuous but using the rule straight theta(t=0)=1 / 2. I show that this prescription fails when the amplitude e(q) is q dependent.
一个带有乘性噪声的朗之万方程是一个形式上大致为dq/dt = -F(q) + e(q)ξ 的方程，其中e(q)ξ 是高斯白噪声，其幅度e(q) 依赖于q 本身。我展示了如何将此类方程转化为路径积分。路径积分的定义关键取决于用于离散时间的约定，并且我特别推导了在使用自然的、时间对称的约定（其时导数为(q(t) - q(t - Δt))/Δt 且坐标为(q(t) + q(t - Δt))/2）时正确的路径积分。（正是这个约定允许对作用量进行标准的微积分操作，比如朴素的分部积分。）文献中有时假定，通过将时间视为连续但使用规则θ(t = 0) = 1/2，斯特拉托诺维奇朗之万方程可以快速转化为路径积分。我表明，当幅度e(q) 依赖于q 时，这个规定是失败的。
