Departamento de Química, Universidad de Los Andes, Mérida 5101, Venezuela.
Phys Rev E. 2018 May;97(5-1):052126. doi: 10.1103/PhysRevE.97.052126.
This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.
本文涉及与在外加简谐场中运动的粒子的动量积分维格纳函数相关的福克-普朗克方程的推导和解。所采用的策略是对 Schramm、Jung 和 Grabert 的唯象方法的简化版本，即将算子解释为实数值以从整个相空间的维格纳函数的两次变换中推导出量子主方程。随机噪声的统计特性来自于 Grabert、Schramm 和 Ingold 的积分泛函理论。通过单个维格纳变换，发现了一个比前面提到的更简单的方程。维格纳函数再现了经典极限的已知结果。这使我们能够将欠阻尼经典朗之万方程重写为具有时变漂移和扩散项的一阶随机微分方程。
