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A Fokker-Planck Model For A Two-Body Problem

机译：两体问题的Fokker-Planck模型

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摘要

The evolution of the state-of-knowledge of system regarding the time-varying state of a particle in the two-body problem is deterministically treated in classical mechanics where no uncertainties in initial conditions or acting forces are assumed. When such uncertainties exist, a Fokker-Planck equation can be formulated that describes the propagation in time of the probability density of possible positions and velocities of a particle. Both dissipative and dispersive forces can be present. This approach allows one to infer the coordinates of a particle in phase space as a function of time (both past and future) subject to acknowledged uncertainties. The Fokker-Planck equation is derived from knowledge of both the deterministic equations of motion and the probability distributions of acting forces. In the absence of uncertainties in acting forces, there are three conserved quantities that correspond to energy, total angular momentum, and the component of angular momentum in the direction of the rotation of the larger body (taken as the earth). The solutions for several cases are discussed. In the first case, gravity is absent while dispersive forces are present. An exact solution of the Fokker-Planck equation is given for this case. In a second case, gravity is present while dispersive forces are not. Here, the solution is time-independent, i.e., a conserved quantity and hence can be normalized. In a third case, both gravity and dispersive effects are present. A solution is derived through a perturbation of the functional corresponding to the exponentiated system energy. The perturbation solution is about the first-order moments of the energy function and thus is a temporal perturbation of the maximum entropy equilibrium distribution.
机译：在经典的力学中确定性地处理了关于二体问题中粒子随时间变化状态的系统知识状态的演变，在这种情况下，假设初始条件或作用力没有不确定性。当存在这种不确定性时，可以制定一个Fokker-Planck方程，该方程描述粒子的可能位置和速度的概率密度随时间的传播。分散力和分散力都可以存在。这种方法允许人们根据公认的不确定性来推断粒子在空间中的坐标随时间的变化（过去和将来）。福克-普朗克方程式是从运动的确定性方程式和作用力的概率分布的知识中得出的。在作用力没有不确定性的情况下，存在三个守恒量，它们分别对应于能量，总角动量以及在较大物体（作为地球）旋转方向上角动量的分量。讨论了几种情况的解决方案。在第一种情况下，不存在重力，而存在分散力。对于这种情况，给出了Fokker-Planck方程的精确解。在第二种情况下，存在重力而没有分散力。在此，解是与时间无关的，即守恒量，因此可以归一化。在第三种情况下，重力和分散作用同时存在。通过扰动对应于指数系统能量的函数可以得出解决方案。摄动解大约是能量函数的一阶矩，因此是最大熵平衡分布的时间摄动。

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