On the motion of classical three-body system with consideration of quantum fluctuations

TL;DR

This paper models the classical three-body problem incorporating quantum fluctuations by formulating stochastic differential equations on a curved space, revealing insights into quantum chaos and transition from quantum to classical chaotic regimes.

Contribution

It introduces a novel approach using stochastic differential equations on curved space to include quantum fluctuations in classical three-body dynamics.

Findings

Derived a second-order PDE for quantum probability distribution.

Demonstrated the topological effects on quantum current tubes.

Provided a framework for quantum-classical transition in chaotic systems.

Abstract

We study the multichannel scattering in the classical three-body system and show that the problem can be formulated as a motion of the point mass on a curved hyper-surface of the energy of the body-system. It is proved that the local coordinate system on curved space produces additional symmetries which along with known integrals of motion allow to reduce the initial problem to the system of the sixth order. Assuming, that the metric of the curved space has a random component, we derive the system of \emph{stochastic differential equations} (SED) describing the classical motion of the three-body system taking into account the influence of random forces of various origin and in particular the quantum fluctuations. Using SDEs of motion, we obtain the partial differential equation of the second order describing the probability distribution of the point mass in the momentum representation. It is shown, that the equation for the probability distribution is solved jointly with the classical equations, which in turn are responsible for the topological peculiarities of tubes of quantum currents and transitions between asymptotic channels. The latter allows to solve the problem of the limiting transition from the quantum region to the region of classical chaotic motion (Poincar\'e region) without violating the analogue of Arnold's theorem on quantum mapping. The expression characterizing a measure of deviation of the quantum probabilistic currents and thus the appearance of quantum chaos in a dynamical system is determined.