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

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 (Poincaré 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.