Classical Mechanics from Stochastic Quantum Dynamics

Abstract

The present study deals with the corresponding stochastic Schringer equation (SSE) leading to the quantum-to-classical transition. This work shows that the stochastic generalisation of the quantum hydrodynamic analogy (QHA) has its corresponding SSE. The SSE owns an imaginary random noise that has a finite correlation distance so that when the physical length of the problem is much smaller than it, the SSE converges to the standard Schringer equation. The model derives the correlation length of the environmental noise, leaving the quantum potential energy of fluctuations finite, and shows that in non-linear (weakly bounded) systems, the term responsible of the non-local interaction in the SSE may have a finite range of efficacy maintaining its non-local effect on a finite distance. A non-linear SSE that describes the related large-scale classical dynamics is derived. The work also shows that at the edge between the quantum and the classical regime the SSE can lead to the semi-empirical Gross-Pitaevskii equation. The SSE can be helpful in describing at larger extent open quantum systems where the environmental fluctuations and the classical effects are both relevant.