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Physical Review A

Logarithmic nonlinearities within the framework of stochastic mechanics

Antnio B. Nassar

Published Mar 1, 1986• 9 citations
Publisher

Abstract

We investigate special types of solutions of the hydrodynamical version of a generalized Schr\"odinger-Langevin equation (GNLSLE), derived in an earlier work, via stochastic mechanics. Within the same scheme of stochastic differential equations, we decompose the stochastic process associated with the GNLSLE into two independent processes: a classical Langevin-type process and a pure quantum Nelson-type process.

DOI
10.1103/physreva.33.2134

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Intellectual Archive•Sep 21, 2026•0 citations
We investigate special types of solutions of the hydrodynamical version of a generalized Schr"odinger-Langevin equation (GNLSLE), derived in an earlier work, via stochastic mechanics. Within the same scheme of stochastic differential equations, we decompose the stochastic process associated with the GNLSLE into two independent processes: a classical Langevin-type process and a pure quantum Nelson-type process.
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A simple scheme of unified description of different physical phenomena by using the Langevin type equations is reviewed. Within this approach much attention is being paid to the study of Brownian and quantum motions. Stochastic equations with a white noise term give all characteristics of the Brownian motion. Some generalization of the Langevin type equations allows us to obtain nonlinear equations of particles' motion, which are formally equivalent to the Schrödinger equation. Thus, we establish Nelson's stochastic mechanics on the basis of the Langevin equation.
Starting from the forward and backward infinitesimal generators of bilateral, time-homogeneous Markov processes, the self-adjoint Hamiltonians of the generalized Schrödinger equations are first introduced by means of suitable Doob transformations. Then, by broadening with the aid of the Dirichlet forms, the results of the Nelson stochastic mechanics, we prove that it is possible to associate bilateral, and time-homogeneous Markov processes to the wave functions stationary solutions of our generalized Schrödinger equations. Particular attention is then paid to the special case of the Lévy-Schrödinger (LS) equations and to their associated Lévy-type Markov processes, and to a few examples of Cauchy background noise.
The quantum-mechanical formulation of a class of stochastic optimal control problems is investigated and two versions of non-linear Schrödinger equation are derived. First, a wave-function is introduced which connects the probability density function of a diffusion process with the minimum cost functional of the stochastic optimal control problem associated with the diffusion process. It is then demonstrated that this wave-function satisfies a kind of Schrödinger-Langevin equation in quantum physics. Finally, the probabilistic interpretation of the wave-function is discussed.