In the general framework of stochastic control theory we introduce a suitable form of stochastic action associated to the controlled process. A variational principle gives all the main features of Nelson's stochastic mechanics. In particular, we derive the expression for the current velocity field as the gradient of the phase action. Moreover, the stochastic corrections to the Hamilton-Jacobi equation are in agreement with the quantum-mechanical form of the Madelung fluid (equivalent to the Schrödinger equation). Therefore, stochastic control theory can provide a very simple model simulating quantum-mechanical behavior. © 1983 The American Physical Society
Abstract. In this work we give a positive answer to the following question: does Stochastic Mechanics uniquely define a three-dimensional stochastic process which describes the motion of a particle in a Bose-Einstein condensate? To this extent we study a system of N trapped bosons with pair interaction at zero temperature under the Gross-Pitaevskii scaling, which allows to give a theoretical proof of Bose-Einstein condensation for interacting trapped gases in the limit of N going to infinity. We show that under the assumption of strictly positivity and continuous differentiability of the many-body ground state wave function it is possible to rigorously define a one-particle stochastic process, unique in law, which describes the motion of a single particle in the gas and we show that, in the scaling limit, the one-particle process continuously remains outside a time dependent random "interaction-set" with probability one. Moreover, we prove that its stopped version converges, in a relative entropy sense, toward a Markov diffusion whose drift is uniquely determined by the order parameter, that is the wave function of the condensate.
In this paper we present some hydrodynamical consequences of a previously proposed stochastic model for superfluid4He. We discuss in particular the possibility of time-dependent evolutions which, starting from a rotational initial state, lead to asymptotic stationary solutions where the vorticity is concentrated in singular regions. An example of such asymptotic stationary solutions is the quantized vortex line solution. We also recall the concept of quantum critical slipping velocity and investigate some possible consequences on the spin-up problem and on the creation of systems of vortex lines
We apply Stochastic Quantization to a system of N interacting identical Bosons in an external potential Φ, by means of a general stationary-action principle. The collective motion is described in terms of a Markovian diffusion on R 3N , with joint densityρ and entangled current velocity fieldV , in principle of non-gradient form, related one to the other by the continuity equation. Dynamical equations relax to those of canonical quantization, in some analogy with Parisi-Wu stochastic quantization. Thanks to the identity of particles, the one-particle marginal densities ρ, in the physical space R 3 , are all the same and it is possible to give, under mild conditions, a natural definition of the single-particle current velocity, which is related to ρ by the continuity equation in R 3. The motion of single particles in the physical space comes to be described in terms of a non-Markovian three-dimensional diffusion with common density ρ and, at least at dynamical equilibrium, common current velocity v. The three-dimensional drift is perturbed by zero-mean terms depending on the whole configuration of the N-bosons interacting system. Finally we discuss in detail under which conditions the one-particle dynamical equations, which in their general form allow rotational perturbations, can be particularized, up to a change of variables, to Gross-Pitaevskii equations.
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