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Some new Conditions for the Existence of Singular non Symmetric Diffusions
No full textSome new conditions which ensure the existence of diffusion processes with values in properly associated to Non-Symmetric Dirichlet Forms are given. The results are extended to the case of diffusions taking values in Wiener Spaces.Some of them can be expressed in terms of dynamical potentials appearing in Schroedinger operators so that they are suitable for application to Stochastic Mechanics both in finite and infinite dimension
Formation of Singularirties in Madelung fluid : a nonconventional application of Ito calculus to foundations of quantum Mechanics
No full textStochastic Quantization is a procedure which provides the equation of motion of a Quantum System starting from its classical description and incorporating quantum effects into a stochastic kinematics. After the pioneering work by E.Nelson in 1966 the method has been developed in the eighties in various different ways. In this communication I revisit an approach based on a lagrangian variational principle were 3/2 order contributions in It\^o calculus are needed. The result is a generalization of Madelung fluid equations where in particular velocity fields with vorticity are allowed. Such a vorticity induces dissipation of the energy so that the irrotational solutions, corresponding to the usual conservative solutions of Schroedinger equation, act as an attracting set.Recent numerical experiments show generation of zeroes of the density with concentration of vorticity and formation of isolated central vortex lines
A relativistically covariant stochastic model for systems with a fluctuating number of particles
No full textSpectral methods for dissipative nonlinear Schrödinger equations
Full text linkIn this technical report we describe an application of spectral methods to numerically solve some nonlinear Schrödinger equations with dissipative terms and carefully study the problem of vortex formation
Dissipation caused by a vorticity field and generation of singularities in Madelung fluid
No full textWe consider a generalization of Madelung fluid equations, whichwas derived in the 1980s by means of a pathwise stochastic calculus of variationswith the classical action functional. At variance with the original ones, the newequations allowus to consider velocity fields with vorticity. Such a vorticity causesdissipation of energy and it may concentrate, asymptotically, in the zeros of thedensity of the fluid. We study, by means of numerical methods, some Cauchyproblems for the bidimensional symmetric harmonic oscillator and observe thegeneration of zeros of the density and concentration of the vorticity close tocentral lines and cylindrical sheets. Moreover, keeping the same initial data, weperturb the harmonic potential by a term proportional to the density of the fluid,thus obtaining an extension with vorticity of the Gross–Pitaevskii equation, andobserve analogous behaviours.
Stochastic quantization for a system of N identical interacting Bose particles
No full textWe apply stochastic quantization to a system of N interacting identical bosons in an external potential phi, by means of a general stationary- action principle. The collective motion is described in terms of a Markovian diffusion on R-3N, with joint density rho. and entangled current velocity field V, in principle of nongradient form, related to one another 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., 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. 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. 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-boson 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 the Gross-Pitaevskii equations
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