Chaotic Quantum Vortexes In A Weakly Non Ideal Bose Gas. Thermodynamic Equilibrium And Turbulence

We study the stochastic behavior of a set of chaotic vortex loops appeared in imperfect Bose gas. Dynamics of Bose-gas is supposed to obey Gross-Pitaevskii equation with additional noise satisfying fluctuation-dissipation relation. The corresponding Fokker-Planck equation for probability functional has solution ${\cal P}(\{{\psi}({\bf r})\})={\cal N}\exp (-H\{{\psi}({\bf r)}\} /T),$ where $H\{{\psi}({\bf r})\} $ is the Ginzburg-Landau free energy. Considering vortex filaments as topological defects of field ${\psi}({\bf r})$ we derive a Langevin-type equation of motion of the line with the correspondingly transformed stirring force. The respective Fokker-Planck equation for probability functional ${\cal P}(\{{\bf s}(\xi)\})$ in vortex loop configuration space is shown to have a solution in the form of ${\cal P}(\{{\bf s}(\xi)\})={\cal N}\exp (-H\{{\bf s}\} /T),$ where ${\cal N}$ is the normalizing factor and $H\{{\bf s}\} $ is energy of vortex line configurations. Analyzing this result we discuss possible reasons for destruction of the thermodynamic equilibrium and follow the mechanisms of transition to the turbulent state