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 $\mathcal{P}(\{\psi (\mathbf{r})\})=\mathcal{N}\exp (-H\{\psi (\mathbf{r})\}\mathrm{/}T),$ where $H\{\psi (\mathbf{r})\}$ is the Ginzburg-Landau free energy. Considering vortex filaments as topological defects of field $\psi (\mathbf{r})$ 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 $\mathcal{P}(\{\mathbf{s}(\xi )\})$ 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 $H\{\mathbf{s}\}$ 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