Stochastic Dynamics of a Vortex Loop. Thermal Equilibrium

We study stochastic behavior of a single vortex loop appeared in imperfect Bose gas. Dynamics of Bose-condensate is supposed to obey Gross-Pitaevskii equation with additional noise satisfying fluctuation-dissipation relation. The corresponding Fokker-Planck equation for probability functional has a solution $$\mathcal{P}$({{\psi}($\mathbf{r}$)})=$\mathcal{N}$\exp (-H{\psi}($\mathbf{r})$ /T),$where$H\psi($\mathbf{r}$)$ is a Ginzburg-Landau free energy. Considering a vortex filaments as a topological defects of the field $\psi (\mathbf{r})$ we derive a Langevin-type equation of motion of the line with 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 of the form $\mathcal{P}(\{\mathbf{s}(\xi )\})=\mathcal{N}\exp (-H\mathbf{s}\mathrm{/}T),$ where $\mathcal{N}$ is a normalizing factor and $H\mathbf{s}$ is energy of vortex line configurations. In other words a thermal equilibrium of Bose-condensate results in a thermal equilibrium of vortex loops appeared in Bose-condensate. Some consequences of that fact and possible violations are discussed.