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 P({psi(r)}) = N exp(-H {psi(r)} /T), where H {psi(r)} is a Ginzburg-Landau free energy. Considering a vortex filaments as a topological defects of the field 0(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 P({s(xi)}) in vortex loop configuration space is shown to have a solution of the form P({s(xi)}) = N exp(-H {s} /T), where N is a normalizing factor and H {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.