Stochastic Dynamics of a Vortex Loop. Thermal Equilibrium

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