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
P({ψ(r)})=Nexp(−H{ψ(r)}/T), where $H\{ψ({\bf r})\} $ is the Ginzburg-Landau free energy. Considering vortex filaments as topological defects of field
ψ(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
P({s(ξ)}) in vortex loop configuration space is shown to have a solution in the form of
P({s(ξ)})=Nexp(−H{s}/T), where
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
We have studied computationally turbulent non-equilibrium dynamics of quasi-two-dimensional superfluid Bose-Einstein condensates. Beginning with a random out-of-equilibrium distribution of vortices and antivortices in the condensate, we have observed emergence of ordered vortex structures corresponding to absolute negative temperature states in this conservative Hamiltonian system. We explain the spontaneous self-organization of the singly quantized vortices into two``Onsager vortex''clusters of like-signed vortices in terms of an``evaporative heating''mechanism of the vortex gas. These results provide a clear pathway to observing Onsager vortex structures and their unusual thermodynamics in a superfluid Bose gas