Chaotic Quantum Vortexes In A Weakly Non Ideal Bose Gas. Thermodynamic Equilibrium And Turbulence

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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)})= N (-H{ψ(r)} /T), where H{ψ(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)})= N (-H{ s} /T), where N is the normalizing factor and H{ 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