Brownian motion theory of the two-dimensional quantum vortex gas

A theory of Brownian motion is presented for an assembly of vortices. The attempt is motivated by a realization of Dyson's Coulomb gas in the context of quantum condensates. By starting with the time-dependent Landau-Ginzburg (LG) theory, the dynamics of the vortex gas is constructed, which is governed by the canonical equation of motion. The dynamics of point vortices is converted to the Langevin equation, which results in the generalized Fokker-Planck (GFP) (or Smolkovski) equation using the functional integral on the ansatz of the Gaussian white noise. The GFP, which possesses a non-Hermitian property, is characterized by two regimes called the overdamping and the underdamping regimes. In the overdamping regime, where the dissipation is much larger that the vortex strength, the GFP becomes the standard Fokker-Planck equation, which is transformed into the two-dimensional many-particle system. Several specific applications are given of the Fokker-Planck equation. An asymptotic limit of small diffusion is also discussed for the two-vortices system. The underdamping limit, for which the vortex charge is much larger than the dissipation, is briefly discussed.