Stochastic theory of quantum vortex on a sphere

A stochastic theory is presented for a quantum vortex in superfluid films coated on a two-dimensional sphere S-2. The starting point is the canonical equation of motion (Kirchhoff equation) for a point vortex, which is derived using the time-dependent Landau-Ginzburg theory. The vortex equation, which is equivalent to the spin equation, turns out to be the Langevin equation in presence of random forces. This is converted to the Fokker-Planck (FP) equation for the distribution function of a point vortex by using a functional integral technique. The FP equation is analyzed with special emphasis on the role of the pinning potential. By considering a typical form of the pinning potential, we address two problems: (i) The one is concerning an interplay between strength of the pinning potential and effective temperature, which discriminates the weak and strong coupling scheme to determine the solutions of the FP equation. (ii) The other is concerning a small diffusion limit, for which an asymptotic analysis is given using the functional integral to lead a compact expression of the distribution function. An extension to the vortex in nonspherical geometry is briefly discussed for the case of vortex on a plane and a pseudosphere.