Stochastic theory of quantum vortex on a sphere

TL;DR

This paper develops a stochastic framework for quantum vortices on a sphere, deriving the Langevin and Fokker-Planck equations from the Landau-Ginzburg theory, and analyzes vortex dynamics with pinning potentials.

Contribution

It introduces a novel stochastic model for quantum vortices on spherical surfaces, deriving equations of motion from fundamental theories and solving for specific vortex behaviors.

Findings

Derived Langevin and Fokker-Planck equations for vortex dynamics

Solved vortex motion equations with various pinning potentials

Discussed extensions to non-spherical vortex geometries

Abstract

A stochastic theory is presented for a quantum vortex that is expected to occur in superfluids coated on two dimensional sphere $ {\rm S}^2 $. The starting point is the canonical equation of motion (the 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, from which the Fokker-Planck equation is obtained by using the functional integral technique. The Fokker-Planck equation is solved for several typical cases of the vortex motion by noting the specific form of pinning potential. An extension to the non-spherical vortices is briefly discussed for the case of the vortex on plane and pseudo-sphere.