Langevin Dynamics of Vortex Lines and Thermodynamic Equilibrium of Vortex Tangle

Abstract

Langevin dynamics—stochastic motion of vortex filaments under action of random force is studied analytically and numerically. We introduce a Langevin-type equation of motion of the line with a stirring force satisfying the fluctuation-dissipation theorem. The respective Fokker-Planck equation for probability functional ℘({s(ξ)}) in vortex loop configuration space is shown to have a solution of the form $\mathcal{P}(\{\mathbf{s}(\xi )\})=\mathcal{N}\exp (-H\{\mathbf{s}\}/T),$ where $\mathcal{N}$ is a normalizing factor and H{s} is energy of vortex line configurations. Numerical calculations are performed on base of the full Biot-Savart law for different intensities of the Langevin force. A new algorithm, which is based on consideration of crossing lines, is used for vortex reconnection procedure. After some transient period the vortex tangle develops into the stationary state characterizing by the developed fluctuations of various physical quantities, such as total length, energy etc. We tested this state to learn whether or not it the thermodynamic equilibrium is reached. With the use of a special treatment, so called method of weighted histograms, we process the distribution energy of the vortex system. The results obtained demonstrate that the thermodynamical equilibrium state is reached.