Evolution of a Network of Vortex Loops in HeII. Exact Solution of the "Rate Equation"

TL;DR

This paper derives an exact power-law solution to the rate equation governing vortex loop evolution in superfluid helium, revealing a non-equilibrium state with dual fluxes and providing insights into vortex tangle structure and decay.

Contribution

It presents an exact analytical solution to the vortex loop rate equation, demonstrating a non-equilibrium flux state and connecting it to vortex tangle dynamics in superfluid helium.

Findings

The solution is a non-equilibrium state with dual fluxes of length/energy.

The mean radius of curvature is comparable to interline space.

Vortex tangle decay follows the Vinen equation.

Abstract

Evolution of a network of vortex loops in HeII due to the fusion and breakdown of vortex loops is studied. We perform investigation on the base of the ''rate equation'' for the distribution function $n(l)$ of number of loops of length $l$ proposed by Copeland with coauthors. By using the special ansatz in the ''collision'' integral we have found the exact power-like solution of ''kinetic equation'' in stationary case. That solution is the famous equilibrium distribution $n(l)\varpropto l^{-5/2}$ obtained earlier in numerical calculations. Our result, however, is not equilibrium, but on the contrary, it describes the state with two mutual fluxes of the length (or energy) in space of the vortex loop sizes. Analyzing this solution we drew several results on the structure and dynamics of the vortex tangle in the superfluid turbulent helium. In particular, we obtained that the mean radius of the curvature is of order of interline space. We also obtain that the decay of the vortex tangle obeys the Vinen equation, obtained earlier phenomenologically. We evaluate also the full rate of reconnection events. PACS-number 67.40