Thermal Equilibrium of Vortex Lines in Counterflowing He II

Abstract

The problem of the statistics of a set of chaotic vortex lines in counterflowing superfluid helium is studied. We introduced a Langevin-type force into the equation of motion of the vortex line in the presence of relative velocity $\mathbf {v_{ns}}$. This random force is supposed to be Gaussian satisfying the fluctuation–dissipation theorem. The corresponding Fokker–Planck equation for probability functional in the vortex loop configuration space is shown to have a solution in the form of Gibbs distribution with the substitution $E\{\mathbf {s\}\rightarrow }E(\{\mathbf {s\}-P(v_{n}-v_{s})}$, where $E\{\mathbf {s\}}$ is the energy of the vortex configuration $\{\mathbf { s\}}$ and $\mathbf {P}$ is the Lamb impulse. Some physical consequences of this fact are discussed.