From Classical Fields to the Two-Fluid Model of Superfluidity: Emergent Kinetics and Local Gauge Transformations

Abstract

The first successful macroscopic theory for the motion of superfluid helium was that of Lev Landau (1941) in which the fluid is modelled phenomenologically as an interpenetrating mixture of a superfluid and a normal fluid. It was later shown that Landau's two-fluid model can be derived from a one-fluid model within the classical field approximation. Assuming a separation of scales exists between the slowly varying, large-scale, background (condensate) field, and the short rapidly evolving excitations, a full description of the kinetics between the condensate and the thermal cloud can be obtained. The kinetics describes three-wave and four-wave interactions that resemble the $C_{12}$ and $C_{22}$ terms, respectively, in the collision integrals of the ZNG theory (Chapter 5). The scale separation assumption precludes analysis of the healing layer and thus does not include the dynamics of quantised vortices. Whilst the analysis required the use of small parameters arising from the scale separation assumption and the assumption of a weakly depleted condensate, we expect the results to hold over a wider range of parameters. This is motivated by the validity of Landau's two-fluid model which can be derived from a one-fluid model using nothing more than the principle of Galilean invariance. Indeed, we argue that similar arguments can be used to recover a two-fluid model directly from a classical field simply by invoking a local gauge transformation. This derivation does not require any small parameters to be introduced suggesting that the results that lead to the kinetic equations may turn out to be more general.