Collision Times and Kinetic Theory for Superfluid Helium

Abstract

We have found that the low-temperature collision-time kinetic theories of Khalatnikov and Chernikova and of Disatnik are not in agreement with all of the predictions of superfluid hydrodynamics. In particular, they overestimate, by a factor of about 30, the attenuation of hydrodynamic first sound. In addition, they use a hydrodynamic collision time to describe both the hydrodynamic and collisionless regimes, causing disagreement with many-body calculations valid for the collisionless regime. We have constructed a kinetic theory which is in agreement with superfluid hydrodynamics, and we have analyzed the existing kinetic theories in order to find the origins of the disagreement with hydrodynamics and with the many-body theory. To obtain correct transport coefficients (and therefore correct attenuation of hydrodynamic first sound) it is found necessary to ensure that the system relaxes to local, rather than to static, equilibrium. We calculate the transport coefficients within a collision-time model, and point out that for generality one must employ two hydrodynamic collision times ${\ensuremath{\tau}}_{l}$ and ${\ensuremath{\tau}}_{t}$ (with ${\ensuremath{\tau}}_{l}\ensuremath{\ge}{\ensuremath{\tau}}_{t}$) associated with longitudinal and transverse processes, respectively. From this we show that ${\ensuremath{\zeta}}_{2}$ is not necessarily equal to zero. In addition, we discuss difficulties involved in extending the theory to higher frequencies, and present a physical argument which requires the use of a wide-angle collision time to define the (low-frequency) hydrodynamic regime and a small-angle collision time to define the (high-frequency) collisionless regime. This reconciles the disagreement between kinetic theory and many-body calculations upon the collision time which defines the collisionless regime. Various experiments are interpreted on this basis, thereby eliminating certain discrepancies with theory. Measurements in the low-temperature hydrodynamic regime are shown to require relatively large chambers (with linear dimensions \ensuremath{\gtrsim} 10 cm).