The Equation of State for Cool Relativistic Two-Constituent Superfluid Dynamics

TL;DR

This paper develops a relativistic superfluid theory for the 'cool' regime where entropy is due solely to phonons, deriving an explicit Lagrangian form linking pressure, entropy, and phonon speed.

Contribution

It provides a relativistic Lagrangian formulation for superfluid dynamics in the cool regime, explicitly relating pressure, entropy, and phonon properties.

Findings

Lagrangian expressed as L = P - 3ψ, linking pressure and phonon entropy contributions.

Explicit algebraic form of the phonon gas grand potential energy in terms of phonon speed.

Connection between cold pressure function and relativistic superfluid thermodynamics.

Abstract

The natural relativistic generalisation of Landau's two constituent superfluid theory can be formulated in terms of a Lagrangian $L$ that is given as a function of the entropy current 4-vector $s^\rho$ and the gradient $\nabla\varphi$ of the superfluid phase scalar. It is shown that in the ``cool" regime, for which the entropy is attributable just to phonons (not rotons), the Lagrangian function $L(\vec s, \nabla\varphi)$ is given by an expression of the form $L=P-3\psi$ where $P$ represents the pressure as a function just of $\nabla\varphi$ in the (isotropic) cold limit. The entropy current dependent contribution $\psi$ represents the generalised pressure of the (non-isotropic) phonon gas, which is obtained as the negative of the corresponding grand potential energy per unit volume, whose explicit form has a simple algebraic dependence on the sound or ``phonon" speed $c_P$ that is determined by the cold pressure function $P$.