Equation of state for cool relativistic two-constituent superfluid dynamics.

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 sρ and the gradient ∇ρ φ 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{~s,∇φ} is given by an expression of the form L = P−3ψ where P represents the pressure as a function just of ∇ρ φ in the (isotropic) cold limit. The entropy current dependent contribution ψ represents the generalised pressure of the (nonisotropic) 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 . PACS numbers: 04.40, 67.90