A theory of first order dissipative superfluid dynamics

Abstract

We determine the most general form of the equations of relativistic superfluid hydrodynamics consistent with Lorentz invariance, time-reversal invariance, the Onsager principle and the second law of thermodynamics at first order in the derivative expansion. Once parity is violated, either because the U(1) symmetry is anomalous or as a consequence of a different parity-breaking mechanism, our results deviate from the standard textbook analysis of superfluids. Our general equations require the specification of twenty parameters (such as the viscosity and conductivity). In the limit of small relative superfluid velocities we find a seven parameter set of equations. In the same limit, we have used the AdS/CFT correspondence to compute the parity odd contributions to the superfluid equations of motion for a generic holographic model and have verified that our results are consistent.

Aspects of hot Galilean field theory

“…As far as upper index data goes, v µ shifts as 4) and h µν is invariant. One can readily check that the simplest Galilean field theory, 5) with D µ Ψ = (∂ µ − imA µ )Ψ, is invariant under all of these symmetries. It is easy to derive Ward identities from these symmetries [11,15].…”

“…Then running through the construction above, we find that x E /c has period β, so x E has period cβ from which we would identify the temperature to be T = 1/(cβ). In general, the physical temperature is 5) or the inverse of the integral of n µ around the thermal circle. Similarly, the physical chemical potential is defined through…”

“…Using standard thermodynamic relations, it is not hard to show that (see e.g. appendix B of [5]) 33) where now the thermodynamic derivatives are taken at fixed ε and ρ respectively. The vector equation is…”

“…By paying careful attention to the entropy argument, Son and Surowka [4] have shown that quantum anomalies modify hydrodynamics at first order in the gradient expansion. The authors of [5] worked out the systematics of hydrodynamic field redefinitions as well as the full theory of first-order dissipative superfluid hydrodynamics. By turning on a slowly varying background metric and/or gauge fields and solving the fluid equations on that background, one accesses the full suite of retarded hydrodynamic correlation functions at the level of linear [2] and non-linear [6] response.…”

Effective actions for anomalous hydrodynamics

“…From these analyses one learns that the family of non-dissipative fluids derivable from an effective action is restricted when compared to those which one would obtain by writing down constitutive relations for conserved charges together with demanding the existence of a conserved entropy current [48]. The latter construction is a variant of conventional approaches to hydrodynamics, generalizing earlier analyses of [49][50][51][52] (to restrict oneself to non-dissipative fluids). While the statement seems natural, in that the existence of an effective action is a more stringent condition than simply the lack of entropy production, the precise details of the effective actions described in the aforementioned references leave something to be desired.…”