Non-dissipative hydrodynamics: Effective actions versus entropy current

TL;DR

This work probes whether non-dissipative hydrodynamics can be captured by an action principle and how that action constraints compare to the entropy-current approach. Using an effective action built from Lagrangian fluid elements with volume-preserving diffeomorphisms, the authors derive a second-order, five-parameter family of non-dissipative, neutral fluids, accompanied by a naturally conserved entropy current. Separately, the most general zero-divergence entropy-current analysis yields a seven-parameter (four-parameter conformal) family of non-dissipative fluids, indicating more freedom than the action-based construction. A frame-invariant comparison then maps many, but not all, parameters between the two approaches, revealing a precise set of identifications (e.g. $A_3=-K_5/T$, $\lambda_2=2(K_1+K_2)/T$ and $\lambda_2=-4B_3$) and highlighting a fundamental mismatch that invites interpretation, such as missing operators in the action or extra microscopic constraints beyond entropy conservation. The results motivate further work on parity-odd sectors, holographic connections, and extensions to charged fluids and superfluids to fully understand the landscape of dissipationless hydrodynamics.

Abstract

While conventional hydrodynamics incorporating dissipative effects is hard to derive from an action principle, it is nevertheless possible to construct classical actions when the dissipative terms are switched off. In this note we undertake a systematic exploration of such constructions from an effective field theory approach and argue for the existence of non-trivial second order non-dissipative hydrodynamics involving pure energy-momentum transport. We find these fluids to be characterized by five second-order transport coefficients based on the effective action (a three parameter family is Weyl invariant). On the other hand since all flows of such fluids are non-dissipative, they entail zero entropy production; one can therefore understand them using the entropy current formalism which has provided much insight into hydrodynamic transport. An analysis of the most general stress tensor with zero entropy production however turns out to give a seven parameter family of non-dissipative hydrodynamics (a four parameter sub-family being Weyl invariant). The non-dissipative fluids derived from the effective action approach are a special case of the fluid dynamics constrained by conservation of the entropy current. We speculate on the reasons for the mismatch and potential limitations of the effective action approach.