Hydrodynamics is the low-energy effective field theory of any interacting quantum theory, capturing the long-wavelength fluctuations of an equilibrium Gibbs density matrix. Conventionally, one views the effective dynamics in terms of the conserved currents, which should be expressed in terms of the fluid velocity and the intensive parameters such as the temperature and chemical potential. However, not all currents allowed by symmetry are physically acceptable; one has to ensure that the second law of thermodynamics is satisfied on all physical configurations. We provide a complete solution to hydrodynamic transport at all orders in the gradient expansion compatible with the second law constraint.
The key new ingredient we introduce is the notion of adiabaticity, which allows us to take hydrodynamics off-shell. Adiabatic fluids are such that off-shell dynamics of the fluid compensates for entropy production. The space of adiabatic fluids admits a decomposition into seven distinct classes. Together with the dissipative class this establishes the eightfold way of hydrodynamic transport. Furthermore, recent results guarantee that dissipative terms beyond leading order in the gradient expansion are agnostic of the second law.
After completing the transport taxonomy, we go on to argue for a new symmetry principle, an Abelian gauge invariance that guarantees adiabaticity in hydrodynamics and serves as the emergent version of microscopic KMS conditions. We demonstrate its utility by explicitly constructing effective actions for adiabatic transport (i.e., seven out of eight classes). The theory of adiabatic fluids, we speculate, provides a useful starting point for a new framework to describe non-equilibrium dynamics. We outline briefly the crucial role of the proposed symmetry of gauged thermal translations in the construction of a Schwinger-Keldysh effective action that encompasses all of hydrodynamic transport
