Adiabatic hydrodynamics: The eightfold way to dissipation

TL;DR

This work develops a complete off-shell framework for hydrodynamic transport, introducing the adiabaticity equation Δ = 0 to define adiabatic fluids whose entropy production is compensated by the dynamics. It partitions transport into eight classes (seven adiabatic plus dissipative Class D), with hydrostatics and anomalies giving rise to concrete Lagrangian and non-Lagrangian sectors; the Eightfold Lagrangian (Class L_T) unifies these via a master action with a new U(1)_{T} symmetry, doubling sources in a Schwinger-Keldysh-inspired construction. The framework recovers known results from hydrostatics, holography, and kinetic theory, and provides a principled method to classify transport at all gradient orders, including Berry-like (Class B), conserved-entropy (Class C), and vector (H_V/ar{H}_V) data. The approach links adiabatic transport to an effective action with enhanced symmetry, offering a potential macroscopic manifestation of KMS invariance and a path toward describing non-equilibrium dynamics with controlled dissipation arising from symmetry breaking. Overall, the paper delivers a comprehensive taxonomy, a unifying Lagrangian formalism, and clear guidance on how adiabatic and dissipative sectors intertwine in relativistic hydrodynamics, with implications for holography and kinetic theory.

Abstract

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 is quite rich, and 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. While this completes a transport taxonomy, we go on to argue for a new symmetry principle, an Abelian gauge invariance that guarantees adiabaticity in hydrodynamics. We suggest that this symmetry is the macroscopic manifestation of the microscopic KMS invariance. We demonstrate its utility by explicitly constructing effective actions for adiabatic transport. The theory of adiabatic fluids, we speculate, provides a useful starting point for a new framework to describe non-equilibrium dynamics, wherein dissipative effects arise by Higgsing the Abelian symmetry.

Figures (4)

• Figure 1: The eightfold way of hydrodynamic transport.
• Figure 2: Flowchart giving the structure of this paper. Light blue sections in the middle column form the main thread of our analysis. The sections in light green contain detailed constructions of the various classes and could be skipped on a first read. The sections in the left column are concerned with various Lagrangian descriptions. The classes in the right column are those that do not fit into a simple Lagrangian framework (without $U(1)_{{\sf T}}$).
• Figure 3: Illustration of the connection between the physical and reference fields for Class L adiabatic fluids. The fields on the physical spacetime manifold ${\cal M}$ are related to those on the reference manifold $\mathbbm M$ by a pull-back using the dynamical fields $\{\varphi^a,c\}$. The constrained variation on ${\cal M}$ which gives the correct equations of motion corresponds to varying $\{\varphi^a,c\}$ while holding $\{\mathbb \bbbeta^a,\Lambda_\mathbb \bbbeta\}$ fixed.
• Figure 4: Illustration of the Schwinger-Keldysh setup. The physical spacetime manifold ${\cal M}$ has been doubled. However, the two copies are not entirely independent as they are both related to the same reference configuration on $\mathbbm M$ via pull-backs using the dynamical fields $\{\varphi,c\}_{\text{\tiny L},\text{\tiny R}}$. Despite the presence of two copies of source fields on $\mathbbm M$ there is only one diffeomorphism and gauge redundancy involved; invariance under this symmetry implies Schwinger-Keldysh Bianchi identities.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2