Fluctuating, Generalized, and Schwinger–Keldysh Hydrodynamics

Hydrodynamics becomes a fluctuating effective theory when thermal noise is retained, and it becomes a generalized hydrodynamics when additional slow structures—Goldstone phases, magnetic flux, integrable charge distributions, spin, or critical modes—must be evolved. This chapter develops those extensions without treating them as interchangeable: each has its own variables, symmetry, constitutive counting, regulator, and evidence ceiling.

Helpful background. The Hydrodynamic Limit and Slow Variables supplies the variable-selection test. Relativistic Dissipative Hydrodynamics fixes the dissipative kernels to which equilibrium noise must be matched.

Enter this chapter

The chapter inherits the $(+---)$ metric and the Fourier pair with forward phase $e^{+ip\cdot x}$. Linear modes therefore use $e^{-i\omega t+i\mathbf k\cdot\mathbf x}$. For Schwinger–Keldysh fields,

$$\phi_r=\frac12(\phi_1+\phi_2), \qquad \phi_a=\phi_1-\phi_2.$$

The $r$ field carries the classical configuration and the $a$ field imposes the equation of motion and generates response. These names do not determine normalization by themselves; every action below states its convention.

Three distinctions organize the material:

1. Constitutive extension: noise, anomalies, superflow, magnetic flux, spin, and critical modes change the local variables or constitutive tensors.
2. Effective-action completion: the Schwinger–Keldysh action enforces normalization, reality, causal response, positivity, and—under additional thermal hypotheses—dynamical KMS.
3. Microscopic specialization: Bethe-ansatz dressing, anomaly coefficients, spin freeze-out, and critical relaxation rates require input that hydrodynamic symmetry alone does not compute.

Route through the chapter

Page	Question answered	Result to carry forward
Hydrodynamic Fluctuations and Noise	How is noise fixed by dissipative response?	A regulated diffusion kernel and the canonical fluctuating/SK consistency table
Schwinger–Keldysh Effective Actions for Fluids	How are response and fluctuations generated by one action?	A quadratic diffusion action satisfying the SK constraints
Dynamical KMS and Topological Symmetries	Which constraints follow from unitarity and which require equilibrium?	Separate normalization/BRST and thermal-KMS structures
Long-Time Tails and Fluctuation Renormalization	How do nonlinear fluctuations defeat an analytic gradient series?	A one-loop $t^{-d/2}$ tail and its frequency nonanalyticity
Charged and Anomalous Hydrodynamics	How do anomalies alter hydrostatic and transport currents?	Consistent/covariant-current translation and bounded anomalous response
Superfluid Hydrodynamics and Goldstone Modes	What changes after a continuous symmetry breaks?	Josephson dynamics, entrainment, and two sound branches
Magnetohydrodynamics and Higher-Form Symmetries	When is magnetic flux a hydrodynamic charge?	The map from a conserved two-form current to ideal MHD and Alfvén waves
Generalized Hydrodynamics of Integrable Systems	How does an infinite charge set close at Euler scale?	Dressing equations, effective velocity, and the partitioning solution
Spin Hydrodynamics, Polarization, and Pseudo-Gauge Dependence	When is spin independently slow and what is convention dependent?	A pseudo-gauge transformation with invariant total angular momentum
Hydro+ and Parametrically Slow Critical Modes	How is a critically slow nonconserved mode retained?	A frequency-dependent sound stiffness and bulk response

The fluctuating-hydrodynamics and SK consistency reference lists the checks that must accompany the first four pages.

The chapter’s three main interfaces—fluctuating fluid EFT, generalized hydrodynamics, and infrared mode coupling—are developed respectively in Crossley, Glorioso, and Liu 2017, §§2–5, Open PDF, Doyon 2020, §§2–6, Open PDF, and Kovtun 2012, §§3–4, Open PDF.

Variable-selection map

Slow structure	Added field	Why ordinary hydrodynamics fails	Controlled stopping condition
Thermal fluctuations	Stochastic stress/current or $a$ field	Mean constitutive equations omit equilibrium variance and loop effects	Coarse-graining scale approaches microscopic mean free path
Broken $U(1)$	Goldstone phase	Phase gradients remain gapless	Explicit breaking or vortex proliferation gaps/disorders the phase
Magnetic flux	Conserved two-form density	Dynamical electromagnetism carries a long-lived flux sector	Monopoles or one-form breaking relax flux rapidly
Integrability	Rapidity occupation	Finitely many charges do not specify local stationary states	Integrability breaking exceeds the observation frequency
Spin	Spin density/potential	Antisymmetric stress can exchange orbital and spin angular momentum slowly	Spin relaxation becomes microscopic
Critical mode	Partial-equilibrium correlator or order parameter	Critical slowing removes the separation used to integrate it out	Move outside the critical scaling window

This map is diagnostic, not a claim that all sectors can be combined without double counting. A theory containing two extensions must match their variables, noise, frames, and conservation laws together.

Review the chapter

You should be able to derive the conserved-noise correlator from a susceptibility and conductivity, reconstruct diffusion response from an $r/a$ action, distinguish SK normalization from KMS, obtain a one-loop long-time tail, translate anomalous and pseudo-gauge currents, derive Alfvén and two-sound mode matrices, solve the Euler-scale GHD closure, and identify the Hydro+ validity window. For every result, state whether it is constitutive, effective-action, microscopic, or inference evidence.

References

• Crossley, Michael, Paolo Glorioso, and Hong Liu. 2017. “Effective Field Theory of Dissipative Fluids.” Journal of High Energy Physics 2017 (9): 095. DOI. Open PDF.

• Doyon, Benjamin. 2020. “Lecture Notes on Generalised Hydrodynamics.” SciPost Physics Lecture Notes 18. DOI. Open PDF.

• Kovtun, Pavel. 2012. “Lectures on Hydrodynamic Fluctuations in Relativistic Theories.” Journal of Physics A: Mathematical and Theoretical 45: 473001. DOI. Open PDF.