Fluctuating, Generalized, and Schwinger–Keldysh Hydrodynamics

Ordinary hydrodynamics must be enlarged when fluctuations or additional slow degrees of freedom matter. This chapter connects stochastic fluids to broken symmetry, anomalies, magnetic flux, integrability, spin, and critical modes.

Enter this chapter

Begin with Hydrodynamic Fluctuations and Noise to include thermal fluctuations consistently with dissipative transport.

Helpful background. Hydrodynamic Fluctuations and Noise and Schwinger–Keldysh Effective Actions for Fluids develop the definitions used by the chapter’s opening pages.

Topics

1. Hydrodynamic Fluctuations and Noise — How do thermal fluctuations enter hydrodynamic variables and constitutive relations, and when is the noise white, local, Gaussian, colored, or bilocal?
2. Schwinger–Keldysh Effective Actions for Fluids — How is a local Schwinger–Keldysh effective action built for hydrodynamic variables while enforcing conservation, unitarity, causality, positivity, and thermal symmetry?
3. Dynamical KMS and Topological Symmetries — Which dynamical KMS and topological or BRST-like structures encode local equilibrium and SK normalization, and what extra assumptions are required beyond microscopic unitarity?
4. Long-Time Tails and Fluctuation Renormalization — How do nonlinear hydrodynamic fluctuations generate nonanalytic long-time tails and renormalize transport, including classical and quantum regimes?
5. Charged and Anomalous Hydrodynamics — How do conserved and anomalous charges modify hydrostatics, constitutive relations, collective modes, and nondissipative transport?
6. Superfluid Hydrodynamics and Goldstone Modes — How does a broken continuous symmetry add a Goldstone phase, Josephson relation, superflow, and additional sound and dissipative channels to hydrodynamics?
7. Magnetohydrodynamics and Higher-Form Symmetries — How can magnetic flux conservation and electromagnetic dynamics be formulated as magnetohydrodynamics or higher-form hydrodynamics, and when are the formulations equivalent?
8. Generalized Hydrodynamics of Integrable Systems — How do infinitely many conserved charges produce generalized hydrodynamics through Bethe/TBA dressing, state densities, effective velocities, and Euler-scale continuity equations?
9. Spin Hydrodynamics, Polarization, and Pseudo-Gauge Dependence — When is spin density an independent slow variable, how do pseudo-gauge transformations reorganize spin and orbital currents, and which polarization observables are invariant?
10. Hydro+ and Parametrically Slow Critical Modes — How does Hydro+ extend hydrodynamics when a critical or parametrically slow nonconserved mode relaxes on the same timescale as conserved densities?

How the ideas fit together

Dissipation and thermal noise cannot be chosen independently in equilibrium. Their relation is encoded by the thermal symmetry of a Schwinger–Keldysh effective action; nonlinear fluctuations then generate long-time tails that a purely deterministic derivative expansion misses Crossley, Glorioso, and Liu 2017, §§ 1–2.

The other extensions follow a common physical test: retain a degree of freedom when it relaxes as slowly as the fluid. A superfluid phase, an integrable rapidity distribution, or a critical fluctuation requires a different enlarged state space. Giving each extension the name “hydrodynamics” does not make its variables or constitutive relations interchangeable.

Where to go next

Continue to Response, Transport, and Inference to derive causal response to an applied source.

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.
• Crossley, Michael, Paolo Glorioso, and Hong Liu. “Effective Field Theory of Dissipative Fluids.” Journal of High Energy Physics 2017, no. 9 (2017): 095. DOI.