Effective Action for Relativistic Hydrodynamics: Fluctuations, Dissipation, and Entropy Inflow

TL;DR

The paper develops a covariant, superspace-based Schwinger-Keldysh effective field theory for relativistic hydrodynamics, unifying fluctuations, dissipation, and entropy inflow via a balanced thermal equivariant framework. It introduces a doubled SK structure, an emergent U(1)_T entropy gauge symmetry, and BRST/topological supersymmetry to enforce fluctuation-dissipation and KMS relations, producing an action that reproduces the eightfold classification of hydrodynamic transport. A central result is the entropy inflow mechanism, whereby entropy production in physical spacetime arises from superspace degrees of freedom and their currents, with positivity guaranteed by the dissipative sector’s structure. The MMO limit yields explicit, tractable calculations for ideal, viscous, and certain second-order transport terms, illustrating how hydrostatic, adiabatic, and dissipative classes emerge from the same superspace action. The framework also suggests deep connections to holography and black hole physics, and outlines several open questions, including U(1)_T gauge dynamics and extensions to flavored (non-neutral) fluids.

Abstract

We present a detailed and self-contained analysis of the universal Schwinger-Keldysh effective field theory which describes macroscopic thermal fluctuations of a relativistic field theory, elaborating on our earlier construction in arXiv:1511.07809. We write an effective action for appropriate hydrodynamic Goldstone modes and fluctuation fields, and discuss the symmetries to be imposed. The constraints imposed by fluctuation-dissipation theorem are manifest in our formalism. Consequently, the action reproduces hydrodynamic constitutive relations consistent with the local second law at all orders in the derivative expansion, and captures the essential elements of the eightfold classification of hydrodynamic transport of arXiv:1502.00636. We demonstrate how to recover the hydrodynamic entropy and give predictions for the non-Gaussian hydrodynamic fluctuations. The basic ingredients of our construction involve (i) doubling of degrees of freedom a la Schwinger-Keldysh, (ii) an emergent thermal gauge symmetry associated with entropy which is encapsulated in a Noether current a la Wald, and (iii) a BRST/topological supersymmetry imposing the fluctuation-dissipation theorem a la Parisi-Sourlas. The overarching mathematical framework for our construction is provided by the balanced equivariant cohomology of thermal translations, which captures the basic constraints arising from the Schwinger-Keldysh doubling, and the thermal Kubo-Martin-Schwinger relations. All these features are conveniently implemented in a covariant superspace formalism. An added benefit is that the second law can be understood as being due to entropy inflow from the Grassmann-odd directions of superspace.

Figures (4)

• Figure 1: An illustration of the hydrodynamic observables in the Schwinger-Keldysh functional integral phrased in the Keldysh average-difference basis. Response functions are correlators of a sequence of difference operators (the external disturbances, denoted as blue dots) followed by an average operator (the measurement, denoted by the black dots) in the future.
• Figure 2: Illustration of the spacetime picture associated with the proposed KMS gauge invariance, with the figure taken from Haehl:2016uah. We upgrade the spacetime manifold (a Lorentzian geometry, depicted in gray, with a typical Cauchy slice indicated in red), on which our quantum system resides to a thermal fibre bundle. We further assume local thermal equilibrium (as in, e.g., hydrodynamics) at each spacetime point which guarantees existence of a thermal vector ${\bm \beta}^\mu$. We draw this vector field as a circle fibration with a thermal circle whose size is set by the local temperature. The KMS transformations we seek implement equivariance with respect to thermal translations along this local imaginary time circle. Restricting to a gauge slice corresponds to picking a Lorentzian section of this fibration. Note that the size of the thermal circle is exaggerated; our arguments assume the high temperature limit where the size of the thermal circle is much smaller than the fluctuation scale.
• Figure 3: Illustration of the data for hydrodynamic sigma models as described in Haehl:2015pja. The physical degrees of freedom are captured in the target space maps $X^\mu(\sigma)$ which are the hydrodynamic pion fields. The worldvolume geometry is equipped with a reference thermal vector field ${\bm \beta}^a$, which pushes forward to the physical thermal vector in spacetime, while the spacetime metric pulls back to the worldvolume metric ${\sf g}_{ab}$.
• Figure 4: Illustration of the data for hydrodynamic sigma models. The physical degrees of freedom are captured in the target space maps $\mathring{X}^\mu(\sigma,\theta,{\bar{\theta}} )$, along with the gauge condition which aligns the Grassmann coordinates in target space with their worldvolume counterparts $\Theta = \theta$ and ${\bar{\Theta}} ={\bar{\theta}}$. The worldvolume geometry is equipped with a reference super-vector field $\mathring{{\bm \beta}}^I$, which pushes forward to the physical thermal vector in spacetime, while the spacetime metric $g_{\mu\nu}$ (with $g_{\Theta {\bar{\Theta}}} = i$) pulls back to the worldvolume metric $\mathring{{\sf g}}_{IJ}$. The pullbacks and pushfowards are $U(1)_{{\sf T}}$ gauge covariant, since the worldvolume dynamics is constrained by this symmetry.

Theorems & Definitions (2)

• Theorem 1
• proof