Lectures on hydrodynamic fluctuations in relativistic theories

TL;DR

These lectures analyze hydrodynamic fluctuations in relativistic fluids by combining linear response, stochastic hydrodynamics, and an effective action viewpoint. They derive retarded correlation functions $G^R_{ab}(\omega,\mathbf{k})$ and associated Kubo formulas for transport coefficients $(\eta,\zeta,\sigma)$, uncovering non-analytic contributions (e.g. $|\omega|^{1/2}$ in $d=3$ and $\ln|\omega|$ in $d=2$) that signal breakdowns of the derivative expansion. A field-theoretic formulation with auxiliary fields and ghosts enables systematic treatment of mode interactions, including one-loop corrections and renormalization-group flows in low dimensions. The results illuminate how hydrodynamic fluctuations alter transport, constrain effective theories, and bear on strongly coupled systems such as the quark-gluon plasma and quantum critical regimes, highlighting the necessity to augment derivative expansions with fluctuation physics.

Abstract

These are pedagogical lecture notes on hydrodynamic fluctuations in normal relativistic fluids. The lectures discuss correlation functions of conserved densities in thermal equilibrium, interactions of the hydrodynamic modes, an effective action for viscous fluids, and the breakdown of the derivative expansion in hydrodynamics.

Figures (7)

• Figure 1: A stationary flow of an ideal fluid with a velocity gradient. Thin arrows represent particles which can transfer $x$-momentum in the $y$-direction, eventually leading to the equilibration of the inhomogeneous velocity profile.
• Figure 2: A fluid flow with an inhomogeneous velocity profile $v_x(y)$ corresponding to the shear mode, represented by thick arrows. Thin wiggly arrows represent sound and shear modes generated by thermal fluctuations in fluid elements in local thermal equilibrium. These collective excitations can transfer $x$-momentum in the $y$-direction, thus contributing to shear viscosity.
• Figure 3: The flow diagram for the differential equations (\ref{['eq:rg-2d']}), pictured in terms of the dimensionless variables $g_\eta = \eta/s$ and $g_\sigma = \sigma T/\chi$, where $\eta$ is the shear viscosity, $\sigma$ is the charge conductivity, $s$ is the equilibrium density of entropy, and $\chi$ is the equilibrium charge susceptibility. The arrows indicate the direction of decreasing $\mu$ (towards lower frequency). The dashed line is a straight line with a unit slope, indicating the infrared asymptotics as $\mu{\to}0$.
• Figure 4: The vertices for the effective action (\ref{['eq:nonlin-hydro-action-1']}). There is an overall sign depending on whether the momentum is flowing in or out of the vertex.
• Figure 5: Three connected one-loop diagrams potentially contributing to $G_{\pi\lambda}$. In addition to the shown diagrams, there are two more diagrams with ghost loops which are completely analogous to the diagrams in the first row.