Hydrodynamic fluctuations at a nonzero temperature can cause slow relaxation toward equilibrium even in observables which are not locally conserved. A classic example is the stress-stress correlator in a normal fluid, which, at zero wave number, behaves at large times as $t^{-3/2}.$ A novel feature of the effective theory of hydrodynamic fluctuations in supersymmetric theories is the presence of Grassmann-valued classical fields describing macroscopic supercharge density fluctuations. We show that hydrodynamic fluctuations in supersymmetric theories generate essentially the same long-time power-law tails in real-time correlation functions that are known in simple fluids. In particular, a $t^{-3/2}$ long-time tail must exist in the stress-stress correlator of $\mathcal{N}=4\mathrm{}$ supersymmetric Yang-Mills theory at non-zero temperature, regardless of the value of the coupling. Consequently, this feature of finite-temperature dynamics can provide an interesting test of the AdS/CFT correspondence. However, the coefficient of this long-time tail is suppressed by a factor of ${1/N}_{\mathrm{c}}^2.$ On the gravitational side, this implies that these long-time tails are not present in the classical supergravity limit; they must instead be produced by one-loop gravitational fluctuations.

• Received 15 March 2003

DOI:https://doi.org/10.1103/PhysRevD.68.025007