Hydrodynamic fluctuations in relativistic superfluids

TL;DR

This work develops a covariant relativistic hydrodynamic description of superfluids using a noncanonical Poisson-bracket framework, showing that identifying the momentum density with the energy-flow leads to the Khalatnikov–Lebedev–Carter formulation. It then employs memory-function techniques to study hydrodynamic fluctuations around equilibrium, deriving sum rules for the response functions and linking the superfluid momentum to the Goldstone mode via $g_s = \xi \mu \\nabla\\varphi$ and $h_s = \xi \mu^2$. The authors compute the full correlation functions, reveal the hydrodynamic modes (first and second sound) with speeds $c_1$ and $c_2$ and their attenuation, and establish Kubo relations for transport coefficients including $\\eta$, $\\zeta$, and $\\kappa$, along with Ward identities. The results connect relativistic superfluid hydrodynamics to standard Landau–Lifshitz theory and provide a practical framework for applications to dense QCD matter and neutron-star contexts, while clarifying the role of the Goldstone mode in the dissipative structure.

Abstract

The Hamiltonian formulation of superfluids based on noncanonical Poisson brackets is studied in detail. The assumption that the momentum density is proportional to the flow of the conserved energy is shown to lead to the covariant relativistic theory previously suggested by Khalatnikov, Lebedev and Carter, and some potentials in this theory are given explicitly. We discuss hydrodynamic fluctuations in the presence of dissipative effects and we derive the corresponding set of hydrodynamic correlation functions. Kubo relations for the transport coefficients are obtained.