Constraints on Fluid Dynamics from Equilibrium Partition Functions

TL;DR

<3-5 sentence high-level summary>The paper develops a framework to constrain relativistic hydrodynamics by requiring consistency with stationary equilibrium partition functions on arbitrary stationary backgrounds, and by enforcing correct anomaly and symmetry properties. By performing explicit analyses in 3+1D charged fluids (first order) and 2+1D parity-odd cases, as well as 3+1D uncharged fluids (second order), it shows that many non-dissipative transport coefficients are fixed by a small set of functions appearing in the partition function, while dissipative coefficients remain unconstrained. The results reproduce known anomaly-induced transport (e.g., Son–Surowka) and closely match entropy-current analyses, supporting a conjecture that partition-function constraints agree with second-law relations to all orders in derivatives. The work also connects these hydrodynamic constraints to conformal limits, CPT invariance, and the AdS/CFT-inspired context, and outlines open questions for higher-order extensions and time-dependent settings.

Abstract

We study the thermal partition function of quantum field theories on arbitrary stationary background spacetime, and with arbitrary stationary background gauge fields, in the long wavelength expansion. We demonstrate that the equations of relativistic hydrodynamics are significantly constrained by the requirement of consistency with any partition function. In examples at low orders in the derivative expansion we demonstrate that these constraints coincide precisely with the equalities between hydrodynamical transport coefficients that follow from the local form of the second law of thermodynamics. In particular we recover the results of Son and Surowka on the chiral magnetic and chiral vorticity flows, starting from a local partition function that manifestly reproduces the field theory anomaly, without making any reference to an entropy current. We conjecture that the relations between transport coefficients that follow from the second law of thermodynamics agree to all orders in the derivative expansion with the constraints described in this paper.