Constraints on Fluid Dynamics from Equilibrium Partition Functions

TL;DR

This paper explores how the requirement of a consistent thermal partition function constrains relativistic hydrodynamics and transport coefficients, linking anomalies and thermodynamics without relying on an entropy current.

Contribution

It establishes a direct connection between partition functions and hydrodynamic constraints, reproducing known anomaly-related results and proposing a conjecture for all-order relations.

Findings

Hydrodynamic equations are constrained by partition function consistency.

Reproduces Son and Surowka's results on chiral magnetic and vorticity flows.

Proposes a conjecture linking second law constraints to all-order transport coefficient relations.

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.