Abstract
Uncharged relativistic fluids in 3+1 dimensions have three independent thermodynamic transport coefficients at second order in the derivative expansion. Fluids with a single global U(1) current have nine, out of which seven are parity preserving. We derive the Kubo formulas for all nine thermodynamic transport coefficients in terms of equilibrium correlation functions of the energy-momentum tensor and the current. All parity-preserving coefficients can be expressed in terms of two-point functions in flat space without external sources, while the parity-violating coefficients require three-point functions. We use the Kubo formulas to compute the thermodynamic coefficients in several examples of free field theories.
Highlights
Fluids in 2+1 dimensions, analogous thermodynamic transport coefficients already appear at first order in the derivative expansion [5]
We will refer to the thermodynamic coefficients that appear in the constitutive relations as “thermodynamic transport coefficients”, and to the coefficients in the equilibrium free energy as “thermodynamic susceptibilities”
The emphasis of this note was on the Kubo formulas for thermodynamic susceptibilities that appear at two-derivative order in the constitutive relations of the energy-momentum tensor and of the global U(1) current
Summary
[22] The third equation in (2.6) says that the electric field induces a charge gradient. As the density F[g, A] is local and gauge-invariant, one can formally consider it to be a function of Aμ and the field strength Fμν. A convenient choice of fixing the ambiguity in the definition of Jfμ is to use (2.6c) to trade the derivatives of the chemical potential in the density F[g, A] for the electric field. This gives Jfμ = ρuμ, where ρ ≡ ∂F/∂μ defines the density of free charges. See [21] for more details about the electric and magnetic contributions to M μν
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