Derivative expansion in hydrodynamics and the definition of the temperature

Question

Thermodynamic quantities are well defined at equilibrium. a fluid at zeroth order can be approximated by a perfect fluid and conserved currents such as electric current and energy momentum tensor is given by: \begin{equation} \begin{aligned} & j^{\mu}=nu^{\mu}\\ &T^{\mu\nu}=(\epsilon +P)u^{\mu}u^{\nu}+Pg^{\mu\nu} \end{aligned} \end{equation} In the next order in derivative expansion at Landau frame which is define by $ u_{\mu} j^{\mu}=-n $ electric current is as follow: \begin{equation} j^{\mu}=nu^{\mu}-\sigma_Q\mathcal P^{\mu\nu}(\partial_{\nu}\mu-\frac{\mu}{T}\partial_{\nu}T) \end{equation} where the project operator $ \mathcal P^{\mu\nu} $ is defined as follow: \begin{equation} \mathcal P^{\mu\nu}=g^{\mu\nu}+u^{\mu}u^{\nu} \end{equation} According to the definition of the temperature, it is only meaningful at equilibrium, so why do we consider the gradients of temperature and chemical potential?

Answer 1

Fluid dynamics is an effective theory for the long wavelength, long time behavior of classical or quantum many body systems. In this limit the system reaches approximate local (but not global) equilibrium. This is the case because local equilibration takes place on a microscopic time scale (the collision rate between atoms, or leptons, quarks, etc), whereas global equilibration requires diffusion or propgation of density or pressure disturbances, which takes place on a macroscopic time scale.

In local thermal equilibrium we can use the coarse grained values of the conserved charges $n$, $\epsilon=T_{00}$ and $\pi_i=T_{0i}$ to define local values of the temperature, chemical potential, and fluid velocity. By assumption these are slowly varying, so we can express the fluxes as a gradient expansion in the local thermodynamic variables.

At zeroth order in gradients we obtain ideal (relativistic) fluid dynamics. at first order we encounter three new terms, two that involve derivatives of the fluid velocity, controlled by bulk and shear viscosity, and one that involves gradients of $T,\mu$, controlled by the thermal conductivity.

P.S.: There is an ambiguity (really, "reparametrization invariance" or "frame dependence") in mapping the densities $(n,\epsilon,\pi_i)$ on the thermodynamic variables $(T,\mu,u_\mu)$. This has to do with the definition of the fluid velocity. Roughly, I can use either the energy current (Landau) or the particle currrent (Eckhardt), or anything in between, to define the fluid velocity. This affects $(T,\mu)$ because I use the fluid rest frame to relate the energy and particle density to ($T,\mu)$. It also affects the dissipative terms, because in the Eckhardt frame (for example) the particle current is defined to be $nu_\mu$, so it cannot have any dissipative corrections. The $\nabla T$ term must then appear in the energy current. Of course, physical quantities must be unaffected (to the order in $\nabla$ that we are working) by this choice.