Definition of entropy for relativistic fluids

Question

In section 5.2 of (https://link.springer.com/article/10.12942/lrr-2007-1) ( version on Arxiv ), the authors mention that the equation of state $\rho=\rho(n)$ for a single relativistic fluid system can be written in terms of a local theory, i.e. we can identify: $$\rho=u^au^b\langle T^{(loc)}_{ab}\rangle$$ and particle number density $n=-u^a\langle J^{(loc)}_a\rangle$, where $u^a$ is the unit time-like vector representing fluid particle velocity. $T^{(loc)}_{ab}$ and $J^{(loc)}_a$ are the stress energy tensor and conserved current associated to some local physical system. However, in chapter 9 they have also considered dependency on entropy density '$s$' in equation of state ($\rho=\rho(s,n)$) without giving a formal definition in terms of a local theory.

For single relativistic fluid system, can we identify our macroscopic entropy density $s$ in terms of local (microscopic) quantities?

Answer 1

There is no operator expression for the entropy in the same way that we have operator expressions for $T_{\mu\nu}$ and $j_\mu$, so that the macroscopic stress tensor and current can be viewed as expectation values of microscopic operators. However, once you are in approximate local equilibrium, then $s$ is fixed by equilibrium thermodynamics, $dE=TdS-pdV+\mu dN$. In other words, the equilibrium equation of state determines $s=s(\rho,n)$, once $\rho$ and $n$ are known.