On the local thermodynamic relations in relativistic spin hydrodynamics

TL;DR

The paper demonstrates, via two explicit counterexamples in free Dirac fields at global thermodynamic equilibrium with rotation and acceleration, that the conventional local thermodynamic differential relations used in relativistic spin hydrodynamics fail: the derivative ${\partial p}/{\partial \omega_{\lambda\nu}}|_{T,\mu}$ contains additional terms of the same order as the spin density $S^{\lambda\nu}$ and cannot be corrected by entropy-gauge transformations. It shows this clearly for massless and massive fermions by computing the pressure, energy, and spin densities to appropriate orders in vorticity and acceleration, and proves that ${\partial p}/{\partial T}|_{\mu,\omega}=s$ and ${\partial p}/{\partial \mu}|_{T,\omega}=n$ can hold, but the key relation with spin density does not. The authors then argue that a simple redefinition of the entropy current cannot restore the desired differential relations, implying that constitutive equations in spin hydrodynamics require a full quantum-statistical treatment. Overall, the work highlights the necessity of going beyond traditional differential thermodynamics to correctly capture spin-related dissipative effects in relativistic fluids.

Abstract

We demonstrate, by providing two specific examples, that the local differential thermodynamic relations used as educated guesses in relativistic hydrodynamics with spin, do not hold even at global thermodynamic equilibrium. We show, by using a rigorous quantum statistical method, that for massless free fermions and massive free fermions with rotation and acceleration at global thermodynamic equilibrium, the derivative of the pressure function with respect to the spin potential differs from the spin density and acquires a correction of the same order. Such correction cannot be eliminated by any redefinition of the entropy current, a so-called entropy-gauge transformation. Therefore, for an accurate determination of the constitutive relations in relativistic spin hydrodynamics, the traditional method of assuming differential thermodynamic relations is not appropriate.