Perfect spinfluid: A divergence-type approach

TL;DR

This work develops a divergence-type relativistic hydrodynamics framework that incorporates spin through a spin potential $\Omega_{\mu\nu}$ and a spin-extended generating function $\chi$. It proves that the resulting equations are symmetric hyperbolic and non-linearly causal to all orders in the spin potential under a thermodynamic convergence constraint, ensuring local well-posedness and stability. By deriving currents from $\chi$ via second derivatives, the authors connect kinetic theory with spin hydrodynamics and perform a linearized stability analysis around nontrivial equilibria, revealing real propagation speeds and spin-wave modes. The framework is designed for numerical simulations of spin-polarized fluids, with applications to heavy-ion collisions, and can be extended to include dissipative effects and electromagnetic fields, broadening the scope of relativistic spin dynamics modeling.

Abstract

We present a new formulation of non-dissipative relativistic spin hydrodynamics that incorporates spin degrees of freedom into the divergence-type theory framework. Due to the divergence-type structure, it is straightforward to enforce non-linear causality and symmetric hyperbolicity of the equations of motion, ensuring local well-posedness of the initial-value problem and stability of the theory. Furthermore, in a specific realization based on spin kinetic theory, we prove that the equations of motion remain non-linearly causal and symmetric-hyperbolic to all orders in the spin potential, provided a specific thermodynamic constraint is satisfied. This framework can be applied for numerical simulations to study the dynamics of spin-polarized fluids, such as the quark-gluon plasma in heavy-ion collisions.

Figures (2)

• Figure 1: Energy density, pressure anisotropy, and spin per particle. In all plots, $z=1$, $\alpha=0$, $\Omega_{03}=0.01$, and the maximum value on the horizontal axis, $\Omega_{12}\approx \tilde{s}^{-1}=2/\sqrt{3}$, saturates the bound \ref{['convergence']}. The solid curves are numerically exact to all orders in $\Omega_{\mu\nu}$, while the dashed lines are obtained from second-order-in-$\Omega_{\mu\nu}$ truncations. (a) Energy density, (b) pressure anisotropy, and (c)$z$-component of spin per particle are shown.
• Figure 2: Propagation speeds. Parameter values are $z=1$, $\alpha=0$, and $\Omega_{03}=0.01$, as in Fig. \ref{['fig:currents']}. (a-b) Propagation speeds $v_q$ as functions of spin potential $\Omega_{12}$ when (a)$\theta_{\bm k}=\pi/2$ and (b)$\theta_{\bm k} = 0$. In (a), the black (solid) curve is triply degenerate, i.e., there are three non-propagating modes. Not pictured are the four modes obtained from the red (long-dashed), blue (dot-dashed), purple (dotted), and orange (short-dashed) curves by flipping the sign of $v_q$, which are present due to time-reversal symmetry and result in a total of $4+3+4=11$ independent modes, as described in the text. When the direction of the wave-vector is tuned from $\theta_{\bm k}=\pi/2$ to its value $\theta_{\bm k}=0$ in (b), the mode corresponding to the orange (short-dashed) curve becomes non-propagating, and the modes corresponding to the blue (dot-dashed) and purple (dotted) curves become degenerate. (c) Propagation speeds as a function of propagation direction $\theta_{\bm k}$ when $\Omega_{12} = 0.5$, demonstrating degeneracies at $\theta_{\bm k} = 0$ and $\theta_{\bm k}=\pi$.