Relativistic Hydrodynamics and Vorticity Dynamics in High-Energy Heavy-Ion Collisions: A Collective Flow Perspective

This article provides a comprehensive overview of the application of relativistic fluid mechanics to describe the collective evolution of the Quark-Gluon Plasma (QGP) formed in ultra-relativistic heavy-ion collisions. We map out the chronological transformation of spatial eccentricities in the initial interaction volume into measurable anisotropic azimuthal momentum distributions, parameterized by the harmonic flow coefficients $v_n$. Utilizing multi-particle correlation techniques developed within the ATLAS experimental framework, we dissect the event-by-event fluctuations of the participant planes and evaluate non-linear hydrodynamic responses across higher harmonics. Furthermore, we embed local rotation fields into this continuous description by solving the covariant transport equations for subatomic vorticity. We demonstrate that while the Helmholtz-Kelvin theorem guarantees the topological conservation of vortex lines within the ideal medium, the collective multi-dimensional expansion forces a systematic 1/t power-law geometric dilution of the local rotational magnitude. Finally, we contrast different pre-equilibrium generation mechanisms and evaluate their final signatures on differential spin alignment observables.