<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="normal">Λ</mml:mi></mml:math> spin polarization in event-by-event relativistic heavy-ion collisions

We present a systematic study of $\mathrm{\ensuremath{\Lambda}}$ hyperon's polarization observables using event-by-event ($3+1$) dimensional relativistic hydrodynamics. The effects initial hot spot size and quark gluon plasma's specific shear viscosity on the are quantified. examine two formulations thermal tensor same hydrodynamic background. With simulations, we make predictions for Fourier coefficients ${\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{'}}\mathrm{s}$ longitudinal ${P}^{z}$ with respect to event planes different orders anisotropic flow. propose new correlations among charged hadron flow further test mapping from fluid velocity gradients polarization. Finally, system scan $\mathrm{Au}+\mathrm{Au}$, $\mathrm{Ru}+\mathrm{Ru}$, $\mathrm{O}+\mathrm{O}$ collisions at $\sqrt{{s}_{NN}}=200\phantom{\rule{0.16em}{0ex}}\mathrm{GeV}$ dependence BNL Relativistic Heavy-ion Collider.