We briefly review a recently proposed formalism of perfect-fluid hydrodynamics with spin, which is a generalization of the standard hydrodynamic framework and provides a natural tool for describing the evolution of spin-polarized systems of particles with spin 1/2. It is based on the de Groot - van Leeuwen - van Weert forms of energy-momentum and spin tensors and conservation laws. Using Bjorken symmetry we show how this formalism may be used to determine observables describing the polarization of particles measured in the experiment.
Relativistic hydrodynamic equations for particles with spin one-half are used to determine the space-time evolution of the spin polarization in a boost-invariant and transversely homogeneous background. The hydrodynamic approach uses the forms of the energy-momentum and spin tensors based on de Groot, van Leeuwen, and van Weert formalism. Our calculations illustrate how the formalism of hydrodynamics with spin can be used to determine physical observables related to the spin polarization and how the latter can be compared with the experimental data.
We present a new solution of relativistic hydrodynamics in 1+3 dimensions which depends on both the transverse coordinate and rapidity. At early times the flow expands dominantly longitudinally in a non-boost-invariant manner, and at late times it expands nearly spherically. These two regimes are shown to be related by symmetry. The effect of viscosity is also discussed.
The one-dimensional non-boost-invariant evolution of the quark-gluon plasma, presumably produced during the early stages of heavy-ion collisions, is analyzed within the frameworks of viscous and anisotropic hydrodynamics. We neglect transverse dynamics and assume homogeneous conditions in the transverse plane but, differently from Bjorken expansion, we relax longitudinal boost invariance in order to study the rapidity dependence of various hydrodynamical observables. We compare the results obtained using several formulations of second-order viscous hydrodynamics with a recent approach to aniso…
We show that the recently formulated causal and stable first-order hydrodynamics has the same dynamics as Israel-Stewart theory for boost-invariant, Bjorken expanding systems with a conformal equation of state and a regulating sector determined by a constant relaxation time. In this case, the general solution of the new first-order formulation can be determined analytically.
Exact correspondence between Israel-Stewart theory and first-order causal and stable hydrodynamics is established for the boost-invariant massive case with zero baryon density and the same constant relaxation times used in the shear and bulk sectors. Explicit expressions for the temperature dependent regulators are given for the case of a relativistic massive gas. The stability and causality conditions known in the first-order approach are applied and one finds that one of them is violated in this case.
We use quasiparticle anisotropic hydrodynamics to study the non-conformal and non-extensive dynamics of a system undergoing boost-invariant Bjorken expansion. To introduce nonextensivity, we use an underlying Tsallis distribution with a time-dependent nonextensivity parameter $q$. By taking moments of the quasiparticle Boltzmann equation in the relaxation-time approximation, we obtain dynamical equations which allow us to determine the time evolution of all microscopic parameters including $q$. We compare numerical solutions for bulk observables obtained using the nonextensive evolution with r…
Space-time evolution of spin polarization within the framework of hydrodynamics with spin based on de Groot - van Leeuwen - van Weert forms of energy-momentum and spin tensors is studied. Due to the non-boost invariant flow in the system the spin polarization components couple to each other implying some effects on the spin polarization observables. We study transverse-momentum and rapidity dependence of mean spin polarization vector for $Λ$ hyperons. Our results show qualitative agreement for rapidity dependence of the global spin polarization with the experiments and other models. The quadru…
We systematically analyze the effects of the derivatives of the hydrodynamic fields on axial Wigner function that describes the spin polarization vector in phase space. We have included all possible first-order derivative contributions that are allowed by symmetry and compute the associated transport functions at one-loop using the linear response theory. In addition to reproducing known effects due to the temperature gradient and vorticity, we have identified a number of potentially significant contributions that are overlooked previously. In particular, we find that the shear strength, the s…
With the PICR hydrodynamic model, we study the polarization splitting between $Λ$ and $\barΛ$ at RHIC BES energy range, based on the meson field mechanism. Our results fit to the experimental data fairly well. Besides, two unexpected effect emerges: (1) the baryon density gradient has non-trivial and negative contribution to the polarization splitting; (2) for 7.7 GeV Au+Au collisions within the centrality range of 20\%-50\%, the polarization splitting surprisingly increases with the centrality decreases. The second effect might help to explain the significant signal of polarization splitting …
We study helicity polarization through the (3+1) dimensional relativistic viscous hydrodynamic models at $\sqrt{s_{NN}}=200$GeV Au+Au collisions. Similar to the local spin polarization, we consider the helicity polarization beyond global equilibrium and investigate the contributions induced by thermal vorticity, shear viscous tensor, and the fluid acceleration. We find that the local helicity polarization induced by thermal vorticity dominates over other contributions. It also implies that in the low-energy collisions, the the fluid vorticity as part of thermal vorticity may play the crucial r…
We compute the longitudinal hydrodynamic flow in ultrarelativistic heavy ion collisions at $\sqrt{s}$ = 5500 GeV by using boost non-invariant initial conditions following from perturbative QCD. The transfer of entropy and energy from the central region to larger rapidities caused by boost non-invariance is determined and the associated decrease in the lifetime of the system is estimated.
Newly introduced equilibrium Wigner functions for particles with spin one-half are used in the semi-classical kinetic equations to study a possible relation between thermal vorticity and spin polarization. It is shown that in global equilibrium both the thermal-vorticity and spin polarization tensors are constant but not necessarily equal. In the case of local equilibrium, we define a procedure leading to hydrodynamic equations with spin. We introduce such equations for the de~Groot, van~Leeuwen, and van~Weert (GLW) formalism as well as for the canonical scheme (these two frameworks differ by …
We consider a family of exact boost invariant solutions of the transport equation for free streaming massless particles, where the one particle distribution function is defined in terms of a function of a single variable. The evolution of second and third moments of the one particle distribution function (the second moment being the energy momentum tensor (EMT) and the third moment the non equilibrium current (NEC)) depends only on two moments of that function. Given those two moments we show how to build a non linear hydrodynamic theory which reproduces the early time evolution of the EMT and…
We propose a generalization of the Bjorken in-out Ansatz for fluid trajectories which, when applied to the (1+1) hydrodynamic equations, generates a one-parameter family of analytic solutions interpolating between the boost-invariant Bjorken picture and the non boost-invariant one by Landau. This parameter characterises the proper-time scale when the fluid velocities approach the in-out Ansatz. We discuss the resulting rapidity distribution of entropy for various freeze-out conditions and compare it with the original Bjorken and Landau results.
We solve the relativistic Navier-Stokes equations with homogeneous boost-invariant boundary conditions, and perform a stability analysis of the solution. We show that, if the bulk viscosity has a peak around $T_c$ as inferred from QCD-based arguments, the background solution "freezes" at $T_c$ to a nearly constant temperature state. This state is however highly unstable with respect to certain inhomogeneous modes. Calculations show that these modes have enough time to blow up and tear the system into droplets. We conjecture that this is how freeze-out occurs in the QGP created in heavy ion col…
By solving a simple kinetic equation, in the relaxation time approximation, and for a particular set of moments of the distribution function, we establish a set of equations which, on the one hand, capture exactly the dynamics of the kinetic equation, and, on the other hand, coincide with the hierarchy of equations of viscous hydrodynamics, to arbitrary order in the viscous corrections. This correspondence sheds light on the underlying mechanism responsible for the apparent success of hydrodynamics in regimes that are far from local equilibrium.
We perform a systematic study of the spin polarization of hyperons in heavy-ion collisions using the MUSIC hydrodynamic model with A Multi-Phase Transport (AMPT) pre-equilibrium dynamics. Our model calculations nicely describe the measured collision-energy, centrality, rapidity, and $p_T$ dependence of $Λ$ polarization. We also study and predict the global spin polarization of $Ξ^-$ and $Ω^-$ as a function of collision energy, which provides a baseline for the studies of the magnetic moment, spin, and mass dependence of the spin polarization. For the local spin polarization, we calculate the r…
We investigate three-dimensional ideal hydrodynamic evolution, with Landau initial conditions, incorporating event-by-event variation with many events and transverse density inhomogeneities. We show that the transition to boost-invariant flow occurs too late for realistic setups, with corrections of $\order{20-30\%}$ expected at freezeout for most scenarios. Moreover, the deviation from boost-invariance is correlated with both transverse flow and elliptic flow, with the more highly transversely flowing regions also showing the most violation of boost invariance. Therefore, if longitudinal flow…
In the present work, we investigate the effects of magnetic field on expanding hot and dense nuclear matter as an ideal fluid. We consider QGP, on the particular case of a (1 + 2) dimensional longitudinally boost-invariant fluid expansion, in the background of an inhomogeneous magnetic field that is generated by external sources. We assume the magnetic field points in the direction perpendicular to the reaction plane, follows the power-law decay in proper time, and has two components on the transverse plane. To simplify the calculation, we suppose the investigated fluid has azimuthal symmetry,…
We solve second order relativistic hydrodynamics equations for a boost-invariant 1+1-dimensional expanding fluid with an equation of state taken from lattice calculations of the thermodynamics of strongly coupled quark-gluon plasma. We investigate the dependence of the energy density as a function of proper time on the values of the shear viscosity, the bulk viscosity, and second order coefficients, confirming that large changes in the values of the latter have negligible effects. Varying the shear viscosity between zero and a few times s/(4 pi), with s the entropy density, has significant eff…
We present results of the application of the anisotropic hydrodynamics (aHydro) framework to (2+1)-dimensional boost invariant systems. The necessary aHydro dynamical equations are derived by taking moments of the Boltzmann equation using a momentum-space anisotropic one-particle distribution function. We present a derivation of the necessary equations and then proceed to numerical solutions of the resulting partial differential equations using both realistic smooth Glauber initial conditions and fluctuating Monte-Carlo Glauber initial conditions. For this purpose we have developed two numeric…
In hydrodynamic approach to relativistic heavy ion collisions, hadrons with nonzero spin, produced out of the hydrodynamic medium, can acquire polarization via spin-vorticity thermodynamic coupling mechanism. The hydrodynamical quantity steering the polarization is the thermal vorticity, that is minus the antisymmetric part of the gradient of four-temperature field. Based on this mechanism there have been several calculations in hydrodynamic and non-hydrodynamic models for non-central heavy ion collisions in the RHIC Beam Energy Scan energy range, showing that the amount of polarization of pro…
Using a holographic method, we further investigate the relaxation towards the hydrodynamic regime of a boost-invariant non-Abelian plasma taken out-of-equilibrium. In the dual description, the system is driven out-of-equilibrium by boundary sourcing, a deformation of the boundary metric, as proposed by Chesler and Yaffe. The effects of several deformation profiles on the bulk geometry are investigated by the analysis of the corresponding solutions of the Einstein equations. The time of restoration of the hydrodynamic regime is investigated: setting the effective temperature of the system at th…
The total angular momentum is conserved in the evolution of the Quark-Gluon Plasma (QGP) created in heavy-ion collision, and consists of two sectors: the orbital angular momentum (OAM) caused by kinetic motion, and the spin, an intrinsic degree of freedom of quarks and gluons. Microscopic scattering processes allow the conversion between these two components, hence the spin density could eventually have non-trivial influence on the QGP evolution. A hydrodynamic theory, with the spin polarization effect properly taken into account, is required to quantitatively study the polarization rate for o…
Background: The near-equilibrium properties of a QCD plasma can be encoded into transport coefficients such as bulk and shear viscosity. In QCD, the ratio of these transport coefficients to entropy density, $ζ/s$ and $η/s$, depends non-trivially on the plasma's temperature. Purpose: We show that in a 0+1D boost-invariant fluid, a temperature-dependent $ζ/s(T)$ or $η/s(T)$ can be described by an equivalent effective viscosity $\left\langle ζ/s \right\rangle_{\textrm{eff}}$ or $\left\langle η/s \right\rangle_{\textrm{eff}}$. We extend the concept of effective viscosity in systems with transverse…
In this article we review the perfect-fluid hydrodynamics with spin framework proposed recently. This framework generalises the standard relativistic hydrodynamics framework to include spin degrees of freedom and provides a natural method to describe the spin polarization evolution of massive spin 1/2 particles. This formalism is based on the GLW (de Groot - van Leeuwen - van Weert) energy-momentum tensor and spin tensor. We show here using Bjorken model that how this spin hydrodynamics framework may be used for the determination of the observables which describes the particle polarization mea…
In this work, we start with chiral kinetic theory and construct the spin hydrodynamic framework for a chiral spinor system. Using the 14-moment expansion formalism, we obtain the equations of motion of second-order dissipative relativistic fluid dynamics with non-trivial spin polarization density. In a chiral spinor system, the spin alignment effect could be treated in the same framework as the Chiral Vortical Effect (CVE). However, the quantum corrections due to fluid vorticity induce not only CVE terms in the vector/axial charge currents but also corrections to the stress tensor. In this fra…
We study the polarization of particles in relativistic heavy-ion collisions at very high energy along the beam direction within a relativistic hydrodynamic framework. We show that this component of the polarization decreases much slower with center-of-mass energy compared to the transverse component, even in the ideal longitudinal boost-invariant scenario with non-fluctuating initial state, and that it can be measured by taking advantage of its quadrupole structure in the transverse momentum plane. In the ideal longitudinal boost-invariant scenario, the polarization is proportional to the grad…
We present some exact solutions to the ideal hydrodynamics of a relativistic superfluid with an almost-conformal equation of state. The solutions have stress tensors which are invariant under Lorentz boosts in one direction, and represent superfluid generalisations of the Bjorken and Gubser flows. We also study corrections to the flows in first-order hydrodynamics, arguing that dissipation is dominated by the shear viscosity. We present some simple numerical solutions for these viscous corrections. Finally, we estimate the size of corrections to the flows arising when the spontaneously broken …
We investigate the phenomenology of freely expanding fluids, with different material properties, evolving through the Israel-Stewart (IS) causal viscous hydrodynamics, and compare our results with those obtained in the relativistic Eckart-Landau-Navier-Stokes (ELNS) acausal viscous hydrodynamics. Through the analysis of scaling invariants we give a definition of thermalization time which can be self-consistently determined in viscous hydrodynamics. Next we construct the solutions for one-dimensional boost-invariant flows. Expansion of viscous fluids is slower than that of one-dimensional ideal…
We report on the approach towards the hydrodynamic regime of boost-invariant N=4 super Yang-Mills plasma at strong coupling starting from various far-from-equilibrium states at tau=0. The results are obtained through numerical solution of Einstein's equations for the dual geometries, as described in detail in the companion article arXiv:1203.0755. Despite the very rich far-from-equilibrium evolution, we find surprising regularities in the form of clear correlations between initial entropy and total produced entropy, as well as between initial entropy and the temperature at thermalization, unde…
Recently proposed thermal-shear and spin-Hall contributions to the particle polarization in heavy-ion collisions are discussed. Alternative definitions of the thermal-shear contribution, i.e. those of Becattini-Buzzegoli-Palermo on the one hand and Liu-Yin on the other, are very similar in the midrapidity region while quite different at forward-backward rapidities, which are measured in fixed-target experiments. It is shown that the thermal-shear contribution to the global polarization with momentum averaging extended to all momenta is very different within these alternative definitions. The s…
The framework of anisotropic hydrodynamics is generalized to include finite particle masses. Two schemes are introduced and their predictions compared with exact solutions of the kinetic equation in the relaxation time approximation. The first formulation uses the zeroth and first moments of the kinetic equation, whereas the second formulation uses the first and second moments. For the case of one-dimensional boost-invariant expansion, our numerical results indicate that the second formulation yields much better agreement with the exact solutions.
We review basic ingredients of the recently introduced perfect-fluid hydrodynamic equations for particles with spin one-half. For a quasi-realistic setup, first numerical solutions for various hydrodynamic variables including the spin polarization tensor are presented. Our results indicate that the initial spin polarization of the hottest region may play a dominant role in the polarization of particles at freeze-out.
We study the spin polarization generated by the hydrodynamic gradients. In addition to the widely studied thermal vorticity effects, we identify an undiscovered contribution from the fluid shear. This shear-induced polarization (SIP) can be viewed as the fluid analog of strain-induced polarization observed in elastic and nematic materials. We obtain the explicit expression for SIP using the quantum kinetic equation and linear response theory. Based on a realistic hydrodynamic model, we compute the differential spin polarization along both the beam direction $\hat{z}$ and the out-plane directio…
The recently formulated framework of anisotropic and dissipative hydrodynamics (ADHYDRO) is used to describe non-boost-invariant motion of the fluid created at the early stages of heavy-ion collisions. Very strong initial asymmetries of pressure are reduced by the entropy production processes. By the appropriate choice of the form of the entropy source we can reproduce realistic scenarios for the isotropization expected in heavy-ion collisions. Our previous results are generalized by including the realistic equation of state as the limit of the isotropization processes.
We summarize the key ingredients of the recently proposed formalism of relativistic perfect-fluid hydrodynamics with spin. Based on the underlying kinetic theory definitions for the equilibrium distribution functions we obtain the evolution equations governing the system's expansion. Employing Bjorken symmetry we study the spin polarization dynamics of the system.
We study local spin polarization in the relativistic hydrodynamic model. Generalizing the Wigner functions previously obtained from chiral kinetic theory by Y. Hidaka et al. [Phys. Rev. D 97, 016004 (2018)] to the massive case, we present the possible contributions up to the order of $\hbar$ from thermal vorticity, shear viscous tensor, other terms associated with the temperature and chemical-potential gradients, and electromagnetic fields to the local spin polarization. We then implement the (3+1)-dimensional viscous hydrodynamic model to study the spin polarizations from these sources with a…
We argue that a detailed analysis of the spin alignement of vector mesons can serve as a probe of two little-understood aspects of spin dynamics in the vortical fluid: The degree of relaxation between vorticity and parton spin polarization, and the degree of coherence of the hadron wavefunction at freeze-out. We illustrate these with a coalescence model.
We compute the fermion spin distribution in the vortical fluid created in off-central high-energy heavy-ion collisions. We employ the event-by-event (3+1)D viscous hydrodynamic model. The spin polarization density is proportional to the local fluid vorticity in quantum kinetic theory. As a result of strong collectivity, the spatial distribution of the local vorticity on the freeze-out hyper-surface strongly correlates to the rapidity and azimuthal angle distribution of fermion spins. We investigate the sensitivity of the local polarization to the initial fluid velocity in the hydrodynamic mode…
We establish a set of equations for moments of the distribution function. In the relaxation time approximations, these moments obey a coupled set of equations that can be truncated order-by-order. Solving the equations of moments, we are able to identify an attractor solution that controls a transition from a free streaming fixed point to a hydrodynamic fixed point. In particular, this attractor solution provides a renormalization of the effective value of the shear viscosity to entropy density ratio, $η/s$, taking into account off-equilibrium effects.
