We introduce a new framework of highly-anisotropic hydrodynamics that includes dissipation effects. Dissipation is defined by the form of the entropy source that depends on the pressure anisotropy and vanishes for the isotropic fluid. With a simple ansatz for the entropy source obeying general physical requirements, we are led to a non-linear equation describing the time evolution of the anisotropy in purely-longitudinal boost-invariant systems. Matter that is initially highly anisotropic approaches naturally the regime of the perfect fluid. Thus, the resulting evolution agrees with the expectations about the behavior of matter produced at the early stages of relativistic heavy-ion collisions. The equilibration is identified with the processes of entropy production.

We exactly solve the one-dimensional boost-invariant Boltzmann equation in the relaxation time approximation for arbitrary shear viscosity. The results are compared with the predictions of viscous and anisotropic hydrodynamics. Studying different non-equilibrium cases and comparing the exact kinetic-theory results to the second-order viscous hydrodynamics results we find that recent formulations of second-order viscous hydrodynamics agree better with the exact solution than the standard Israel-Stewart approach. Additionally, we find that, given the appropriate connection between the kinetic and anisotropic hydrodynamics relaxation times, anisotropic hydrodynamics provides a very good approximation to the exact relaxation time approximation solution.

We exactly solve the relaxation-time approximation Boltzmann equation for a system which is transversely homogeneous and undergoing boost-invariant longitudinal expansion. We compare the resulting exact numerical solution with approximate solutions available in the anisotropic hydrodynamics and second order viscous hydrodynamics frameworks. In all cases studied, we find that the anisotropic hydrodynamics framework is a better approximation to the exact solution than traditional viscous hydrodynamical approaches.The application of relativistic viscous hydrodynamics is important in a wide variety of situations including, for example, the dynamics of high energy astrophysical plasmas and the quark-gluon plasma created in relativistic heavy-ion collisions. Since the seminal work of Israel and Stewart [1,2] there have been many papers that have addressed the questions of how to apply and systematically improve relativistic viscous hydrodynamics [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. Recently, a new framework called anisotropic hydrodynamics (aHydro) has emerged for describing the nonequilibrium dynamics of relativistic systems [22][23][24][25][26][27][28][29].

A R T I C L E I N F O Keywords: relativistic heavy-ion collisions relativistic hydrodynamics spin polarization pseudo-gauge transformations semi-classical expansion Pauli-Lubański vector A B S T R A C T Recent progress in the formulation of relativistic hydrodynamics for particles with spin one-half is reviewed. We start with general arguments advising introduction of a tensor spin chemical potential that plays a role of the Lagrange multiplier coupled to the spin angular momentum. Then, we turn to a discussion of spin-dependent distribution functions that have been recently proposed to construct a hydrodynamic framework including spin and serve as a tool in phenomenological studies of hadron polarization. Distribution functions of this type are subsequently used to construct the equilibrium Wigner functions that are employed in the semi-classical kinetic equation. The semi-classical expansion elucidates several aspects of the hydrodynamic approach, in particular, shows the ways in which different possible versions of hydrodynamics with spin can be connected by pseudo-gauge transformations. These results point out at using the de Grootvan Leeuwen -van Weert versions of the energy-momentum and spin tensors as the most natural and complete physical variables. Finally, a totally new method is proposed to design hydrodynamics with spin, which is based on the classical treatment of spin degrees of freedom. Interestingly, for small values of the spin chemical potential the new scheme brings the results that coincide with those obtained before. The classical approach also helps us to resolve problems connected with the normalization of the spin polarization three-vector. In addition, it clarifies the role of the Pauli-Lubański vector and the entropy current conservation. We close our review with several general comments presenting possible future developments of the discussed frameworks.

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 oneparticle 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 numerical implementations: one which is based on straightforward integration of the resulting partial differential equations supplemented by a two-dimensional weighted Lax-Friedrichs smoothing in the case of fluctuating initial conditions; and another that is based on the application of the Kurganov-Tadmor central scheme. For our final results we compute the collective flow of the matter via the lab-frame energy-momentum tensor eccentricity as a function of the assumed shear viscosity to entropy ratio, proper time, and impact parameter.

We derive a system of moment-based dynamical equations that describe the 1+1d space-time evolution of a cylindrically symmetric massive gas undergoing boost-invariant longitudinal expansion. Extending previous work, we introduce an explicit degree of freedom associated with the bulk pressure of the system. The resulting form generalizes the ellipsoidal one-particle distribution function appropriate for massless particles to massive particles. Using this generalized form, we obtain a system of partial differential equations that can be solved numerically. In order to assess the performance of this scheme, we compare the resulting anisotropic hydrodynamics solutions with the exact solution of the 0+1d Boltzmann equation in the relaxation time approximation.We find that the inclusion of the bulk degree of freedom improves agreement between anisotropic hydrodynamics and the exact solution for a massive gas.

We compute the QGP suppression of Upsilon(1s), Upsilon(2s), Upsilon(3s), chi_b1, and chi_b2 states in sqrt(s_NN)=2.76 TeV Pb-Pb collisions. Using the suppression of each of these states, we estimate the inclusive R_AA for the Upsilon(1s) and Upsilon(2s) states as a function of N_part, y, and p_T including the effect of excited state feed down. We find that our model provides a reasonable description of preliminary CMS results for the N_part-, y-, and p_T-dependence of R_AA for both the Upsilon(1s) and Upsilon(2s). Comparing to our previous model predictions, we find a flatter rapidity dependence, thereby reducing some of the tension between our model and ALICE forward-rapidity results for Upsilon(1s) suppression.Comment: 5 pages, 4 figures; tsv files for all plots included in "anc" folder of the submissio

We derive the form of the viscous corrections to the phase-space distribution function due to the bulk viscous pressure and shear stress tensor using the iterative Chapman-Enskog method. We then calculate the transport coefficients necessary for the second-order hydrodynamic evolution of the bulk viscous pressure and the shear stress tensor. We demonstrate that the transport coefficients obtained using the Chapman-Enskog method are different than those obtained previously using the 14-moment approximation for a finite particle mass. Specializing to the case of boost-invariant and transversally homogeneous longitudinal expansion, we show that the transport coefficients obtained using the Chapman-Enskog method result in better agreement with the exact solution of the Boltzmann equation in the relaxation-time approximation compared to results obtained in the 14-moment approximation. Finally, we explicitly confirm that the time evolution of the bulk viscous pressure is significantly affected by its coupling to the shear stress tensor.