In this work we present a general derivation of relativistic fluid dynamics from the Boltzmann equation using the method of moments. The main difference between our approach and the traditional 14-moment approximation is that we will not close the fluid-dynamical equations of motion by truncating the expansion of the distribution function. Instead, we keep all terms in the moment expansion. The reduction of the degrees of freedom is done by identifying the microscopic time scales of the Boltzmann equation and considering only the slowest ones. In addition, the equations of motion for the dissipative quantities are truncated according to a systematic power-counting scheme in Knudsen and inverse Reynolds number. We conclude that the equations of motion can be closed in terms of only 14 dynamical variables, as long as we only keep terms of second order in Knudsen and/or inverse Reynolds number. We show that, even though the equations of motion are closed in terms of these 14 fields, the transport coefficients carry information about all the moments of the distribution function. In this way, we can show that the particle-diffusion and shear-viscosity coefficients agree with the values given by the Chapman-Enskog expansion.

The correct thermodynamic relation has a negative sign that was missing in Eq. (12) and, therefore, n _ dso, h -I > ao 0 € n dSy dn ( 12) In the first line of Eq. (62), the sign of the second term, ((, -Q^Co). was incorrect. This term should have a positive sign and the corrected equation reads m~P i --Q|0 0)n + (£,. -n%JCo)0 = -A-ojn + O(Kn). >(0) -_ o (°) t (62) The remaining two equations listed as part of Eq. (62) have no mistakes. In Eq. (72), the term [the first term on the right-hand side of the second equation listed in Eq. (72)] and the term [the first term on the right-hand side of the third equation listed in Eq. (72)] should be multiplied by t n and x", respectively. The corrected form for Eq. (72) then reads J* = -5nnn^6 -^V^II + ^-X nnnv&lv + XnYiIiF -krmnfwI v,

Paatelainen, R. (2016). Event-by-event fluctuations in a perturbative QCD + saturation + hydrodynamics model: Determining QCD matter shear viscosity in ultrarelativistic heavy-ion collisions. Physical Review C, 93 (2) We introduce an event-by-event perturbative-QCD + saturation + hydro ("EKRT") framework for ultrarelativistic heavy-ion collisions, where we compute the produced fluctuating QCD-matter energy densities from next-to-leading-order perturbative QCD using a saturation conjecture to control soft-particle production and describe the space-time evolution of the QCD matter with dissipative fluid dynamics, event by event. We perform a simultaneous comparison of the centrality dependence of hadronic multiplicities, transverse momentum spectra, and flow coefficients of the azimuth-angle asymmetries against the LHC and RHIC measurements. We compare also the computed event-by-event probability distributions of relative fluctuations of elliptic flow and event-plane angle correlations with the experimental data from Pb + Pb collisions at the LHC. We show how such a systematic multienergy and multiobservable analysis tests the initial-state calculation and the applicability region of hydrodynamics and, in particular, how it constrains the temperature dependence of the shear viscosity-to-entropy ratio of QCD matter in its different phases in a remarkably consistent manner.

Relativistic dissipative fluid dynamics is a common tool to describe the space-time evolution of the strongly interacting matter created in ultrarelativistic heavy-ion collisions. For a proper comparison to experimental data, fluid-dynamical calculations have to be performed on an event-by-event basis. Therefore, fluid dynamics should be able to reproduce, not only the event-averaged momentum anisotropies, v n , but also their distributions. In this paper, we investigate the event-by-event distributions of the initial-state and momentum anisotropies n and v n , and their correlations. We demonstrate that the event-by-event distributions of relative v n fluctuations are almost equal to the event-by-event distributions of corresponding n fluctuations, allowing experimental determination of the relative anisotropy fluctuations of the initial state. Furthermore, the correlation c(v 2 , v 4 ) turns out to be sensitive to the viscosity of the fluid providing an additional constraint to the properties of the strongly interacting matter.

We investigate the influence of a temperature-dependent shear viscosity over entropy density ratio η/s on the transverse momentum spectra and elliptic flow of hadrons in ultrarelativistic heavy-ion collisions. We find that the elliptic flow in √S(NN)=200  GeV Au+Au collisions at RHIC is dominated by the viscosity in the hadronic phase and in the phase transition region, but largely insensitive to the viscosity of the quark-gluon plasma (QGP). At the highest LHC energy, the elliptic flow becomes sensitive to the QGP viscosity and insensitive to the hadronic viscosity.

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We show how the linearized equations of motion of any dissipative current are determined by the analytical structure of the associated retarded Green's function. If the singularity of the Green's function, which is nearest to the origin in the complex-frequency plane, is a simple pole on the imaginary frequency axis, the linearized equations of motion can be reduced to relaxation-type equations for the dissipative currents. The value of the relaxation time is given by the inverse of this pole. We prove that, if the relaxation time is sent to zero, or equivalently, the pole to infinity, the dissipative currents approach the values given by the standard gradient expansion.

We study the hadron spectra in nearly central A+A collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) in a broad transverse momentum range. We cover the low-p T spectra using longitudinally boost-invariant hydrodynamics with initial energy and net-baryon number densities from the perturbative QCD (pQCD)+saturation model. Buildup of the transverse flow and sensitivity of the spectra to a single decoupling temperature T dec are studied. Comparison with RHIC data at √ s NN = 130 and 200 GeV suggests a rather high value T dec = 150 MeV. The high-p T spectra are computed using factorized pQCD cross sections, nuclear parton distributions, fragmentation functions, and describing partonic energy loss in the quark-gluon plasma by quenching weights. Overall normalization is fixed on the basis of p+p(p) data and the strength of energy loss is determined from RHIC Au+Au data. Uncertainties are discussed. With constraints from RHIC data, we predict the p T spectra of hadrons in 5% most central Pb+Pb collisions at the LHC energy √ s NN = 5500 GeV. Because of the closed framework for primary production, we can also predict the net-baryon number at midrapidity, as well as the strength of partonic energy losses at the LHC. Both at the LHC and RHIC, we recognize a rather narrow crossover region in the p T spectra, where the hydrodynamic and pQCD fragmentation components become of equal size. We argue that in this crossover region the two contributions are to a good approximation mutually independent. In particular, our results suggest a wider p T region of applicability for hydrodynamical models at the LHC than at RHIC.