Abstract: We propose a new scenario characterizing the transition of the quark–gluon plasma (QGP) produced in heavy-ion collisions from a highly non-equilibrium state at early times toward a fluid described by hydrodynamics at late times. We develop an analogy to the evolution of a quantum mechanical system that is governed by the instantaneous ground states. In the simplest case, these slow modes are “pre-hydrodynamic” in the sense that they are initially distinct from, but evolve continuously into, hydrodynamic modes. For a class of collision integrals, the pre-hydrodynamic mode represents the angular distribution (in momentum space) of those gluons that carry most of the energy. We illustrate this scenario using a kinetic description of weakly-coupled Bjorken expanding plasma. Rapid longitudinal expansion drives a reduction in the degrees of freedom at early times. In the relaxation time approximation for the collision integral, we show quantitatively that the full kinetic theory evolution is dominated by the pre-hydrodynamic mode. We elaborate on the criterion for the dominance of pre-hydrodynamic slow modes and speculate that adiabatic hydrodynamization may describe the pre-equilibrium behavior of the QGP produced in heavy-ion collisions.

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Topics: Quark–gluon plasma (56%), Adiabatic process (54%), Degrees of freedom (physics and chemistry) (51%) ... read more

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11 results found

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Abstract: We present a concise review of the recent development of relativistic hydrodynamics and its applications to heavy-ion collisions. Theoretical progress on the extended formulation of hydrodynamics toward out-of-equilibrium systems is addressed, with emphasis on the so-called attractor solution. Moreover, recent phenomenological improvements in the hydrodynamic modeling of heavy-ion collisions with respect to the ongoing beam energy scan program, the quantitative characterization of transport coefficients in three-dimensionally expanding quark–gluon plasma, the fluid description of small colliding systems, and certain other interdisciplinary connections are discussed.

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36 Citations

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Abstract: Hydrodynamic attractors have recently gained prominence in the context of early stages of ultrarelativistic heavy-ion collisions at the RHIC and LHC. We critically examine the existing ideas on this subject from a phase space point of view. In this picture the hydrodynamic attractor can be seen as a special case of the more general phenomenon of dynamical dimensionality reduction of phase space regions. We quantify this using principal component analysis. Furthermore, we adapt the well known slow-roll approximation to this setting. These techniques generalize easily to higher dimensional phase spaces, which we illustrate by a preliminary analysis of a dataset describing the evolution of a five-dimensional manifold of initial conditions immersed in a 16-dimensional representation of the phase space of the Boltzmann kinetic equation in the relaxation time approximation.

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Topics: Phase space (61%), Attractor (59%), Manifold (51%)

17 Citations

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Abstract: We present an analytic solution of a simple set of equations that govern the expansion of boost-invariant plasmas of massless particles. These equations describe, approximately, the early time, collisionless regime, and the transition to hydrodynamics at late time. Their mathematical structure encompasses all versions of second order hydrodynamics when applied to Bjorken flows. The analytic solution provides an explicit expression for the attractor solution that connects the collisionless regime to hydrodynamics. It also constitutes a neat example of an application of the theory of resurgence in asymptotic series.

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Topics: Attractor (57%), Asymptotic expansion (52%)

14 Citations

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Aleksi Kurkela^{1}, Aleksi Kurkela^{2}, Seyed Farid Taghavi^{3}, Urs Achim Wiedemann^{1} +1 more•Institutions (3)

Abstract: The observation of fluid-like behavior in nucleus-nucleus (AA), proton-nucleus (pA) and high-multiplicity proton-proton (pp) collisions motivates systematic studies of how different measurements approach their fluid-dynamic limit. We have developed numerical methods to solve the ultra-relativistic Boltzmann equation for systems of arbitrary size and transverse geometry. Here, we apply these techniques for the first time to the study of azimuthal flow coefficients v n including non-linear mode-mode coupling and to an initial condition with realistic event-by-event fluctuations. We show how both linear and non-linear response coefficients extracted from v n develop as a function of opacity from free streaming to perfect fluidity. We note in particular that away from the fluid-dynamic limit, the signal strength of linear and non-linear response coefficients does not reduce uniformly, but that their hierarchy and relative size shows characteristic differences.

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Topics: Boltzmann equation (53%), Initial value problem (53%), Numerical analysis (50%) ... read more

10 Citations

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Jorge Casalderrey-Solana^{1}, José Guilherme Milhano^{2}, Daniel Pablos^{3}, Krishna Rajagopal^{4} +1 more•Institutions (4)

Abstract: We explore how to improve the hybrid model description of the particles originating from the wake that a jet produced in a heavy ion collision leaves in the droplet of quark-gluon plasma (QGP) through which it propagates, using linearized hydrodynamics on a background Bjorken flow. Jet energy and momentum loss described by the hybrid model become currents sourcing linearized hydrodynamics. By solving the linearized hydrodynamic equations numerically, we investigate the development of the wake in the dynamically evolving droplet of QGP, study the effect of viscosity, scrutinize energy-momentum conservation, and check the validity of the linear approximation. We find that linearized hydrodynamics works better in the viscous case because diffusive modes damp the energy-momentum perturbation produced by the jet. We calculate the distribution of particles produced from the jet wake by using the Cooper-Frye prescription and find that both the transverse momentum spectrum and the distribution of particles in azimuthal angle are similar in shape in linearized hydrodynamics and in the hybrid model. Their normalizations are different because the momentum-rapidity distribution in the linearized hydrodynamics analysis is more spread out, due to sound modes. Since the Bjorken flow has no transverse expansion, we explore the effect of transverse flow by using local boosts to add it into the Cooper-Frye formula. After including the effects of transverse flow in this way, the transverse momentum spectrum becomes harder: more particles with transverse momenta bigger than $2$ GeV are produced than in the hybrid model. Although we defer implementing this analysis in a jet Monte Carlo, as would be needed to make quantitative comparisons to data, we gain a qualitative sense of how the jet wake may modify jet observables by computing proxies for two example observables: the lost energy recovered in a cone of varying open angle, and the fragmentation function. We find that linearized hydrodynamics with transverse flow effects added improves the description of the jet wake in the hybrid model in just the way that comparison to data indicates is needed. Our study illuminates a path to improving the description of the wake in the hybrid model, highlighting the need to take into account the effects of both transverse flow and the broadening of the energy-momentum perturbation in spacetime rapidity on particle production.

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Topics: Wake (51%)

5 Citations

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37 results found

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Abstract: We argue that the distribution functions for quarks and gluons are computable at small $x$ for sufficiently large nuclei, perhaps larger than can be physically realized. For such nuclei, we argue that weak coupling methods may be used. We show that the computation of the distribution functions can be recast as a many-body problem with a modified propagator, a coupling constant which depends on the multiplicity of particles per unit rapidity per unit area, and for non-Abelian gauge theories, some extra media-dependent vertices. We explicitly compute the distribution function for gluons to lowest order, and argue how they may be computed in higher order.

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Topics: Quark–gluon plasma (56%), Color-glass condensate (55%), Gluon (54%) ... read more

1,530 Citations

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Abstract: We compute the Green's function for scalars, fermions, and vectors in the color field associated with the infinite momentum frame wave function of a large nucleus. Expectation values of this wave function can be computed by integrating over random orientations of the valence quark charge density. This relates the Green's functions to correlation functions of a two-dimensional, ultraviolet finite, field theory. We show how one can compute the sea quark distribution functions and explicitly compute them in the kinematic range of transverse momenta, ${\mathrm{\ensuremath{\alpha}}}_{\mathit{s}}^{2}$${\mathrm{\ensuremath{\mu}}}^{2}$\ensuremath{\ll}${\mathit{k}}_{\mathit{t}}^{2}$\ensuremath{\ll}${\mathrm{\ensuremath{\mu}}}^{2}$, where ${\mathrm{\ensuremath{\mu}}}^{2}$ is the average color charge squared per unit area. When ${\mathit{m}}_{\mathrm{quark}}^{2}$\ensuremath{\ll}${\mathrm{\ensuremath{\mu}}}^{2}$\ensuremath{\sim}${\mathit{A}}^{1/3}$, the sea quark contribution to the infinite momentum frame wave function saturates at a value that is the same as that for massless sea quarks.

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Topics: Distribution function (67%)

744 Citations

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Jamal Jalilian-Marian^{1}, Alex Kovner^{1}, Larry McLerran^{1}, Heribert Weigert^{1}•Institutions (1)

Abstract: We compute the distribution functions for gluons at very small x and not too large values of transverse momenta. We extend the McLerran-Venugopalan model by using renormalization group methods to integrate out effects due to those gluons which generate an effective classical charge density for Weizs{umlt a}cker-Williams fields. We argue that this model can be extended from the description of nuclei at small x to the description of hadrons at yet smaller values of x. This generates a Lipatov-like enhancement for the intrinsic gluon distribution function and a nontrivial transverse momentum dependence as well. We estimate the transverse momentum dependence for the distribution functions, and show how the issue of unitarity is resolved in lepton-nucleus interactions. {copyright} {ital 1997} {ital The American Physical Society}

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Topics: Distribution function (60%), Color-glass condensate (54%), Quantum chromodynamics (53%) ... read more

531 Citations

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Abstract: We describe how thermalization occurs in heavy ion collisions in the framework of perturbative QCD When the saturation scale Qs is large compared to ΛQCD, thermalization takes place during a time of order α−13/5Qs−1 and the maximal temperature achieved is α2/5Qs

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Topics: Perturbative QCD (52%)

477 Citations

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Abstract: 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.

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Topics: Lattice Boltzmann methods (64%), Boltzmann equation (62%), Direct simulation Monte Carlo (60%) ... read more

330 Citations