Far-from-equilibrium attractors with a realistic non-conformal equation of state Mubarak Alqahtani

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Far-from-equilibrium attractors with a realistic non-conformal

equation of state

Mubarak Alqahtani

Department of Physics, College of Science,

Imam Abdulrahman Bin Faisal University, Dammam 31441, Saudi Arabia

(Dated: May 1, 2023)

Abstract

Using anisotropic hydrodynamics, we examine the existence of early-time attractors of non-

conformal systems undergoing Bjorken expansion. In the case of a constant mass, we ﬁnd that

the evolution of the scaled longitudinal pressure is insensitive to variations of initial conditions

converging onto an early-time universal curve and eventually merging with the late-time Navier-

Stokes attractor (the hydrodynamic attractor). On the other hand, the bulk and the shear viscous

corrections do not show an early-time attractor behavior. These results are consistent with previous

studies considering a constant mass. When a realistic equation of state is included in the dynamics

with a thermal mass, we demonstrate for the ﬁrst time the absence of strict late-time universal

attractors. However, a semi-universal feature of the evolution at very late times remains.

arXiv:2210.06712v2 [nucl-th] 28 Apr 2023

I. INTRODUCTION

Relativistic hydrodynamics has been a successful tool in modeling relativistic heavy-ion

collisions [1–7]. Throughout the years, diﬀerent dissipative hydrodynamics approaches have

been used to describe ﬁnal-state observables [8–18] (For a recent review, see Ref. [19]).

This success in predicting and describing the experimental results was not limited to large

systems such as Pb-Pb and Au-Au, but was also seen in small systems such as p-Pb [20–25].

This applicability in small systems seems to contradict the assumption that hydrodynamics

applicability is limited to systems with near thermal equilibrium which is not expected in

small systems at all. This puzzle triggered great interest in understanding the domain of

applicability of hydrodynamics [26].

The applicability of hydrodynamics could be explained by the existence of the attrac-

tor solutions in far-from-equilibrium systems. In such systems, the evolution of solutions-

irrespective of their initial conditions- converge quickly to a universal solution (the attrac-

tor) at very early times. This universality seems to be a property of hydrodynamics where

diﬀerent solutions lose information about their initial conditions. There have been many

interesting works where attractor solutions have been found and examined using diﬀerent

approaches and symmetries see e.g., [26–36]. We note that almost all of the attractor stud-

ies cited above have focused on simple setups where the systems under consideration were

conformal. Recent reviews about attractors can be found in [6,37].

Recently, the eﬀect of nonconformality on the attractors for diﬀerent approaches has

been studied in [38–45]. In Ref. [43] speciﬁcally, a modiﬁed anisotropic hydrodynamics

approach is introduced and compared to the exact kinetic theory solutions. The agreement

found using this approach to the kinetic theory solutions is excellent, as pointed out by the

authors, especially at early times and for the largest possible initial negative bulk pressures.

Such an agreement between anisotropic hydrodynamics and exact solutions motivates us to

study the existence of early-time attractors of anisotropic hydrodynamics using a realistic

non-conformal equation of state for quasiparticles.

This work is an attempt to study the eﬀect of using a realistic non-conformal equation of

state on hydrodynamic attractor solutions. To do so, we will use anisotropic hydrodynamics

approach of systems undergoing Bjorken expansion of quasiparticles having a thermal mass

m(T) [46]. This model is called quasiparticle anisotropic hydrodynamics and has shown a

good phenomenological agreement with experimental results in the 3+1D case, see e.g. [47–

51]. In this work, however, we limit ourselves to the simplest case of 0+1D and try to

examine if early-time attractors survive when a realistic non-conformal equation of state is

assumed in the dynamics.

The structure of the manuscript is as follows. In Sec. II, we introduce anisotropic hydro-

dynamics generally, then obtain the dynamical equations needed for the system’s evolution

in the quasiparticle approach. In Sec. III, we show our results of the early-time attractors for

the scaled longitudinal pressure and the shear stress pressure. For comparison, we also show

the results of systems with a constant mass using anisotropic hydrodynamics as well. Unlike

the constant mass case, where strict early-time attractors exist for the scaled longitudinal

pressure, by using a realistic equation of state, we ﬁnd a semi-universal attractor at very

late times. Conclusions and a future outlook are summarized in Sec. IV.

II. ANISOTROPIC HYDRODYNAMICS

A. 3+1D anisotropic hydrodynamics

The anisotropic hydrodynamics (aHydro) approach is motivated by the fact that the

quark-gluon plasma (QGP) dynamics is highly momentum-space anisotropic [52,53] (see

Ref. [54] for an introduction of aHydro). In this framework, the one-particle distribution

function is assumed to be momentum-space anisotropic in the local-rest frame (LRF) [55]

fLRF(x, p) = feq 1

λsX

α2

+m2!,(1)

with αibeing the anisotropy parameters (i∈ {x, y, z}) and λbeing a parameter that is

identiﬁed with the temperature in the isotropic equilibrium limit. In the limit where αi= 1

and λ=T, one recovers the isotropic distribution function and in the case of Boltzmann

statistics, feq = exp(−E/T ).

From kinetic theory, the distribution function of a gas of particles having mass m(T)

obeys the Boltzmann equation [46,56]

pµ∂µf+1

2∂im2∂i

(p)f=−C[f],(2)

with fbeing the assumed single-particle distribution function and C[f] is the collisional

kernel. In this work, we assume fas been given in Eq. (1) and C[f] to be in the relaxation-

time approximation given by C[f] = pµuµ(f−feq)/τeq where τeq is position dependent and

given by

τeq(T) = 15¯η

κ( ˆmeq)T1 + Eeq(T)

Peq(T),(3)

where κcan be expressed in terms of modiﬁed Bessel functions of the second kind and

modiﬁed Struve functions as deﬁned in Ref. [46] where ˆmeq =m/T . Moreover, ¯ηis shear

viscosity to entropy density ratio η/Seq, which is held constant during the evolution of the

system. Eeq and Peq are the equilibrium energy density and pressure, respectively.We note

that in the conformal limit, Eq. (3) becomes

τeq(T) = 5η

4Peq

=5¯η

T.(4)

The dynamical equations can be obtained by taking the lower moments of the Boltzmann

equation, Eq. (2)

∂µJµ=−ZdP C[f],(5)

∂µTµν =−ZdP pνC[f],(6)

∂µIµνλ −J(ν∂λ)m2=−ZdP pνpλC[f],(7)

which are the zeroth, ﬁrst, and second moments of the Boltzmann equation, respectively.

Jµis the particle four-current, Tµν is the energy-momentum tensor, and Iµνλ is a rank-three

tensor. They are given by

Jµ≡ZdP pµf(x, p),(8)

Tµν ≡ZdP pµpνf(x, p) + Bgµν ,(9)

Iµνλ ≡ZdP pµpνpλf(x, p),(10)

where dP is the Lorentz invariant momentum-space integration measure given by ˜

Nd3p

Ewith

N≡Ndof /(2π)3where Ndof is the number of degrees of freedom.

We note here that a background contribution Bis added to the deﬁnition of the energy-

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Far-from-equilibriumattractorswitharealisticnon-conformalequationofstateMubarakAlqahtaniDepartmentofPhysics,CollegeofScience,ImamAbdulrahmanBinFaisalUniversity,Dammam31441,SaudiArabia(Dated:May1,2023)AbstractUsinganisotropichydrodynamics,weexaminetheexistenceofearly-timeattractorsofnon-conformalsyst...

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Far-from-equilibrium attractors with a realistic non-conformal equation of state Mubarak Alqahtani

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