Towards a full solution of the relativistic Boltzmann equationfor quark-gluon matter on GPUs

Jun-Jie Zhang,1 Hong-Zhong Wu,1 Shi Pu ,1 Guang-You Qin,2,3 and Qun Wang11Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China2Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), Central China

Normal University, Wuhan, Hubei 430079, China3Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

(Received 15 January 2020; accepted 28 September 2020; published 19 October 2020)

Many studies for nonequilibrium systems, e.g., the pre-equilibration puzzle in heavy-ion collisions,require solving the relativistic Boltzmann equation (BE) with the full collisional kernel to high precision. Itis challenging to solve relativistic BE due to its high dimensional phase-space integrals and limitedcomputing resources. We have developed a numerical framework for a full solution of the relativisticBoltzmann equations for the quark-gluon matter using the multiple graphics processing units (GPUs) ondistributed clusters. Including all the 2 → 2 scattering processes of 3-flavor quarks and gluons, we computethe time evolution of distribution functions in both coordinate and momentum spaces for the cases of puregluons, quarks, and the mixture of quarks and gluons. By introducing a symmetrical sampling method onGPUs which ensures the particle number conservation, our framework is able to perform the space-timeevolution of quark-gluon system toward thermal equilibrium with high performance. We also observe thatthe gluons naturally accumulate in the soft region at the early time, which may indicate the gluoncondensation.

DOI: 10.1103/PhysRevD.102.074011

I. INTRODUCTION

Relativistic Boltzmann equation (BE), an effectivetheory of many-body systems, is a profound and widelyused tool to study the properties of the systems out ofequilibrium or in thermal equilibrium. Recently, BE is oftenapplied to study the problem of early thermalization, whichremains to be one of the “greatest unsolved problems” [1]in relativistic heavy-ion collisions, which collide twoaccelerated nuclei to create a hot and dense deconfinednuclear matter, named quark-gluon plasma (QGP). Thespace-time evolution of QGP has been well described byrelativistic hydrodynamics simulations. The success ofhydrodynamical models on soft hadron production andcollective flows provides strong evidence for the rapidthermalization of the quark-gluon system to create astrongly-interacting QGP [2–6]. The timescale expectedfor thermalization is estimated to be less than 1 fm=c [4] oreven shorter than 0.25 fm=c [7,8] in nucleus-nucleuscollisions at Relativistic Heavy-Ion Collider (RHIC) andthe Large Hadron Collider (LHC). However, it remains to

be a puzzle how an overoccupied gluonic system with weakcoupling can reach thermal equilibrium within such a shorttimescale [5].

A. Background

The study of the systems under the framework of BEwith suitable initial conditions, also referred to as thekinetic approach, is a well-established method for probingthe real-time quark and gluon dynamics in the dilute regimeat weakly coupling limit [9–11]. However, a full solution ofthe relativistic BE involving all parton species, e.g., u, d, squarks, their antiparticles, and gluons, is still challengingboth analytically and numerically due to the complexity ofthe collision integral, higher dimensions and computingresources.The typical initial condition for relativistic heavy-ion

collisions [12] is an overpopulated gluonic state namedcolor glass condensate (CGC) [13–20], which is formed inthe dynamical balance between the splitting and fusion ofgluons in the small-x region. The occupation number ofsmall-x gluons is of order 1=αs [21]. The gluon numbergrows until the gluon size is larger than 1=Qs [22], with Qsbeing the saturation scale. After the CGC state, the glasma,a state of color electromagnetic fields, may be formed[22–27]. Currently, how the glasma transits to a therma-lized QGP in a short timescale is not well understood yet,which is often referred to as the early thermalization puzzle.

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There have been various studies focusing on the evolu-tion of the glasma stage. For example, the “bottom-upscenario” [28] estimates a thermalization time of orderτth ∼ α−13=5s =Qs, which is unfortunately too large comparedto the timescale required by the hydrodynamics. Toreconcile this discrepancy, many other possible mecha-nisms have been considered. One interesting mechanism isthe so-called plasma instability [29–35]. It originates fromthe anisotropic momentum distribution in the plasma andmay drastically speed up the process of glasma equilibra-tion [36–38]. However, some studies also imply that theplasma instability may not play a significant role at theearly stage [39,40], and a scaling solution may also berequired [41]. Another mechanism is the Bose-Einsteincondensation of gluons. It has been suggested that thegluon condensation at the early stage [42,43] may accel-erate the thermalization process [44–48]. The influence offermions and masses in forming condensation has also beendiscussed in Refs. [49–53]. However, it is argued that theinelastic scattering processes may strongly hinder the effectof gluon condensation [54,55]. In fact, the role of inelasticscatterings in the thermalization is still not quite clear so far.It has been suggested a long time ago that the inelasticprocesses might be essential for thermalization [56]. Someworks by solving BE with the test particle method includ-ing 2 → 3 processes has obtained the results close to the“bottom-up” scenario [57]. Another simulation from theBoltzmann approach of multiparton scattering (BAMPS), apackage for solving BE using the test particle method,suggests that the bremsstrahlung from 2 → 3 processesincreases the efficiency of thermalization [58]. Later studiesfrom BAMPS imply that the inverse processes, i.e., the3 → 2 processes, inhibit the 2 → 3 processes. With bothprocesses included, the timescale of thermal equilibrationfrom the BAMPS is of order α−2s lnðαsÞ−2Q−1

s [59]. It hasalso been argued that the inclusion of all next-to-leadingorder processes may make the equilibration considerablyfaster than the simple 2 → 3 processes [60].

B. Motivation

Some of the above studies require solving relativistic BEnumerically. Historically, a full numerical solution of thenonrelativistic BE has always been a challenge due toits high dimensions and the intrinsic physical properties[61–63]. Even in today’s petascale clusters, BE stillpresents a substantial computational challenge [63]. Inthe real application of nonrelativistic BE, one usually needsvery dense spatial grids to describe complicated effectsrelated to pressure, temperature, and turbulence, etc.The main difference between the relativistic and non-

relativistic BE lies in the collision term and the coupledequations. In the relativistic BE, the collision integrals areusually much more complicated than those in the non-relativistic case. For example, in our relativistic BE, there

are seven particle species with complicated scatteringmatrix, while in the nonrelativistic case, one usually dealswith single species of particles in a gas or liquid. Despitethe heavy workload of the collision term, our relativistic BEalso contains coupled equations of seven species, whichneed to be solved simultaneously. Furthermore, the dis-tributions for fermions should not exceed unity due to thePauli exclusion principle. This limitation requires that thetime step of the evolution be sufficiently small, which inturn drastically slows down the speed of the code.Though complicated, several numerical tools based on

the parton cascade model [64,65] have been developed inthe market, e.g., BAMPS [66] and ZPC [67,68]. Anotherway is the lattice Boltzmann approach [69–71], which hasbeen utilized as a fast lattice Boltzmann solver for relativ-istic hydrodynamics with relaxation time approximation[62,72,73]. In addition, the straightforward implementa-tions of effective kinetic theory (e.g., see Ref. [9] for thetheoretical framework) for pure gluons [74,75] and quark-gluon systems [51,52,76] also provide us physical insightsfor prethermalization. While these models and approacheshave succeeded in describing the nonequilibrium evolutionof quark-gluon matter with a set of parameters given by thephysical consideration for simplicity, a full solution of theBE is still demanded for a comprehensive understanding ofthe thermalization puzzle.

C. Our numerical framework

In this study, we develop a numerical framework for thefull numerical solutions of the relativistic BE with the helpof the state-of-the-art GPUs. GPU, known for its high clockrate, high instruction per cycle [77], and multiple cores,makes more and more contributions to computationalphysics nowadays [78]. Some calculations, which areextremely difficult in the old-times, are now within thescope [79–87]. Some physical phenomena could be under-stood by the simulations via GPUs [88]. With the newGPU techniques, various attempts have been done to tacklethe BE from different aspects for nonrelativistic cases[63,89–93]. The GPU techniques also motivate us todevelop the framework for solving the relativistic BE inthe context of relativistic heavy-ion collisions, despite aseries of methodological challenges that have no counter-parts in the nonrelativistic realm. As a first step, we onlyconsider the 2 → 2 scattering processes in the current work.The difficulties in solving the relativistic BE originate

from two aspects. First, the collision terms are highdimensional integrals. In this work, we use the packageZMCintegral 5.0 [94,95] to perform these high dimensionalcollisional integrals. ZMCintegral is an open-source Python

package developed by some of the authors in this work.The second difficulty is the issue of particle numbernonconservation due to the discrete Monte Carlo (MC)integration, see also Refs. [96,97]. To achieve a strictparticle number conservation in the CPU framework will

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usually cost lots of computing time. In our work, wepropose a “symmetrical sampling” method for the colli-sional integrals via GPUs. With the help of new features ofGPUs, named CUDA atomic operations [98,99], we canachieve the strict particle number conservation with accept-able computing time.Our numerical framework provides a full solution of the

BE with complete 2 → 2 scattering processes. The programis developed with the combination of Python library Numba[100] and Ray [101], which enable the manipulation ofGPU devices on distributed clusters. We will test the codein many aspects, for instance, the stability of collisionalintegrals, the particle number conservation, and the totalenergy conservation. We will also show the time evolutionof the distribution functions in both coordinate andmomentum spaces.The structure of this paper is as follows. In Sec. II, we

briefly review the ordinary BE for thermal quarks andgluons. Next we introduce our numerical framework inSec. III. We first discuss how to get stable numerical resultsin collision integral in Sec. III A and then introduce themethod to keep the particle numbers conserved in Sec III B.We will present our numerical results in Sec. IV. In Sec. IVA and IV B, we test the stability of the collisional integralsand the particle number conservation. We discuss the totalenergy conservation in Sec. IV C. With some physicalinitial conditions, we show the time evolution of the systemin both coordinate and momentum spaces in Secs. IV D andIV E, respectively. The summary of our paper will bepresented in Sec. V.Throughout this work, we choose the metric gμν ¼

diagfþ;−;−;−g and the space and momentum fourvectors as xμ ¼ ðt;xÞ and pμ ¼ ðEp;pÞ. For a momentumkμa, we choose μ ¼ ðt; x; y; zÞ for the space-time index anda ¼ 1, 2, 3 for u, d, s flavors.

II. BOLTZMANN EQUATION FOR QUARKGLUON MATTER

In this section, we will briefly review the ordinary BE forthe thermal quarks and gluons in the leading-log order.More details can be found in our previous systematicstudies in Refs. [10,102].The relativistic BE, which is an effective theory for

relativistic many-body systems, describes the evolution ofsystem in the phase space. The general expression for theBE reads,

ddt

fpðt;x;pÞ≡ ∂∂t fp þ

∂x∂t ·∇xfp þ

∂p∂t · ∇pfp ¼ C½fp�;

ð1Þ

where fpðt;x;pÞ is the distribution function and C½f� iscalled the collision term. The ∂x=∂t and ∂p=∂t are theeffective velocity and effective force for the particles,

respectively. One can derive the ∂x=∂t and ∂p=∂t fromthe equation of motion of the action for a single particle. Inour study, the action in the classical level, i.e., up to theorder of ℏ0, is S ¼ R

dtðp · dxdt − EpÞ, with Ep being theparticle’s energy. For simplicity, we neglect the particle’sphysical mass, but the nonvanishing thermal mass mðxÞdepends on the space-time in general. Accordingly, theparticle’s energy EpðxÞ≡

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þm2ðxÞ

pdepends on the

space-time.For thermal quark-gluon matter, the BE has the follow-

ing general structure [9,10,103]:

∂fapðt;x;pÞ∂t þ p

Eap·∇xfapðt;x;pÞ

−∇xEap · ∇pfapðt;x;pÞ ¼ Ca; ð2Þ

where fapðt;x;pÞ denotes the color and spin averageddistribution function for particle a, and a ¼ q; q; g standsfor quarks, antiquarks, and gluons. Ea

pðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þm2

aðxÞp

and Ca are the energy and collision term for particle a,respectively. −∇xEa

p is an effective force, which comesfrom the equation of motion of ∂p=∂t [9,10,103].In the present work, we only consider the 2 → 2

scatterings. The collision term for a quark of flavor acan be obtained,

NqCqa ¼1

2Cqaqa↔qaqa þ Cqaqa↔qaqa þ

1

2Cgg↔qaqa

þ Cqag↔qag þXb;b≠a

ðCqaqb↔qaqb

þ Cqaqb↔qaqb þ Cqbqb↔qaqaÞ; ð3Þ

where Nq ¼ 2 × 3 ¼ 6 is the quark helicity and colordegeneracy factor and the factor 1=2 is included whenthe initial state is composed of two identical particles. For agluon, the collision term reads

NgCg ¼1

2Cgg↔gg þ

Xa

ðCgqa↔gqa þ Cgqa↔gqa þ Cqaqa↔ggÞ;

ð4Þ

where Ng ¼ 2 × 8 ¼ 16 is the gluon helicity and colordegeneracy factor.The collision term for 2 → 2 scatterings, aðk1Þ þ

bðk2Þ → cðk3Þ þ dðpÞ, has the following generalexpression,

Cab→cd ≡Z Y3

i¼1

d3kið2πÞ32Eki

×jMab↔cdj2

2Ep

× ð2πÞ4δð4Þðk1 þ k2 − k3 − pÞ× ½fak1fbk2Fc

k3Fdp − Fa

k1Fbk2fck3f

dp�; ð5Þ

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where Fgp ¼ 1þ fgp, F

qðqÞp ¼ 1 − fqðqÞp and Mab→cd is the

matrix element in which all colors and helicities of theinitial and final states are summed over. We summarizeall 2 → 2 scattering matrix elements in Table I. In ournumerical calculation, the tree-level matrix elements forall 2 ↔ 2 scattering processes are set as the defaultconfiguration. We also provide an application program-ming interface (API) for users to define their own matrixelements for some specific purposes.When the system reaches the global thermal equilibrium,

the distribution functions should satisfy the ordinary Bose-Einstein or Fermi-Dirac distributions,

fgp ¼ 1

eðEp−μgÞ=T − 1; ð6Þ

fqðqÞp ¼ 1

eðEp∓μqÞ=T þ 1; ð7Þ

where T is the temperature and μa is the chemical potentialfor particle a.The thermal masses of gluon and quark (anti-quark) are

usually written as [10,103],

m2g ¼

2g2

dA

Zd3p

ð2πÞ32Ep

�NgCAf

gp þ

XNf

i¼1

NqiCFðfqip þ fqip Þ�;

ð8Þ

m2qi ¼ m2

qiðxÞ ¼ 2CFg2Z

d3pð2πÞ32Ep

ð2fgp þ fqp þ fqpÞ;

ð9Þ

where Nf is taken to be 3 a we only consider qi ¼ u, d, squarks and their antiparticles. Note that, when adding thecontribution of 1 → 2 scattering, the thermal mass can beintroduced in a systematic way, and the invariantmomentum differential piece d3p=½ð2πÞ3Ep� could bereplaced by d3p=½ð2πÞ3jpj� in all momentum integrals,see, e.g., in [9,51,52]. In the first time step of evolution,since we have no prior information of particle masses, wewill use Ep ¼ jpj to perform the calculation in Eqs. (8)and (9). In later time steps, we will use the normalEapðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þm2

aðxÞp

. This iteration approach is rea-sonable since the difference jEp − pj from nonzeromasses is of higher order [10].In our calculations, we need to check the total particle

number conservation. The particle number is defined as:

NðgÞ ¼Z

d3xd3pð2πÞ3 fgNg

NðqiÞ ¼Z

d3xd3pð2πÞ3 f

qiNqi : ð10Þ

Note that, in general, the total number for each type ofparticles (e.g., gluons, quarks, and antiquarks) is notconserved due to the strong interaction. However, sincewe only consider 2 → 2 scatterings in this work, the total

particle number Ng þPNf

i¼1ðNqi þ NqiÞ is conserved. Wewill check and confirm the total particle number conserva-tion at each time step of our numerical simulations.Since the thermal masses of particles depend on the space

and time, the ordinary kinetic energy-momentum tensor,

TμνkinðxÞ ¼

Xa

Zd3p

ð2πÞ3EapNapμpνfapðxÞ; ð11Þ

is not conserved [103]. Instead, we have:

TABLE I. Matrix elements squared for all 2 → 2 partonscattering processes in QCD. The helicities and colors of allinitial and final state particles are summed over. q1 (q1) and q2(q2) represent quarks (antiquarks) of different flavors, and grepresents the gluon. dF and dA denote the dimensions of thefundamental and adjoint representations of SUcðNÞ gauge groupwhile CF and CA are the corresponding quadratic Casimirs. Thes,u,t are Mandelstam variables. In a SUcð3Þ theory withfundamental representation fermions, dF ¼ CA ¼ 3, CF ¼ 4=3,and dA ¼ 8. The infrared divergence is suppressed by introducinga regulator in the denominator [9,10,104].

ab → cd jMaðk1Þbðk2Þ→cðk3ÞdðpÞj2

q1q2 → q1q2q1q2 → q1q2q1q2 → q1q2q1q2 → q1q2

8g4 d2FC2F

dA½ s2þu2

ðt−m2gÞ2�

q1q1 → q1q1q1q1 → q1q1

8g4 d2FC2F

dA

hs2þu2

ðt−m2gÞ2 þ

s2þt2

ðu−m2gÞ2i

þ16g4dFCFðCF − CA2Þ s2

ðt−m2gÞðu−m2

gÞ

q1q1 → q1q1 8g4 d2FC2F

dA

hs2þu2

ðt−m2gÞ2 þ

u2þt2

s2

iþ16g4dFCFðCF − CA

2Þ u2

ðt−m2gÞs

q1q1 → q2q2 8g4 d2FC2F

dAt2þu2

s2

q1q1 → gg 8g4dFC2F

hu

ðt−m2gÞ þ

tðu−m2

gÞi

−8g4dFCFCAðt2þu2

s2 Þq1g → q1gq1g → q1g

−8g4dFC2F½us þ s

ðu−m2gÞi

þ8g4dFCFCA½ s2þu2

ðt−m2gÞ2�

gg → gg 16g4dAC2A½3 − su

ðt−m2gÞ2 −

stðu−m2

gÞ2 −tus2�

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∂μTμνkinðxÞ ¼ SνexðxÞ ¼

1

2

Xa

∂νm2a

Zd3p

ð2πÞ3EapNafap; ð12Þ

where SνexðxÞ is a source term due to the mass variations.As we have shown, the total energy is not conserved if

the thermal mass is space dependent. It is because thegradient of thermal mass is an effective force in Eq. (2). It isalso shown in the pioneer work from the thermal fieldtheory done by Ref. [103]. Usually, for simplicity, one canalso set the thermal mass be homogeneous in space. Then,the total energy becomes conserved. Here, in our work, wewill consider the dynamical thermal mass.

III. NUMERICAL FRAMEWORK FOR SOLVINGBOLTZMANN EQUATION

In order to keep the total particle number conserved, wefirst rewrite the BE in Eq. (2) as follows,

∂fapðxÞ∂t þ∇x ·

�pEapfapðxÞ

�−∇p · ½ð∇xEa

pÞfapðxÞ� ¼ Ca½f�;

ð13Þ

where we have used the following identity,

−�∇x ·

pEap

�þ ½∇p · ð∇xEa

pÞ� ¼ 0: ð14Þ

The form of Eq. (13) ensures the conservation of totalparticle numbers when periodical boundary conditions areapplied in phase space. Then using central difference, wecan express the left-hand side of Eq. (13) into discrete formas follows,

fapðxþ ΔtÞ − fapðxÞ△t

þX

i¼1;2;3

�1

2△xi

�pifapðxi þ△xiÞEapðxi þ△xiÞ

−pifapðxi −△xiÞEapðxi −△xiÞ

�

−1

2△pi

�fapiþ△piðxÞ

2△xiðEa

piþ△piðxi þ△xiÞ − Ea

piþ△piðxi −△xiÞÞ

−fapi−△pi

ðxÞ2△xi

ðEapi−△pi

ðxi þ△xiÞ − Eapi−△pi

ðxi −△xiÞÞ��

: ð15Þ

A. The δ-function and the collision term

Now we look at the collision term in the right-hand sideof Eq. (2) or Eq. (13). Usually, one can integrate over themomentum d3ki with the δ-function,

δð4Þðk1 þ k2 − k3 − pÞ¼ δð3Þðk1 þ k2 − k3 − pÞδðE1 þ E2 − E3 − EpÞ; ð16Þwhich can reduce the number of integral variables. Here, wehave two choices: either expressing k2 by k3 þ p − k1 orexpressingk3 byk1 þ k2 − p. These twochoices can lead todifferent numerical behaviors. In this work, we choose thefirst choicewhichwillmake our numerical integrationsmorestable than the second one (as we will show later).With the help of the δ-function, we can integrate over

d3k2 and obtain,Z Y3i¼1

d3ki

ð2πÞ32Eki

δð4Þðk1 þ k2 − k3 − pÞ

¼Z

d3k1

ð2πÞ32E1

d3k3

ð2πÞ32E3

1

ð2πÞ32E2

δðE1 þ E2 − E3 − EpÞ

¼ 1

ð2πÞ9Z

d3k3dkx1dky1

2E12E22E3

Xi¼�

1

jJðki1zÞj; ð17Þ

where we have used

δðE1 þ E2 − E3 − EpÞ ¼Xi¼�

1

jJðki1zÞjδðk1z − ki1zÞ; ð18Þ

with

Jðk�1zÞ ¼k�1zE1

−−k�1z þ k3z þpz

E2

;

k�1z ¼ Root½E1 þE2 −E3 −Ep ¼ 0�: ð19ÞThere are two roots for k1z from the equation E1 þ E2−E3 − Ep ¼ 0, and k1z has the form of k�1z ≡ A� ffiffiffi

HpB , where

A, B, H are functions of kx1; ky1 and k3.

Substituting Eq. (17) into Eq. (5), we obtain the collisionterm, which consists of a 5-dimensional integration andmay be calculated numerically by using the direct MCmethod on GPU. With the help of the packages Ray andNumba, we can solve the Boltzmann equation (2) for all2 ↔ 2 scattering processes on the distributed GPU clusters.Before we present our numerical results, we would like

to discuss a little more about the difference betweenintegrating over k2 or k3 when we use the δ-function.The difference comes from solving k1z from the equationE1 þ E2 ¼ E3 þ Ep. In our first choice in Eq. (17), k1z is afunction ofE3 þ Ep. SinceE3 þ Ep is always positive for allk3 and p, the integration of the collision term in Eq. (2) isstable. If we integral out k3 in the δ-function, then k1z will

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become a function of E2 − Ep, which could flip its signwhen we change k2 and p. Therefore, the integral in Eq. (2)using the second setup is not as stable as using the first setup.We will discuss more on the stability of collision term inSec. IVA.

B. Particle number conservationand symmetrical sampling

Since we use the direct MC method to compute thecollision integral, the total particle numbers are not strictlyconserved in each time step due to the randomness of MCsampling. Such nonconservation of particle numbers canaccumulate with time and may affect the result at later timesteps. To ensure a strict particle number conservation, weintroduce a method named “symmetrical sampling”on GPUs. Here we use the process of gluon scatteringgðk1Þ þ gðk2Þ → gðk3Þ þ gðpÞ as an example to illustratethe basic idea of our “symmetrical sampling” method.To calculate the collision term Cgg→ggðx;pÞ in Eq. (5), weneed to sample a series values of the integration vari-ables ðk1x; k1y; k3x; k3y; k3zÞ. The collision term can bewritten as,

Cgðk1Þþgðk2Þ→gðk3ÞþgðpÞðx;pÞ

¼Z

cgpðk1x; k1y; k3x; k3y; k3zÞdk1xdk1ydk3xdk3ydk3z

≃Vdomain

N

XNsi¼1

cgpðsample siÞ; ð20Þ

where Vdomain is the volume of the integration domain, andthe kernel cp denotes,

cpðk1x; k1y; k3x; k3y; k3zÞ¼ ½fak1fbk2Fc

k3Fdp − Fa

k1Fbk2fck3f

dp� × sym

×Xi¼1;2

1

jJaðki1zÞj1

ð2πÞ52E12E22E32EpjMab↔cdj2; ð21Þ

where the symmetry factor sym ¼ 1=2 when the initialstate is composed of two identical particles, and other-wise equals 1. Given each of ðk1x; k1y; k3x; k3y; k3zÞ andðpx; py; pzÞ, we can obtain the corresponding ðk1z; k2x;k2y; k2zÞ and cpðk1x; k1y; k3x; k3y; k3zÞ.Let us consider one specific sample cgpðsample siÞ in

Eq. (20) (also see Fig. 1). In the usual MC sampling,one will only add the contribution of cgpðsample siÞ toCk1þk2→k3þpðx;pÞ. This sample will not influence thevalues of Ck3þp→k1þk2

ðx;k1Þ, Ck3þp→k1þk2ðx;k2Þ and

Ck1þk2→k3þpðx;k3Þ, for which one will compute theircorresponding cgk1

, cgk2and cgk3

separately. Due to suchindependence of Cgg→ggðx;pÞ at each grid, one cannotachieve a strict particle number conservation in the MCapproach.To fix the issue of the particle number nonconservation,

we reuse the value of cgpðsample siÞ. Actually, for a givenset of p, k1, k2, k3 satisfying k1 þ k2 ¼ k3 þ p, thekernels cgp, c

gk1, cgk2

and cgk3in different collisional integrals

are related to each other due to the symmetry in thescattering amplitude jMab↔cdj2

FIG. 1. Illustration of the “symmetrical sampling” method. The sample cgpðsample siÞ will be used by four integrations. Thissymmetrical reuse of samples leads to a conserved particle number. We call this trick of using sample cgpðsample siÞ for all fourmomentum grids as “symmetrical sampling.” The sample points have been quadrupled.

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cgp ¼ cgk3¼ −cgk2

¼ −cgk1 ;

for givenp;k1;k2;k3 and k1 þ k2 ¼ k3 þ p: ð22Þ

The above relation can be easily seen from Eq. (21). Ifwe switch particle 1 and p in Eq. (21), we only changethe sign of the term ½fak1fbk2Fc

k3Fdp − Fa

k1Fbk2fck3f

dp�.

Therefore, in our program, we will also use the value ofcgpðsample siÞ for Ck3þp→k1þk2

ðx;k1Þ, Ck3þp→k1þk2ðx;k2Þ

and Ck1þk2→k3þpðx;k3Þ as well. We call this trick, to usethe sample cgpðsample siÞ for all four momentum grids, the“symmetrical sampling” method. This means that thesample points have been quadrupled.With our “symmetrical sampling” trick, we can avoid the

errors from the related samples in the collisional integrals atdifferent momentum grids. Accordingly, we can obtain astrict particle number conservation. In principle, one canapply this method in the direct MC sampling based on CPUapproaches. However, it usually takes lots of computingtime and is hard to implement. Fortunately, with the helpof the feature in GPUs, named CUDA atomic operation[98,99], the extra time for implementing symmetricalsampling is almost negligible. As explained by the officialdocuments [105], “Atomic operations are operations whichare performed without interference from any other threads.Atomic operations are often used to prevent race condi-tions, which are common problems in multithreadedapplications.” In our case, each value of the array cgp,whose element represents a specific cgp value at momenta p,is saved in the global GPU memory. During the process ofparallel evaluations, at the same momenta p, we obtain“simultaneously”many values for cgp from different threads[we will also obtain the value of cgp from cgk3

;−cgk2;−cgk1

asdiscussed in Eq. (22)]. Since the accumulation of thesevalues can only be performed sequentially, when one valueof cgp in a GPU thread is calculated and being accumulatedto this global memory array cgp, all the other threads do nothave the access of cgp at p. These processes in GPUs refer toCUDA “atomic operation.” Compared with parallel CPUmanipulation, CUDA is extremely fast at performing thisatomic operation, which enables a fast “symmetricalsampling.” Similar strategy has also been used in CPUsimplementation to ensure particle number conservation,e.g., see Refs. [51,52,74–76].We note that while the condition k1 þ k2 ¼ k3 þ p

ensures that the integration always preserves energy andmomentum conservation in all microscopic processes, thetotal energy as computed via Eq. (11) might not be strictlyconserved due to the discrete grids. To explain the possiblenonconservation of the total energy, we can take a closelook at Eq. (13). By integrating both sides of Eq. (13) overd3xd3p=ð2πÞ3 and using the definition of total particlenumber in Eq. (10), we obtain the time variation of particlenumber

ddt

N ¼ Na

Zd3x

d3pð2πÞ3 C½f�; ð23Þ

where a ¼ g; qðqÞ. Our symmetrical sampling method inthe collisional integral C½f� guarantees the time reversalsymmetry in all microscopic processes, e.g., in Eq. (22).Such time reversal symmetry in collisional integral C½f�makes the integral

R d3pð2πÞ3 C½f� exactly zero numerically

[106,107], which ensures the strict particle numberconservation.Similarly, by integrating Eq. (13) over d3xd3p=ð2πÞ3

with the multiplication of p0 on both sides and using thedefinition of energy-momentum tensor in Eq. (11), we getthe time variation of total energy,

ddt

T00 ¼ Na

Zd3x

d3pð2πÞ3 p

0C½f� þZ

d3xS0ex; ð24Þ

where Sμex is the source term in Eq. (12). For simplicity,let us assume the mass is constant, then the source termvanishes. Although the time reversal symmetry still holds,the errors could come from the discrete grids for p0.

Therefore, eventually, the integralR d3p

ð2πÞ3 p0C½f� is not

strictly zero numerically. On the other hand, from theabove analysis, we could expect that the errors for thenonconservation of total energy will decrease if we increasethe number of grids. We will address this point in moredetail in Sec. IV C.

IV. TEST OF PROGRAM AND TIME EVOLUTIONIN COORDINATE AND MOMENTUM SPACE

In this section, we will first test several aspects of ourprogram, and then show the time evolution of the distri-bution in both coordinate and momentum spaces. Thestability of the collision integrals will be tested in Sec. IVA.The check of particle number conservation will be pre-sented in Sec. IV B, and our results show that the particlenumber is strictly conserved. The total energy conservationis checked in Sec. IV C. It is found that even though thetotal energy is not strictly conserved, the numerical errorswill decrease very fast with increasing the number of thegrid. Finally, we will present the time evolution of thesystems in both coordinate and momentum spaces for puregluons, pure quarks, and the mixture of quarks and gluonsin Secs. IV D and IV E. As is expected, the system tends tobe homogenous in coordinate space and become thermal-ized in momentum space. We also find indications of gluoncondensation in the soft region.

A. Test of the stability of the collision integrals

As mentioned in Sec. III, there are two ways to integrateover the δ-function in the collision term. Differentapproaches of handling the δ-function can affect the

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numerical stability of the collision term. Here we use gluon-gluon scatterings to illustrate such difference.In Fig. 2, we show the results from two approaches using

the same initial distribution for gluons. Each data pointrepresents the numbers of the gluons associated with thedistribution function fgðt0 þ dt;x;pÞ where t0 ¼ 0 fm anddt ¼ 0.01 fm. The blue triangle points, labeled as “morestable,” stand for particle numbers obtained by using k2 ¼k3 þ p − k1. The green circle points, labeled as “lessstable,” denote the results by using k3 ¼ k1 þ k2 − p.We can see that the green circle points spread over arelatively larger area than the blue triangle ones, whichmeans that the errors in the “less stable” method arerelatively larger than the “more stable” ones. Therefore,using k2 ¼ k3 þ p − k1 to integrate over dk2 is a morestable method, consistent with our previous argument inSec. III.

B. Test of particle number conservation

In Fig. 3, we show the particle numbers for differentparton species. For simplicity, we label the results obtainedby our symmetrical sampling method in Sec. III B as withsymmetrical sampling, while the results from direct MCsimulations are labelled as without symmetrical sampling.In the figures, the blue triangle and green circle pointsstand for the cases with and without symmetrical sampling,respectively. In the upper panel, we show the gluon num-bers for cases of pure gluon scattering and quark-gluon

FIG. 2. Test for the stability of two approaches dealing with thedelta function. Each data point represents the gluon numbers forthe distribution function fgðt0 þ dt;x;pÞ, where t0 ¼ 0 fm anddt ¼ 0.01 fm. The horizontal axis denotes the ith independentevaluation of the evolution from t0 to t0 þ dt. The initial distri-bution function fgðt0;x;pÞ, drawn from random values between[0,1], is kept the same for all data points in the figure. The phasespace box is of size ½−3 fm; 3 fm�3 × ½−2 GeV; 2 GeV�3. Wetake 40 grids for each momentum freedom, the spatial grid is setto 1, and αs ¼ 0.3. For each integration point Cgðx;pÞ, we havesampled 100 points to perform a direct MC integration. Thematrix element for gluon-gluon scattering is in Table I. The bluetriangle and green circle points stand for the results of integratingover k2 and k3 in Eq. (16), respectively.

(a)

(b)

(c)

(d)

FIG. 3. Number of particles for the cases with and without“symmetrical sampling.” Each data point represents the particlenumbers for the distribution function faðt0 þ dt; x;pÞ usingthe same initial configuration as in Fig. 2. The horizontal axisdenotes the ith independent evaluation of the evolution from t0 tot0 þ dt. The blue triangle and green circle points denote theresults with and without symmetrical sampling, respectively.The panel (a) is for the pure gluon case and the panels (b), (c),and (d) are for the cases including the scatterings of quarks andgluons.

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scattering in Fig. 3(a) and Fig. 3(b), respectively. We cansee that for pure gluon case, the gluon number is conservedwhen using our symmetrical sampling method. In Fig. 3(b),since the gluons can be converted to quarks and antiquarks,there are some variations of the gluon numbers. Figure 3(d)shows that there are also variations for the numbers ofquarks and antiquarks. However, as shown in Fig. 3(c), thetotal number of particles, including quarks, antiquarks andgluons is conserved for 2 → 2 scatterings when using oursymmetrical sampling method.In Fig. 4, we show the numbers of gluons and quarks

(antiquarks) as a function of the evolution time. Since weinitialize the system with all gluons, we find that gluonstend to convert into quarks and antiquarks during theevolution, see in Fig. 4(a). After some time, both gluonand quark numbers tend to achieve equilibrium. While theindividual parton numbers are changing with time, the totalparticle number is strictly conserved during the evolutionwhen our symmetrical sampling method is employed.Without the symmetrical sampling, the conservation of

the total particle number can be violated by a small amount(about 0.03%) as shown in Fig. 4(b).

C. Test of energy conservation

In implementing the MC integration of the collisionterm, we can use the symmetrical sampling method toensure the strict particle number conservation. However,the conservation of total particle number does not guaranteethe strict conservation of total energy numerically due tothe discrete grids in the numerical calculation. When wecalculate the total energy with Eq. (11), the smooth p0 isapproximated by discrete grids. This discretization willaffect the numerical evaluation of the total energy. Sucheffect can be seen in Fig. 5, where a constant gluon massm2

g ¼ 0.5 GeV2 is used. To quantify the violation of energyconservation, here we introduce the following quantityδErelative,

δErelative ¼jEt¼1 fm − Et¼0 fmj

Et¼0 fm: ð25Þ

In Fig. 5, we plot δErelative as a function of the number ofmomentum grids. In this test, we choose the initial gluondistribution as a step function, fg;initialðpÞ ¼ 0.5 × θð1 −jpj=QsÞj with Qs ¼ 1.5 GeV. We find that the deviationδErelative decreases fast as a function of the number of thegrids. From the figure, we can see that when the number ofgrids in momentum space is taken as 30, δErelative ∼ 0.015,i.e., the fluctuation of total energy computed from Eq. (11)is about 1.5% at t ¼ 1 fm=c. Such small deviation ismainly caused by the discretization of momentum. Ofcourse, the total energy is still conserved physically for thiscase since we have used the condition k1 þ k2 ¼ k3 þ p inthe computation of the collision term.

(a)

(b)

FIG. 4. Number of particles of different species as a function ofevolution time for the cases with and without symmetricalsampling. The parameters for fg=qðqÞðt; x;pÞ are the same asin Figs. 2 and 3. Panel (a) plots the particle numbers for bothfermions and gluons using the symmetrical sampling. Panel(b) compares the total particle number as a function of evolutiontime with and without using the symmetrical sampling, as shownby the blue and green lines, respectively. On one Nvidia TeslaV100 card, the entire evaluations take 6456 and 8198 seconds forthe cases without and with the symmetrical sampling.

FIG. 5. The deviation of the total energy as a function of thenumber of momentum grids, for a pure gluon system. Here, thenumber for spatial grid is taken to be one, αs ¼ 0.3, the phasespace box is of size ½−3 fm; 3 fm�3 × ½−2 GeV; 2 GeV�3 and thetime step dt ¼ 0.0001 fm. The initial gluon distribution is chosenas a step function, fg;initialðpÞ ¼ 0.5 × θð1 − jpj=QsÞj with Qs ¼1.5 GeV and m2

g ¼ 0.5 GeV.

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Now we consider the case of dynamical mass ascomputed by Eqs. (8), (9). For simplicity, we focus onthe systems of pure gluons or u quarks that are homogenousin coordinate space. Then Eq. (12) reduces to,

∂0T00kin ¼ S0ex ¼

1

2∂0m2

a

Zd3p

ð2πÞ3EapNafap: ð26Þ

Here the term ∂0T00kinðxÞ is evaluated directly via

½T00ðtþ dtÞ − T00ðt − dtÞ�=ð2dtÞ. In Fig. 6, we show thetime variation of the kinetic energy ∂0T00

kin and the zerocomponent of source term S0ex. We can see that except forthe first few steps, two terms (∂0T00

kin and S0ex) are almostidentical. For gluons as shown by 6(a), the value of∂0T00

kinðxÞ oscillates drastically for the first few time steps,but after t ¼ 20 fm, it slightly fluctuates around the zerovalue, which indicates the reach of nearly thermal equili-bration. This result demonstrates that the change of the totalenergy comes from the source term.The above consistency check in Fig. 6 means that our

results satisfy Eq. (26) automatically. But this does notmean that the total energy is physically conserved since the

masses are dynamically changing with space and time. Inprinciple, for a single flavor case, one can rewrite the right-hand side of Eq. (12) as a total derivative term, then define amodified conserved total energy-momentum tensor asfollows [103],

Tμνtotal ¼ Tμν

kin −1

4m2

a

Zd3p

ð2πÞ3EapNafap; ð27Þ

where the definition of squared mass in Eq. (8), (9) is usedto derive the second term. For a multiple flavor case,one usually cannot get a modified conserved energy-momentum tensor.For the case of dynamical mass, there may be another

source of numerical error originating from the definitionof mass in Eqs. (8), (9), apart from the discrete grids. Toobtain the squared mass, we need to integrate over p withEp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 þm2

p, which depends on the dynamical mass.

At the first step of the time evolution, we use jpj instead ofEp to derive the dynamic mass. This tiny difference may alsobe a source of numerical errors. Despite the above mentionednumerical errors, the conservation of the total energycomputed via Eq. (11) can be archived up to 99.8% in Fig. 6.

D. Evolution of distribution functions and dynamicalmasses in coordinate space

When the spatial part of the BE is taken into account, theevolution of the phase space distributions becomes morecomplicated. Since the dynamical masses now also dependon the spatial grids as in Eqs. (8) and (9), all differentialterms in Eq. (13) contribute to the evolution. The initialconditions for the distribution function must be physical,e.g., the fermion distribution functions should be smallerthan unity, and all phase distributions should be positivedefinite. Here for simplicity, we set the initial gluondistribution as follows:

fg;initialðx;pÞ ¼�fg;max − fg;min

2jxx;maxj�ðxþ jxx;maxjÞ

þ�−fg;max − fg;min

2jpx;maxj�ðpx þ jpx;maxjÞ

þ 2fg;min; ð28Þwhere fg;max and fg;min are two parameters, which stand forthe maximum and minimum values of the distribution.2jpx;maxj is the size of the momentum box in the x direction,and 2jxx;maxj is the size of the spatial box in the x direction.One can also choose other types of initial conditions, andthe main results will be similar. The distribution functionfgðx;pÞ in Eq. (28) linearly increases in the x direction andlinearly decreases in the px direction. For simplicity, we setthe initial quark distribution to be zero,

fqðqÞ;initial ¼ 0: ð29Þ

(a)

(b)

FIG. 6. Time variation of the kinetic energy ∂0T00kin and the zero

component of source term S0ex, as a function of evolution time.Panels (a) and (b) show the results for pure gluon and quarksystems, respectively. The initial distribution function is given byfg=u;initialðpÞ ¼ 0.5 × θð1 − jpj=QsÞj with Qs ¼ 1.5 GeV. We setαs ¼ 0.3, dt ¼ 0.00005 fm for gluons and dt ¼ 0.001 fm forquarks.

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In Fig. 7, we show the gluon and u-quark distributions asa function of the spatial directions x and y at differentevolution times. Initially, the gluon distribution is linear in

spatial x and u quark distribution is vanishing. As the timeevolves, the gluons tend to convert to quarks, similar toFig. 4. In the end, the distributions of both u quarks andgluons become approximately uniform in all spatial direc-tions (Here we show the distribution functions in the x andy-directions). Other fermions have a similar pattern as well.In Fig. 8, we show the dynamical masses for gluons and

u quarks in the spatial x and y-directions at differentevolution times. Initially, both masses squared m2

g and m2u

increase linearly with x. Then as time evolves, they tend tobecome homogenous in spatial x direction. The distribu-tions for other flavors of quarks and antiquarks are similar.

(a)

(b)

(c)

(d)

FIG. 7. The gluon and u-quark distributions in spatial x and ydirections at different times. The phase space grid is taken as½nx; ny; nz; npx; npy; npz� ¼ ½10; 10; 1; 10; 10; 1�, with n beingthe number of grids. We have also chosen the coupling constantαs ¼ 0.3, phase space box is of size ½−3 fm;3 fm�3× ½−2GeV;2GeV�3 and dt ¼ 0.0005 fm. The initial gluon distribution isgiven by Eq. (28) with jpx;maxj ¼ 2 GeV, jxx;maxj ¼ 3 fm,fg;max ¼ 0.2 and fg;min ¼ 0.1. At each x or y, we plot fg orfu which is averaged in ðy; z;pÞ or ðx; z;pÞ, respectively. Thecalculation takes 3312 seconds on one Nvidia Tesla V100 card.

(a)

(b)

(c)

(d)

FIG. 8. Distribution of mass squaredm2g andm2

u in spatial x andy directions at different times. The parameters are chosen to bethe same as in Fig 7.

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In the future, we will extend our discussion here. It isinteresting to study the evolution of the system from a faraway equilibrium and to learn how the system reaches thelocal equilibrium before the global one.

E. Evolution of distribution functions in momentumspace and gluon condensation

Now we show the evolution of systems in momentumspace. For simplicity, we neglect the spatial dependenceand set the systems to be homogenous in coordinate space.As a first attempt, we investigate the time evolution of puregluon or pure quark systems in momentum space. Theinitial gluon distribution is chosen to be a step function,

fgðpÞ ¼ fg;0θ

�1 −

jpjQs

�; ð30Þ

where f0 and Qs are parameters. This initial condition is tomimic the distribution from color glass condensation(CGC) [22], with Qs being the saturation scale. Theevolution of the distribution function for a pure gluonsystem is shown in Fig. 9(a) and (c). Note that for gluonevolution, the time dt is set very small (in our casedt ¼ 0.00005 fm) to ensure that the distribution functionsare positive. Similarly for a pure quark system, we choosethe initial quark distribution function as,

fqðpÞ ¼ fq;0θ

�1 −

jpjQs

�: ð31Þ

The time evolution of the pure u quark distributionfunction is shown in Figs. 9(b) and 9(d). For pure quarksystem, the time step dt can be relative larger (in our casedt ¼ 0.001 fm).From Fig. 9, we find that the thermalization of gluons is

quite different from that of quarks. The gluons will firstaccumulate in the soft region, where the energy is smallerthan 1.0 GeV. This phenomenon may indicate the gluoncondensation. While for quarks, we have not observed suchphenomenon. Since the system will reach to the thermalequilibrium at the end, the distribution function in lowmomentum region then turns back and finally matches tothermal one. The similar behavior has also been discussedin a simplified scalar field model in Ref. [108]. The gluoncondensation may be of importance to the prethermaliza-tion of quark-gluon plasma created in the heavy-ioncollisions. It is impossible to understand the full evolutionof the system in the presence of gluon condensation in thiswork. We will investigate such issue in our future studiesbased on our program. It is noted that at the thermalequilibrium, the gluon chemical potential is negative forpure gluon system while the quark chemical potential ispositive for pure quark system in Fig. 9.

FIG. 9. Time evolution of the parton distribution functionsfrom initial step functions to thermal distributions. Panels (a) and(c) show the results for pure gluons while panels (b) and (d)for pure quarks. The phase space grid is taken as ½nx; ny;nz; npx; npy; npz� ¼ ½1; 1; 1; 30; 30; 30�. The coupling constantαs ¼ 0.3, the phase space box is of size ½−3 fm; 3 fm�3×½−2 GeV; 2 GeV�3, with f0 ¼ 0.5 and Qs ¼ 1.5 GeV. For puregluons, the time step is taken as dt ¼ 0.00005 fm and for purequarks dt ¼ 0.001 fm. On one Nvidia Tesla V100 card, theevaluation from 0 fm to 50 fm takes 60 hours for pure gluon caseand 2 hours for pure fermions case from 0 fm to 20 fm.

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In Fig. 10, we show the evolution of the dynamicalmasses squared for gluons and quarks. For a pure gluonsystem, the gluon’s thermal mass decreases with time andeventually becomes stable as the system reaches thermalequilibrium. For a pure quark system, the quark’s thermalmass oscillates with time, but the averaged value tends to beconstant.We now extend our simulation to a system composed of

both quarks and gluons. The initial condition for gluons is setthe same in Eq. (30), and for quarks we choose fqðpÞ ¼ 0.In Fig. 11, we show the distribution functions of gluons andu quarks as a function of parton energy at different evolutiontimes. For gluons, the distribution function in the soft region(≲1.0 GeV) increases very fast at the beginning and thendecreases to the thermal equilibration. The accumulation ofgluons in the soft region implies possible gluon condensa-tion. For quarks, the distribution function reaches the Fermi-Dirac distribution gradually. We also notice that the chemicalpotentials for gluons and quarks are both negative in thethermal equilibrium. Compared with the result in Figs. 9(b)and 9(d), the negative chemical potential for quarks mightcome from different initial condition and their interactionwith gluons.

(a)

(b)

FIG. 10. Time evolution of masses squared for pure gluons andpure quarks from initial step functions to thermal distributions.Panels (a) and (b) show the results for gluons and quarks, res-pectively. The parameters are chosen to be the same as in Fig. 9.

FIG. 11. Time evolution of the parton distribution functionsfor quark-gluon systems. The initial condition is fqðpÞ ¼ 0 forquarks and fgðpÞ ¼ fg;0θð1 − jpj=QsÞ for gluons, with f0 ¼ 0.5and Qs ¼ 1.5 GeV. The phase space grid is taken as ½nx; ny;nz; npx; npy; npz� ¼ ½1; 1; 1; 30; 30; 30�. The coupling αs ¼ 0.3,the phase space box is of size ½−3 fm;3 fm�3× ½−2GeV;2GeV�3,and the time step is taken as dt ¼ 0.00005 fm. On one NvidiaTesla V100 card, the evaluation from 0 fm to 50 fm takesaround 2 days.

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In Fig. 12, we show the dynamical masses squared m2g;u

and the particle numbers as a function of the evolution time.It is interesting that there is no oscillation behavior for m2

u

here, as opposed to the results in Fig. 10. Instead, m2u

decreases with time and then reaches to a constant,similarly to m2

g. In Fig. 12(b), we also confirm that thetotal particle number is strictly conserved. The gluonsconvert to quarks in a very short time (≲10 fm), and thenboth gluon and quark numbers tend to be constant. We alsonote that the number of gluons is larger than the one offermions. It is caused by the initial conditions. Since thereis only one grid in space direction, i.e., the thermal mass ishomogeneous in space, both the total energy in Eq. (12)and particle number are conserved. We can solve thetemperature and chemical potential in the equilibrium fromthe initial conditions. Then, we have checked that in theequilibrium, the number of gluons are larger than the one offermions.

V. CONCLUSION AND DISCUSSION

In this work, we have developed a new numericalframework for obtaining the full solutions of relativisticBE on GPUs. Our main equation, i.e., the complete relati-vistic BE, is of form Eq. (2). We have considered thethermal systems of 3 flavor quarks, antiquarks and gluons.For simplicity, we only consider 2 → 2 scattering proc-esses, in which the total particle numbers are conserved.

Since the quarks and gluons have dynamical masses inEqs. (8), (9), there is an external force in Eq. (2). Also thekinetic energy-momentum tensor in Eq. (12) is not con-served due to the external force.To solve BE numerically, we first rewrite the main

equation as Eq. (13) with its discrete form in Eq. (15).There are two ways to handle the δ-function in the colli-sional integral, either integrating out k2 or integrating outk3. In this work, we have chosen the first choice inEq. (17), which is more stable than the second one asshown in Fig. 2. Next we introduce the symmetricalsampling method to ensure the conservation of the totalparticle number. We have also investigated the energyconservation which is not strictly conserved numericallydue to the discrete grids. However, the numerical errors willdecrease very fast if we increase the numbers of the grids,and the conservation of the total energy can be achieved upto 99.8% in our calculation.We have studied the time evolution of the distribution

functions in both coordinate and momentum spaces.Figure 7 has shown the gluon and u-quark distributionsas a function of spatial direction x at different evolutiontime, given the initial condition in Eqs. (28), (29). It isfound that the distributions of both u quarks and gluonsbecome homogeneous in the coordinate space at a latertime, implying the reach of thermal equilibrium. Figures 9and 11 show the evolution of the distribution functions inthe momentum space for pure gluons, pure quarks andquark-gluon mixtures. It is interesting that the thermal-ization process of gluons is different from that of quarks.The fermions reach the thermal distribution smoothly,while the distribution functions of gluons at an early timeincrease very fast in the soft region and then decrease tothermal distributions.In summary, we have provided a full numerical solution

to the BE with complete 2 → 2 scattering processes withhigh computing performance. Our framework may serve asa basis to study the prethermalization stage in heavy-ioncollisions in the future. Currently, our program can use thegrid sizes up to nx; ny; nz ¼ 20 and npx; npy; npz ¼ 40.Very large grid sizes such as nx; ny; nz; npx; npy; npz ≥ 50

are still challenging, even with the help of GPU clusters.We will continue to improve our framework and algorithmsalong this direction.In the future, we will include 2 → 3 processes and study

the interplay between elastic and inelastic scatterings. With2 → 3 processes, the particle number is not conserved,therefore, we will not implement the symmetric sampling inthat case. We may also include the external electromagneticfields to study the quantum transport phenomena under thestrong electromagnetic fields. Another important and nec-essary extension is to add the expanding effect. We mayfollow the famous trick proposed by Ref. [109] to add theBjorken expansion effect (also see Ref. [110] for theapplication for the quantum kinetic theory).

(a)

(b)

FIG. 12. Time evolution of masses squaredm2g;u and the particle

numbers for quark-gluon systems. The initial condition andparameters are take as the same in Fig. 11.

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ACKNOWLEDGMENTS

The authors thank Jean Paul Blaizot, AleksasMazeliauskas, Xin-nian Wang, Shuai Liu and Ren-hongFang for useful comments and discussions. S. P. and Q.W.are grateful to the Yukawa Institute for Theoretical Physics(YITP) for its hospitality. This work is supported in part by

the Natural Science Foundation of China (NSFC)under Grants No. 11535012 and No. 11890713. S. P. issupported by NSFC under Grants No. 12075235.G.-Y. Q. is supported by NSFC under Grants No. 11775095,No. 11890711, No. 11375072 and by the China ScholarshipCouncil (CSC) under Grant No. 201906775042.

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