The iS3D Particlization Module for Heavy-Ion Collisions

Mike McNelis, Derek Everett, and Ulrich HeinzThe Ohio State University

1/10/2019

1

Overview • What is the best choice for !"?• Linearized !"

• 14 moment approximation • Chapman Enskog expansion

• Modified equilibrium distribution• McNelis (OSU) • Bernhard (Duke)

• JETSCAPE will compare these viscous corrections to the particle spectra using iS3D

• Discuss particle sampler in iS3D and test its performance

2

#$ = #&',$ + *#$+,-.$-/0 = ℎ2/ 3

40 5 -/6 #$

14 moment approximation

• Moments expansion truncated to the hydrodynamic moments

• Assume irreducible coefficients are species independent• Fix to satisfy Landau matching conditions and reproduce viscous part of !"#

3A. Monnai and T. Hirano, Phys. Rev. C80 (2009) 054906

$%&,( =*(

exp . / 0! ± 13$( = $%&,($

4%&,(5"60"06

3$78,( = $%&,($4%&,( 59:(

; + 5= . / 0 ; + 5>"# 0⟨"0 ⟩#

$4%&,( = 1 ± *(47$%&,(

5>"# =

A"#

2 ℰ + D !;59 = E9(!)Π 5= = E=(!)Π

Chapman Enskog expansion

• Gradient expansion of the RTA Boltzmann equation

! " #$% = −(") *+,*-.,+

01→ 3$% ≈ −

01)"5*-.,+(")

• Assume relaxation time 67 is momentum and species independent

• (89, 8:) = shear / bulk viscosity : relaxation time ratios

• ; = =̇/?

4A. Jaiswal, R. Ryblewski and M. Strickland, Phys. Rev. C90 (2014) 044908

8: =;(=) ℰ + D

=+5FGH(=)3=

; = −ℰ + D =H

FGJ(=)3$KL,% = $MN,%$

,MN,%

Π8:

P " ! ;=H

+P " ! H − Q%

H

3 P " ! =+

RST!⟨S! ⟩T

289 P " ! =

89 =FGH(=)=

Pratt McNelis Distribution

• Construct a map ! = #(!%) in the LRF • Momentum space deformations ∝ viscous corrections

• Effective temperature increases with negative Π• Renormalize particle density to the Chapman Enskog one

• In small Π, )*+ limit → Chapman Enskog expansion5

-*+ = 1 +Π

3123*+ +

)*+

215

67 =89:,7 + 382,7

87(;<=)

!* = -*+!+%

S. Pratt and G. Torrieri, Phys. Rev. C82 (2010) 044901

?@A,7 =67 B7

expF7G + !% G

H + 12IJΠK

± 1

!* = −N* O ! (LRF momentum)

)*+ = N* O ) O N+ (LRF shear stress)

px

py

pp ′

πxx

2 βπ

πyy

2 βπ

Jonah Bernhard’s Distribution

• Momentum rescaling:

• Renormalization:

• Same shear stress modification, no modified temperature• (!", #") parameterized to reproduce bulk pressure without

changing the energy density (exactly, in absence of $%&)• Particle abundance ratios do not change with bulk pressure

6

Λ%& = 1 + !" +%& +$%&2-.

/ = #"detΛ

px

py

pp ′

πxx

2 βπ

πyy

2 βπ

3% = Λ%&3&4

J.E. Bernhard, arxiv:1804.06469 (2018)

567,9 =/:9

exp =9> + 34 >? ± 1

3% = −B% C 3 (LRF momentum)

$%& = B% C $ C B& (LRF shear stress)

Sampling Particles from Cooper Frye

• Need to enforce positivity with Θ " # $%& Θ '(• Sampled particle spectra will slightly deviate from the physical ones

• Methodology:1) Sample the number of hadrons emitted from each freezeout cell2) For each hadron, sample its type and momentum

7

)*$+($%" = ℎ.% /

0" # $%& '( Θ " # $%& Θ '(

J.E. Bernhard, arxiv:1804.06469 (2018)

L.G. Pang, H. Peterson, and X.N. Wang, Phys. Rev. C97 (2018) 064918

S. Pratt and G. Torrieri, Phys. Rev. C 82, 044901 (2010)

C. Shen et al., Comput. Phys. Commun. 199 (2016) 61-85

QGPQGP

hadron resonance gas

freezeout hypersurface Σ

Particles

Number of Hadrons

• Assume hadron number of each type follow Poisson statistics

• Time(null)-like cells: ! " #$% > 0 → Δ*+ = -+#$%.• Space-like cells: momentum regions with ! " #$% < 0 cut out

• Enforcing outflow Θ ! " #$% increases particle yield

8

Δ* =1+Δ*+ = 2

3! " #$% 4+ Θ ! " #$% Θ 4+

23= 2 #$!

ℎ$63

https://slideplayer.com/slide/14963596/

P(*) = exp(−Δ*) Δ* >

*!

Δ*+ = Δ*.,+ + Δ*B,+ = -.,+#$%. + -B,+ #$%C ≥ -+#$%.

d3σμ

E+ =Δ*+Δ*

(hadron number distribution)

(hadron type distribution)

(mean number of hadrons emitted

from freezeout cell)

#$%. = F " #$%#$%C = −GC " #$%

Momentum

• Sample local-rest-frame momentum !" from the distribution #$(!")• Here only show method for linear '($

• Draw !" from distribution )$(!")*+!,-. and accept sample with probability /$(!")• Choose )$(!") such that 0 ≤ /$ ≤ 1

9

#$(!")*+!,-. ~ 45678

! 9 *+: (;<,$ +'($ Θ ! 9 *+: Θ (;<,$ +'($

d3σμ

~ )$(!")*+!,-. × /$(!")

= )$(!")*+!,-.×(;<,$)$(!")

*+:B −!"*+:"D Θ *+:B −

!"*+:"D 1 + '($

(;<,$Θ (;<,$ + '($

/$ !" = /;< × /4E × /FG

Acceptance-Rejection Method

10

!"($%)'($)*+ =exp(−$/2)exp(−3/2) ×$5 '$ 'cos 9 ':

;" $% = ;<= × ;>? × ;@A

;<=,C =exp($/2)

exp 3/2 − 1

;>? = '(E FG '(EH −$%'(E%3

Θ '(EH −$%'(E%3 '(EH −

$%'(E%3

≤ '(EH + '(E% = '(E

;@A = 1 +LM"M<=,"

Θ M<=," + LM" →121 +

LMP<Q,"M<=,"

(;<=,C ≤ 1 for m/2 ≥ 0.8554 or 2 ≤ 157.8 MeV)

LMP<Q," ≤ M<=,"(lower bound from Θ function)

(place additional upper bound)

;<=,Z<[\] =$3

exp(3/2)exp 3/2 ±1

pionsheavy hadrons

(;<=,Z<[\] ≤ 1 for m/2 ≥ 1.008)

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Accepted?

Start event for loop

Compute the total particle yield to determine the number of events

Start sampler routine

Set the seed and make random number engines for the distributions

Compute mean number of each hadron type in the freezeout cell

Sample the number of hadrons in the event

Sample the particle's type

Add particle's type, position and momentum to the event list

Start rejection while loop

Draw a momentum sample and compute the weight

no

yes

Start freezeout cell for loop

Start sampled hadron for loop

no

no

yes

no

End sampler routine

yes

yes

Finish loop?

Finish loop?

Finish loop?

Program Flow Chart

!"#"$%&~!&()*+",!)"($

• !&()*+", = statistical ensemble• user input parameter

• !)"($ = total mean particle yield

Performance • Central Pb+Pb collision

• Smooth Glauber initial conditions

• !"#$% = 0.6 GeV• &' = 0.25 fm/c

• 2+1d hydrodynamics (GPU-VH)• (/* = 0.2• !+#,- = 180 MeV (ζ/* peak)

• Cooper Frye Formula (iS3D)• !/0 = 150 MeV• Sample 12, 42, 5 separately

• 6/,7+8#9 = 107 particles

• Δ5; = 0.03 GeV

• Average over <, =+

12

!/0 = 150 MeV!/0 = 150 MeV!+#,- = 155 MeV!+#,- = 180 MeV

D. Bazow, U. Heinz and M. Strickland, Comput. Phys. Commum. 225 (2018) 92-113

0.001

0.01

0.1

1

10

100

0.0 0.5 1.0 1.5 2.0 2.5 3.0

(a)π+

K+

p

ideal discrete ΔN = (u·d3σ)neqideal continuous w/o Θ(p·d3σ)

0.5 1.0 1.5 2.0 2.5 3.0

(b)

ideal discrete ΔN = (u·d3σ)neqideal continuous w/ Θ(p·d3σ)

0.5 1.0 1.5 2.0 2.5 3.0

0.001

0.01

0.1

1

10

100(c)

ideal discrete ΔN = ΔNt + ΔNx

ideal continuous w/ Θ(p·d3σ)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.9

0.95

1.0

1.05

1.1

discrete

/continuous

0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0

0.9

0.95

1.0

1.05

1.1

⟨ dN2π pT dpT dy

⟩ y

pT (GeV)

Effect of Outflow

(a) Both discrete and continuous spectra have same yield but different momentum distributions.

(b) Better agreement but sampler underestimates yield by a few percent.

(c) Θ % & '() correction to sampled yield necessary for self-consistency and precision testing of the sampler.

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d3σμ

0.001

0.01

0.1

1

10

100

0.0 0.5 1.0 1.5 2.0 2.5 3.0

ideal

14 momentπ+

K+

p

0.5 1.0 1.5 2.0 2.5 3.0

Chapman Enskog

0.5 1.0 1.5 2.0 2.5 3.0

0.001

0.01

0.1

1

10

100Modified

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.9

0.95

1.0

1.05

1.1

discrete

/continuous

0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0

0.9

0.95

1.0

1.05

1.1

⟨ dN2π pT dpT dy

⟩ y

pT (GeV) 14

d3σμ

Viscous Corrections

• Shear + bulk viscous corrections to the spectra• 14 moment• Chapman Enskog• Modified (McNelis)

• Overall, good agreement but sampled particle yield slightly underestimated

• Difficulties in calculating Δ"# with outflow and viscous corrections • Especially modified

distribution

15

d3σμ

Non-central collision (b = 5 fm)

0 0.5 1.0 1.5 2.0 2.5 3.00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.5 1.0 1.5 2.0 2.5 3.0

14 moment

π+

K++ 0.1

p + 0.2

0.5 1.0 1.5 2.0 2.5 3.0

0.5 1.0 1.5 2.0 2.5 3.0

Chapman Enskog

ideal

0.5 1.0 1.5 2.0 2.5 3.0

0.5 1.0 1.5 2.0 2.5 3.0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Modified

v2(pT)

pT (GeV)

Achievements and Outlook

• Incorporated a variety of modes in iS3D:• Continuous and sampled spectra (boost invariant or non-boost invariant)

• Viscous !" corrections (linear and modified)

• Tested the particle sampler for central and non-central collisions• Event-averaged pT spectra and v2 in good agreement with the Cooper Frye formula

• To do list:• Test the sampled spacetime distributions

• Continue testing particle sampler integration in JETSCAPE

• Add Jonah’s modified distribution to iS3D

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