1. Thermalization of Gluons with Bose-Einstein Condensation
Sept. 24, 2014, Wuhan
Thermalization of Gluons with Bose-Einstein Condensation
Zhe Xu
with
Kai Zhou, Pengfei Zhuang, Carsten Greiner
Zhe Xu

2. Outline Motivations Transport equation for Bose-Einstein condensation
Bose statistics in BAMPS
Recent results
Summary and Outlook
Zhe Xu
Zhe Xu

3. Motivations Ultrarelativistic Heavy Ion Collisions
CGC leads to overpopulated QGP
Blaizot, Gelis, Liao, McLerran, Venugopalan, NPA 873 (2012)
𝜖 0 ~ 𝑄 𝑠 4 / 𝛼 𝑠 ,
𝑛 0 ~ 𝑄 𝑠 3 / 𝛼 𝑠 ;
𝑛 0 𝜖 0 −3/4 ~ 𝛼 𝑠 −1/4
For CGC
𝑓 𝑒𝑞 = 1 𝑒𝑥𝑝 𝑝/𝑇 −1 ,
In equilibrium
𝑛 𝑒𝑞 𝜖 𝑒𝑞 −3/4 ~1
For small 𝛼 𝑠 CGC leads to overpopulated QGP with BEC assuming
number conservation
𝑓= 𝑓 𝑒𝑞 + 2𝜋 3 𝑛 𝑐 𝛿 (3) 𝑝
Zhe Xu
Zhe Xu

4. Motivations Studies within the kinetic theory Onset of BEC, BUT no BEC
Blaizot, Gelis, Liao, McLerran, Venugopalan, NPA 873 (2012)
Blaizot, Liao, McLerran, NPA 920 (2013)
Huang, Liao, arXiv:
Blaizot, Wu, Yan, arXiv:
Scardina, Perricone, Plumari, Ruggieri, Greco, arXiv:
Studies within the classical field theory (no number conservation)
Epelbaum, Gelis, NPA 872 (2011)
Berges, Sexty, PRL 108 (2012)
Kurkela, Moore, PRD 86 (2012)
Berges, Boguslavski, Schlichting, Venugopalan, arXiv:
Zhe Xu
Zhe Xu

5. Motivations
We study the thermalization of gluons with BE condensation by using
a transport model BAMPS.
Assumptions:
Static case: homogenous in space
Gluon number conservation by considering only elastic scatterings
Initial distribution: 𝑓 𝑖𝑛𝑖𝑡 𝑝 = 𝑓 0 𝜃 𝑄 𝑠 −𝑝
the condensation occurs when 𝑓 0 >0.154
We know the final distribution
𝑓 𝑒𝑞 𝑝 = 1 𝑒 𝑝/𝑇 −1 + 2𝜋 3 𝑛 𝑐 𝛿 (3) 𝑝
We want to know how fast the thermalization occurs.
Zhe Xu

6. Transport Equation for BEC
Boltzmann Equation
𝜕 𝑡 + 𝑝 1 𝐸 1 ∙ 𝛻 𝑓 1 𝑥, 𝑝 1 = 1 2 𝐸 𝑑 3 𝑝 𝜋 3 2 𝐸 2 𝑑 3 𝑝 𝜋 3 2 𝐸 3 𝑑 3 𝑝 𝜋 3 2 𝐸 𝜈 ℳ 12→34 2
× 2𝜋 4 𝛿 (4) 𝑝 1 + 𝑝 2 − 𝑝 3 − 𝑝 4
× 𝑓 3 𝑓 4 𝟏+ 𝒇 𝟏 𝟏+ 𝒇 𝟐 − 𝑓 1 𝑓 2 𝟏+ 𝒇 𝟑 𝟏+ 𝒇 𝟒
𝑓= 𝑓 𝑔𝑎𝑠 + 𝑓 𝑐 , 𝑓 𝑐 = 2𝜋 3 𝑛 𝑐 𝛿 (3) 𝑝
Included
𝑔+𝑔↔𝑔+𝑔
𝑔+𝑔↔𝑔+𝑐
Not included
𝑔+𝑐↔𝑔+𝑐
𝑐+𝑐↔𝑐+𝑐
𝑔: gas particle
𝑐: condensate particle
Zhe Xu
Zhe Xu

7. Transport Equation for BEC
For gas particles:
𝜕 𝑡 + 𝑝 1 𝐸 1 ∙ 𝛻 𝑓 1 𝑔𝑎𝑠 𝑥, 𝑝 1 = 1 2 𝐸 𝑑 3 𝑝 𝜋 3 2 𝐸 2 𝑑 3 𝑝 𝜋 3 2 𝐸 3 𝑑 3 𝑝 𝜋 3 2 𝐸 ℳ 12→34 2
× 2𝜋 4 𝛿 (4) 𝑝 1 + 𝑝 2 − 𝑝 3 − 𝑝 4
× [ 1 2 𝑓 3 𝑔𝑎𝑠 𝑓 4 𝑔𝑎𝑠 1+ 𝑓 1 𝑔𝑎𝑠 1+ 𝑓 2 𝑔𝑎𝑠 − 1 2 𝑓 1 𝑔𝑎𝑠 𝑓 2 𝑔𝑎𝑠 1+ 𝑓 3 𝑔𝑎𝑠 1+ 𝑓 4 𝑔𝑎𝑠
+ 𝑓 3 𝑔𝑎𝑠 𝑓 4 𝑔𝑎𝑠 1+ 𝑓 1 𝑔𝑎𝑠 𝑓 2 𝑐 − 1 2 𝑓 1 𝑔𝑎𝑠 𝑓 2 𝑐 1+ 𝑓 3 𝑔𝑎𝑠 1+ 𝑓 4 𝑔𝑎𝑠
𝑓 3 𝑐 𝑓 4 𝑔𝑎𝑠 1+ 𝑓 1 𝑔𝑎𝑠 1+ 𝑓 2 𝑔𝑎𝑠 − 𝑓 1 𝑔𝑎𝑠 𝑓 2 𝑔𝑎𝑠 𝑓 3 𝑐 1+ 𝑓 4 𝑔𝑎𝑠 ]
Zhe Xu
Zhe Xu

8. Transport Equation for BEC
For condensate particles:
𝜕 𝑡 𝑓 1 𝑐 𝑥, 𝑝 1 = 1 2 𝐸 𝑑 3 𝑝 𝜋 3 2 𝐸 2 𝑑 3 𝑝 𝜋 3 2 𝐸 3 𝑑 3 𝑝 𝜋 3 2 𝐸 ℳ 12→34 2
× 2𝜋 4 𝛿 (4) 𝑝 1 + 𝑝 2 − 𝑝 3 − 𝑝 4
× 𝑓 3 𝑔𝑎𝑠 𝑓 4 𝑔𝑎𝑠 𝑓 1 𝑐 1+ 𝑓 2 𝑔𝑎𝑠 − 1 2 𝑓 1 𝑐 𝑓 2 𝑔𝑎𝑠 1+ 𝑓 3 𝑔𝑎𝑠 1+ 𝑓 4 𝑔𝑎𝑠
𝑓 𝑐 = 2𝜋 3 𝑛 𝑐 𝛿 (3) 𝑝
𝑑 3 𝑝 𝜋 3 𝜕 𝑡 𝑓 1 𝑐 = 𝜕 𝑛 𝑐 𝜕𝑡 = 𝑅 𝑐 𝑔𝑎𝑖𝑛 − 𝑅 𝑐 𝑙𝑜𝑠𝑠
A small phase space volume competes a 𝛿 function.
Zhe Xu
Zhe Xu

9. Transport Equation for BEC
Whether the condensation occurs?
𝑅 𝑐 𝑔𝑎𝑖𝑛 = 𝑛 𝑐 4𝜋 𝑑 𝑝 3 𝑑 𝑝 4 𝑓 3 𝑓 𝑓 2 𝑝 3 𝑝 4 𝐸 3 𝐸 4 𝐸 ℳ 34→ 𝑠 𝐸− 𝑝 2 + 𝑚 2 =𝑚
𝐸= 𝐸 3 + 𝐸 4 , 𝑝= 𝑝 𝑝 4 , 𝑠= 𝐸 2 − 𝑝 2
𝑚: particle mass
The kinematic constraint 𝐸− 𝑝 2 + 𝑚 2 =𝑚 leads to 𝑠=2𝐸𝑚.
For massless particles 𝑠=0, i.e., 𝑝 3 and 𝑝 4 are parallel.
ℳ 34→ 𝑠 𝑠=0 can be zero, infinity, or has a finite value, which
depends on the form of the scattering matrix.
Zhe Xu
Zhe Xu

10. Transport Equation for BEC
Whether the condensation occurs?
𝐹= ℳ 34→ 𝑠 𝑠=0 = ?
Case 1: interactions with the isotropic distribution of the collision angle.
ℳ 34→ ~𝑠𝜎, 𝐹 can be a finite value, if the cross section is
not diverge at 𝑠=0.
Case 2: interactions with the pQCD cross section of gluons
ℳ 34→ ~ 𝑠 2 𝑡 2 ≈ 𝑠 2 𝑡− 𝑚 𝐷 , 𝐹=0 at 𝑠=0.
ℳ 34→ ~ 𝑠 2 𝑡 2 ≈ 𝑠 2 𝑡 𝑡− 𝑚 𝐷 2 , 𝐹 has a finite value at 𝑠=0.
Aurenche, Gelis, Zaraket, JHEP 05 (2002)
Zhe Xu
Zhe Xu

11. Bose statistics in BAMPS
BAMPS: Boltzmann Approach of MultiParton Scatterings
solves the semi-classical, relativistic Boltzmann equation
in the framework of pQCD by Monte Carlo simulations.
ZX and C. Greiner, PRC 71, (2005)
𝜕 𝑡 + 𝑝 1 𝐸 1 ∙ 𝛻 𝑓 1 𝑥, 𝑝 1 =𝐶
test particle representation of 𝑓
stochastic interpretation of the
collision rates
Zhe Xu
Zhe Xu

12. Bose statistics in BAMPS
Collision scheme: stochastic method
𝑃 22 = 𝑣 𝑟𝑒𝑙 𝜎 22 ′ 𝑁 𝑡𝑒𝑠𝑡 ∆𝑡 ∆𝑉
For 𝑔+𝑔→𝑔+𝑔
𝑔+𝑐→𝑔+𝑔
𝜎 22 ′ = 1 4𝑠 𝑑 3 𝑝 𝜋 3 2 𝐸 1 𝑑 3 𝑝 𝜋 3 2 𝐸 ℳ 34→ 𝜋 4 𝛿 (4) ⋯ 𝟏+ 𝒇 𝟏 𝟏+ 𝒇 𝟐
= 𝑑Ω 𝑑 𝜎 22 𝑑Ω 𝟏+ 𝒇 𝟏 𝟏+ 𝒇 𝟐 = 𝑑Ω 𝑑 𝜎 22 ′ 𝑑Ω
used by Scardina, Perricone, Plumari, Ruggieri, Greco, arXiv:
Disadvantages:
Uncertainties from the extraction of 𝑓, and the integration in 𝜎 22 ′
2. Time consuming due to the integrations in 𝜎 22 ′ for each particle pairs,
whether or not they collide.
Zhe Xu

13. Bose statistics in BAMPS
𝑑 𝑃 22 𝑑Ω = 𝑣 𝑟𝑒𝑙 1 𝑁 𝑡𝑒𝑠𝑡 𝑑 𝜎 22 ′ 𝑑Ω ∆𝑡 ∆𝑉 1 P
collision
no collision
New Scheme:
sample first the collision angle Ω
2. sample a random number between 0 and 𝑑𝑃/𝑃 /𝑑Ω. If this number is
smaller than 𝑑𝑃/𝑑Ω, a collision occurs.
Zhe Xu

14. Bose statistics in BAMPS
𝑃 22 = 𝑣 𝑟𝑒𝑙 𝜎 22 𝑐 𝑁 𝑡𝑒𝑠𝑡 ∆𝑡 ∆𝑉
For 𝑔+𝑔→𝑔+𝑐
𝜎 22 𝑐 = 1 2𝑠 𝑑 3 𝑝 𝜋 3 2 𝐸 1 𝑑 3 𝑝 𝜋 3 2 𝐸 ℳ 34→ 𝜋 7 𝛿 (4) ⋯ 𝒏 𝒄 𝜹 (𝟑) 𝒑 𝟏 1+ 𝑓 2
= 𝜋 2 𝑛 𝑐 (1+ 𝑓 2 ) 1 𝑝 ℳ 34→ 𝑠 𝛿 𝐸−𝑝 2
Approximation:
Particles with 𝒑<𝜺 are regarded as condensate particles.
𝜀 should be small to reduce the numerical uncertainty.
The rate for producing BEC does not depend on 𝜀.
𝛿 (3) 𝑝 1 ≈ 𝜃 𝜀− 𝑝 1 4𝜋 𝑝 1 2 𝜀 ⇒ 𝜎 22 𝑐 ≈ 𝜋 2 𝑛 𝑐 1+ 𝑓 𝑝 ℳ 34→ 𝑠
× 1 4𝜀 2 𝐸−𝑝 − 1 𝜀 𝜃 𝜀− 𝐸−𝑝 2
Zhe Xu

15. Recent results
ℳ 34→ ≈ 12𝜋 2 𝛼 𝑠 2 𝑠 2 𝑡 𝑡− 𝑚 𝐷 2 , 𝑚 𝐷 2 =16𝜋 𝑁 𝑐 𝛼 𝑠 𝑑 3 𝑝 (2𝜋) 𝑝 𝑓 𝑔𝑎𝑠
Onset of BEC: without 𝑔+𝑔→𝑔+𝑐 (BUT, there are still particles
with energy smaller than 𝜀.)
𝑓 0 =0.4
The distribution is
frozen out at 4 fm/c
(onset of BEC
happens earlier)
and
is far from the
equilibrium one.
Zhe Xu

16. Recent results Onset of BEC: without 𝑔+𝑔→𝑔+𝑐
𝑓≈ 1 𝑒𝑥𝑝 𝑝− 𝜇 ∗ 𝑇 ∗ −1 ≈ 𝑇 ∗ 𝑝− 𝜇 ∗
Assume: for small p,
effective temperature
effective chemical potential
Zhe Xu

17. Recent results
BE Condensation: with 𝑔+𝑔→𝑔+𝑐, when 𝜇 ∗ is sufficiently close to zero
and is positive due to numerical fluctuation.
𝑓 0 =0.4
The condensation
begins at 1.6 fm/c.
The distribution at
small p is increasing
until 2.2 fm/c.
Zhe Xu

18. Recent results
Thermlization with BE Condensation
Zhe Xu

19. Recent results
Thermlization with BE Condensation
Zhe Xu

20. Recent results Thermlization with BE Condensation perfect agreement !
Zhe Xu

21. Recent results Thermlization with BE Condensation perfect agreement !
Zhe Xu

22. Recent results Scaling behaviour of BE condensation BEC completion
Zhe Xu

23. Recent results
Scaling behaviour of entropy production during BE condensation
Two step entropy
production
Full thermalization
at 𝑓 0 𝑡≈3 𝑓𝑚/𝑐
Zhe Xu

24. Recent results
Entropy production before BE condensation
Zhe Xu

25. Recent results BE condensation slows down the entropy production.
Zhe Xu

26. Summary and Outlook
show the onset and full process of BE condensation in BAMPS
BE condensation is complete at 𝑓 0 𝑡≈6 𝑓𝑚/𝑐
almost full thermalization at 𝑓 0 𝑡≈3 𝑓𝑚/𝑐
BE condensation slows down the entropy production.
Further studies:
Influence of momentum anisotropy, 2<->3 processes,
quarks, and expansion on BE condensation
Measurable effects
Zhe Xu

27. Backup
Zhe Xu

28. Transport Equation for BEC
𝑅 𝑐 𝑔𝑎𝑖𝑛 = 𝑛 𝑐 4𝜋 𝑑 𝑝 3 𝑑 𝑝 4 𝑓 3 𝑓 𝑓 2 𝑝 3 𝑝 4 𝐸 3 𝐸 4 𝐸 ℳ 34→𝟏 𝑠 𝑠=0
𝑅 𝑐 𝑙𝑜𝑠𝑠 = 𝑛 𝑐 4𝜋 𝑑 𝑝 3 𝑑 𝑝 𝑓 𝑓 4 𝑓 2 𝑝 3 𝑝 4 𝐸 3 𝐸 4 𝐸 ℳ 𝟏2→ 𝑠 𝑠=0
𝜕 𝑛 𝑐 𝜕𝑡 ~ 𝑛 𝑐
need an initial seed for the condensation
𝑓 2,3,4 = 1 𝑒𝑥𝑝 𝑝 2,3,4 𝑇 −1
At thermal equilibrium,
In this case 𝑅 𝑐 𝑔𝑎𝑖𝑛 − 𝑅 𝑐 𝑙𝑜𝑠𝑠 =0, but, both 𝑅 𝑐 𝑔𝑎𝑖𝑛 and 𝑅 𝑐 𝑔𝑎𝑖𝑛 are divergent.
The divergence is logarithmically with a lower cutoff 𝜀 for 𝑝.
Zhe Xu
Zhe Xu

29. Recent results
Onset of BEC: without 𝑔+𝑔→𝑔+𝑐 (BUT, there are still particles
with energy smaller than 𝜀.)
𝜀= 𝐺𝑒𝑉
Density of particles with
energy smaller than 𝜀,
about 1% of the density
of the real condensate
Particles ( 𝑛 𝑐 =2.98 𝑓𝑚 −3 ).
Zhe Xu

30. Recent results BEC: with 𝑔+𝑔→𝑔+𝑐 𝑓≈ 1 𝑒𝑥𝑝 𝑝− 𝜇 ∗ 𝑇 ∗ −1 ≈ 𝑇 ∗ 𝑝− 𝜇 ∗
𝑓≈ 1 𝑒𝑥𝑝 𝑝− 𝜇 ∗ 𝑇 ∗ −1 ≈ 𝑇 ∗ 𝑝− 𝜇 ∗
Assume: for small p,
effective temperature
effective chemical potential
Zhe Xu

31. Recent results
Onset of BEC: without 𝑔+𝑔→𝑔+𝑐
Zhe Xu

32. Recent results BE Condensation is complete at 10 fm/c for 𝑓 0 =0.4.
perfect
agreement !
Zhe Xu

33. Recent results
Thermlization with BE Condensation ( 𝑓 0 =2)
Zhe Xu

34. Recent results Computation setups: Tests:
Box with volume: 3 𝑓𝑚×3 𝑓𝑚×3 𝑓𝑚
Cells with volume: 𝑓𝑚×0.125 𝑓𝑚×0.125 𝑓𝑚
Number of test particles per each real particle: 𝑁 𝑡𝑒𝑠𝑡 =6000
Total particle number in computation: 2.23× 10 6
MPI: 216 CPUs
Tests:
new scheme
collision rates
case of underpopulated gluons (negative chemical potential)
critical case (zero chemical potential and no condensation)
Zhe Xu

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