Sept. 24, 2014, Wuhan
Thermalization of Gluons with
Bose-Einstein Condensation
Zhe Xu
[email protected]
with
Kai Zhou, Pengfei Zhuang, Carsten Greiner
Outline
• Motivations
• Transport equation for Bose-Einstein condensation
• Bose statistics in BAMPS
• Recent results
• Summary and Outlook
Zhe Xu
2/26
Motivations
Ultrarelativistic
Heavy Ion
Collisions
CGC leads to overpopulated QGP
Blaizot, Gelis, Liao, McLerran, Venugopalan, NPA 873 (2012)
For CGC
𝜖0 ~𝑄𝑠4 /𝛼𝑠 , 𝑛0 ~𝑄𝑠3 /𝛼𝑠 ;
1
In equilibrium 𝑓𝑒𝑞 = 𝑒𝑥𝑝 𝑝/𝑇 − 1 ,
−3/4
𝑛0 𝜖0
−3/4
𝑛𝑒𝑞 𝜖𝑒𝑞
−1/4
~𝛼𝑠
~1
For small 𝛼𝑠 CGC leads to overpopulated QGP with BEC assuming
number conservation
𝑓 = 𝑓𝑒𝑞 + 2𝜋 3 𝑛𝑐 𝛿 (3) 𝑝
Zhe Xu
3/26
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:1303.7214
Blaizot, Wu, Yan, arXiv:1402.5049
Scardina, Perricone, Plumari, Ruggieri, Greco, arXiv:1408.1313
• 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:1408.1670
Zhe Xu
4/26
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: 𝑓𝑖𝑛𝑖𝑡 𝑝 = 𝑓0 𝜃 𝑄𝑠 − 𝑝
the condensation occurs when 𝑓0 > 0.154
We know the final distribution
𝑓𝑒𝑞 𝑝 =
1
𝑒 𝑝/𝑇 − 1
+ 2𝜋 3 𝑛𝑐 𝛿 (3) 𝑝
We want to know how fast the thermalization occurs.
Zhe Xu
5/26
Transport Equation for BEC
Boltzmann Equation
𝑝1
1
𝜕𝑡 + ∙ 𝛻 𝑓1 𝑥, 𝑝1 =
𝐸1
2𝐸1
𝑑 3 𝑝2
𝑑 3 𝑝3
𝑑 3 𝑝4 1
ℳ12→34
3
3
3
2𝜋 2𝐸2 2𝜋 2𝐸3 2𝜋 2𝐸4 𝜈
2
× 2𝜋 4 𝛿 (4) 𝑝1 + 𝑝2 − 𝑝3 − 𝑝4
× 𝑓3 𝑓4 𝟏 + 𝒇𝟏 𝟏 + 𝒇𝟐 − 𝑓1 𝑓2 𝟏 + 𝒇𝟑 𝟏 + 𝒇𝟒
𝑓 = 𝑓 𝑔𝑎𝑠 + 𝑓 𝑐 ,
𝑓 𝑐 = 2𝜋 3 𝑛𝑐 𝛿 (3) 𝑝
𝑔: gas particle
𝑐: condensate particle
Included
Not included
𝑔+𝑔 ↔𝑔+𝑔
𝑔+𝑐 ↔𝑔+𝑐
𝑔+𝑔 ↔𝑔+𝑐
𝑐+𝑐 ↔𝑐+𝑐
Zhe Xu
6/26
Transport Equation for BEC
For gas particles:
𝑝1
1
𝑔𝑎𝑠
𝜕𝑡 + ∙ 𝛻 𝑓1
𝑥, 𝑝1 =
𝐸1
2𝐸1
𝑑 3 𝑝2
𝑑 3 𝑝3
𝑑 3 𝑝4
ℳ12→34
3
3
3
2𝜋 2𝐸2 2𝜋 2𝐸3 2𝜋 2𝐸4
2
× 2𝜋 4 𝛿 (4) 𝑝1 + 𝑝2 − 𝑝3 − 𝑝4
1
2
𝑔𝑎𝑠 𝑔𝑎𝑠
𝑓4
× [ 𝑓3
𝑔𝑎𝑠 𝑔𝑎𝑠
𝑓4
+𝑓3
1
2
𝑔𝑎𝑠
+ 𝑓3𝑐 𝑓4
𝑔𝑎𝑠
1 + 𝑓1
𝑔𝑎𝑠
1 + 𝑓2
1
2
𝑔𝑎𝑠
𝑓2𝑐 − 𝑓1
𝑔𝑎𝑠
1 + 𝑓2
1 + 𝑓1
1 + 𝑓1
1
2
𝑔𝑎𝑠 𝑐
𝑓2
𝑔𝑎𝑠
𝑔𝑎𝑠 𝑔𝑎𝑠
𝑓2
− 𝑓1
𝑔𝑎𝑠
1 + 𝑓3
𝑔𝑎𝑠 𝑔𝑎𝑠 𝑐
𝑓2 𝑓3
− 𝑓1
𝑔𝑎𝑠
1 + 𝑓3
𝑔𝑎𝑠
1 + 𝑓4
𝑔𝑎𝑠
1 + 𝑓4
𝑔𝑎𝑠
1 + 𝑓4
]
Zhe Xu
7/26
Transport Equation for BEC
For condensate particles:
𝜕𝑡 𝑓1𝑐 𝑥, 𝑝1
𝑑 3 𝑝2
𝑑 3 𝑝3
𝑑 3 𝑝4
ℳ12→34
3
3
3
2𝜋 2𝐸2 2𝜋 2𝐸3 2𝜋 2𝐸4
1
=
2𝐸1
2
× 2𝜋 4 𝛿 (4) 𝑝1 + 𝑝2 − 𝑝3 − 𝑝4
𝑔𝑎𝑠 𝑔𝑎𝑠 𝑐
𝑓4 𝑓1
× 𝑓3
𝑔𝑎𝑠
1 + 𝑓2
1
2
𝑔𝑎𝑠
− 𝑓1𝑐 𝑓2
𝑔𝑎𝑠
1 + 𝑓3
𝑔𝑎𝑠
1 + 𝑓4
𝑓 𝑐 = 2𝜋 3 𝑛𝑐 𝛿 (3) 𝑝
𝑑 3 𝑝1
𝜕𝑛𝑐
𝑔𝑎𝑖𝑛
𝑐
𝑙𝑜𝑠𝑠
𝜕
𝑓
=
=
𝑅
−
𝑅
𝑡
𝑐
1
𝑐
2𝜋 3
𝜕𝑡
A small phase space volume competes a 𝛿 function.
Zhe Xu
8/26
Transport Equation for BEC
Whether the condensation occurs?
𝑔𝑎𝑖𝑛
𝑅𝑐
𝑛𝑐
=
4𝜋
3
𝑝3 𝑝4
𝑑𝑝3 𝑑𝑝4 𝑓3 𝑓4 1 + 𝑓2
𝐸
𝐸3 𝐸4
𝐸 = 𝐸3 + 𝐸4 ,
𝑝 = 𝑝3 + 𝑝4 ,
ℳ34→12
𝑠
2
𝐸− 𝑝2 +𝑚2 =𝑚
𝑠 = 𝐸 2 − 𝑝2
𝑚: particle mass
The kinematic constraint 𝐸 − 𝑝2 + 𝑚2 = 𝑚 leads to 𝑠 = 2𝐸𝑚.
For massless particles 𝑠 = 0, i.e., 𝑝3 and 𝑝4 are parallel.
ℳ34→12 2
can be zero, infinity, or has a finite value, which
𝑠
𝑠=0
depends on the form of the scattering matrix.
Zhe Xu
9/26
Transport Equation for BEC
Whether the condensation occurs?
𝐹=
ℳ34→12
𝑠
2
=?
𝑠=0
Case 1: interactions with the isotropic distribution of the collision angle.
ℳ34→12 2 ~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
2
𝑠
2
ℳ34→12 ~ 2
𝑡
2
𝑠
ℳ34→12 2 ~ 2
𝑡
≈
≈
𝑠2
2 2
𝑡−𝑚𝐷
𝑠2
2
𝑡 𝑡−𝑚𝐷
, 𝐹 = 0 at 𝑠 = 0.
, 𝐹 has a finite value at 𝑠 = 0.
Aurenche, Gelis, Zaraket, JHEP 05 (2002)
Zhe Xu
10/26
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, 064901 (2005)
𝑝1
𝜕𝑡 + ∙ 𝛻 𝑓1 𝑥, 𝑝1 = 𝐶
𝐸1
test particle representation of 𝑓
stochastic interpretation of the
collision rates
Zhe Xu
11/26
Bose statistics in BAMPS
Collision scheme: stochastic method
For 𝑔 + 𝑔 → 𝑔 + 𝑔
𝑔+𝑐 →𝑔+𝑔
′
𝜎22
1
=
4𝑠
=
𝑃22
′
𝜎22
∆𝑡
= 𝑣𝑟𝑒𝑙
𝑁𝑡𝑒𝑠𝑡 ∆𝑉
𝑑 3 𝑝1
𝑑 3 𝑝2
ℳ34→12
2𝜋 3 2𝐸1 2𝜋 3 2𝐸2
𝑑𝜎22
𝑑Ω
𝟏 + 𝒇𝟏 𝟏 + 𝒇𝟐 =
𝑑Ω
2
2𝜋 4 𝛿 (4) ⋯ 𝟏 + 𝒇𝟏 𝟏 + 𝒇𝟐
′
𝑑𝜎22
𝑑Ω
𝑑Ω
used by Scardina, Perricone, Plumari, Ruggieri, Greco, arXiv:1408.1313
Disadvantages:
′
1. Uncertainties from the extraction of 𝑓, and the integration in 𝜎22
′
2. Time consuming due to the integrations in 𝜎22
for each particle pairs,
whether or not they collide.
Zhe Xu
12/26
Bose statistics in BAMPS
′
𝑑𝑃22
1 𝑑𝜎22
∆𝑡
= 𝑣𝑟𝑒𝑙
𝑑Ω
𝑁𝑡𝑒𝑠𝑡 𝑑Ω ∆𝑉
1
P
collision
no collision
New Scheme:
1. 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
13/26
Bose statistics in BAMPS
For 𝑔 + 𝑔 → 𝑔 + 𝑐
𝑐
𝜎22
1
=
2𝑠
𝑃22
𝑐
𝜎22
∆𝑡
= 𝑣𝑟𝑒𝑙
𝑁𝑡𝑒𝑠𝑡 ∆𝑉
𝑑 3 𝑝1
𝑑 3 𝑝2
ℳ34→12
2𝜋 3 2𝐸1 2𝜋 3 2𝐸2
2
𝜋
1 ℳ34→12 2
= 𝑛𝑐 (1 + 𝑓2 )
𝛿 𝐸−𝑝
2
𝑝
𝑠
2𝜋 7 𝛿 (4) ⋯ 𝒏𝒄 𝜹(𝟑) 𝒑𝟏 1 + 𝑓2
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) 𝑝1
𝜃 𝜀 − 𝑝1
≈
4𝜋𝑝12 𝜀
⇒
𝑐
𝜎22
𝜋
1 ℳ34→12 2
≈ 𝑛𝑐 1 + 𝑓2
2
𝑝
𝑠
1
2
1
𝐸−𝑝
×
−
𝜃 𝜀−
4𝜀 𝐸 − 𝑝 𝜀
2
Zhe Xu
14/26
Recent results
ℳ34→12
2
≈ 12𝜋
2 2
𝛼𝑠
𝑠2
2 ,
𝑡 𝑡 − 𝑚𝐷
2
𝑚𝐷
= 16𝜋𝑁𝑐 𝛼𝑠
𝑑 3 𝑝 1 𝑔𝑎𝑠
𝑓
(2𝜋)3 𝑝
Onset of BEC: without 𝑔 + 𝑔 → 𝑔 + 𝑐 (BUT, there are still particles
with energy smaller than 𝜀.)
𝑓0 = 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
15/26
Recent results
Onset of BEC: without 𝑔 + 𝑔 → 𝑔 + 𝑐
Assume: for small p,
effective temperature
1
𝑇∗
𝑓≈
≈
∗
𝑝 − 𝜇∗
𝑝
−
𝜇
𝑒𝑥𝑝
−1
𝑇∗
effective chemical potential
Zhe Xu
16/26
Recent results
BE Condensation: with 𝑔 + 𝑔 → 𝑔 + 𝑐, when 𝜇∗ is sufficiently close to zero
and is positive due to numerical fluctuation.
𝑓0 = 0.4
The condensation
begins at 1.6 fm/c.
The distribution at
small p is increasing
until 2.2 fm/c.
Zhe Xu
17/26
Recent results
Thermlization with BE Condensation
Zhe Xu
18/26
Recent results
Thermlization with BE Condensation
Zhe Xu
19/27
Recent results
Thermlization with BE Condensation
perfect
agreement !
Zhe Xu
20/26
Recent results
Thermlization with BE Condensation
perfect
agreement !
Zhe Xu
21/26
Recent results
Scaling behaviour of BE condensation
BEC completion
at 𝑓0 𝑡 ≈ 6 𝑓𝑚/𝑐
Zhe Xu
22/26
Recent results
Scaling behaviour of entropy production during BE condensation
Two step entropy
production
Full thermalization
at 𝑓0 𝑡 ≈ 3 𝑓𝑚/𝑐
Zhe Xu
23/26
Recent results
Entropy production before BE condensation
Zhe Xu
24/26
Recent results
BE condensation slows down the entropy production.
Zhe Xu
25/26
Summary and Outlook
• show the onset and full process of BE condensation in BAMPS
• BE condensation is complete at 𝑓0 𝑡 ≈ 6 𝑓𝑚/𝑐
• almost full thermalization at 𝑓0 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
26/26
Backup
Zhe Xu
27/26
Transport Equation for BEC
𝑔𝑎𝑖𝑛
𝑅𝑐
𝑅𝑐𝑙𝑜𝑠𝑠
•
𝑛𝑐
=
4𝜋
𝑛𝑐
=
4𝜋
𝑝3 𝑝4
𝑑𝑝3 𝑑𝑝4 𝑓3 𝑓4 1 + 𝑓2
𝐸
𝐸3 𝐸4
3
𝑑𝑝3 𝑑𝑝4 1 + 𝑓3
3
𝜕𝑛𝑐
~ 𝑛𝑐
𝜕𝑡
ℳ34→𝟏2
𝑠
𝑝3 𝑝4
1 + 𝑓4 𝑓2
𝐸
𝐸3 𝐸4
2
𝑠=0
ℳ𝟏2→34
𝑠
2
𝑠=0
need an initial seed for the condensation
• At thermal equilibrium,
𝑔𝑎𝑖𝑛
In this case 𝑅𝑐
𝑓2,3,4 =
1
𝑝2,3,4
𝑒𝑥𝑝
−1
𝑇
𝑔𝑎𝑖𝑛
− 𝑅𝑐𝑙𝑜𝑠𝑠 = 0, but, both 𝑅𝑐
𝑔𝑎𝑖𝑛
and 𝑅𝑐
are divergent.
The divergence is logarithmically with a lower cutoff 𝜀 for 𝑝.
Zhe Xu
28/26
Recent results
Onset of BEC: without 𝑔 + 𝑔 → 𝑔 + 𝑐 (BUT, there are still particles
with energy smaller than 𝜀.)
𝜀 = 0.0025 𝐺𝑒𝑉
Density of particles with
energy smaller than 𝜀,
about 1% of the density
of the real condensate
Particles (𝑛𝑐 = 2.98 𝑓𝑚−3 ).
Zhe Xu
29/26
Recent results
BEC: with 𝑔 + 𝑔 → 𝑔 + 𝑐
Assume: for small p,
effective temperature
1
𝑇∗
𝑓≈
≈
∗
𝑝 − 𝜇∗
𝑝
−
𝜇
𝑒𝑥𝑝
−1
𝑇∗
effective chemical potential
Zhe Xu
30/26
Recent results
Onset of BEC: without 𝑔 + 𝑔 → 𝑔 + 𝑐
Zhe Xu
31/26
Recent results
BE Condensation is complete at 10 fm/c for 𝑓0 = 0.4.
perfect
agreement !
Zhe Xu
32/26
Recent results
Thermlization with BE Condensation (𝑓0 = 2)
Zhe Xu
33/26
Recent results
Computation setups:
Box with volume: 3 𝑓𝑚 × 3 𝑓𝑚 × 3 𝑓𝑚
Cells with volume: 0.125 𝑓𝑚 × 0.125 𝑓𝑚 × 0.125 𝑓𝑚
Number of test particles per each real particle: 𝑁𝑡𝑒𝑠𝑡 = 6000
Total particle number in computation: 2.23 × 106
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
34/26
Zhe Xu
35/26
Zhe Xu
36/26
Zhe Xu
37/26
Zhe Xu
38/26