Thermalization of Gluons with Bose-Einstein Condensation Sept. 24, 2014, Wuhan Thermalization of Gluons with Bose-Einstein Condensation Zhe Xu xuzhe@mail.tsinghua.edu.cn with Kai Zhou, Pengfei Zhuang, Carsten Greiner Zhe Xu

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

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

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 Zhe Xu

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

Transport Equation for BEC Boltzmann Equation 𝜕 𝑡 + 𝑝 1 𝐸 1 ∙ 𝛻 𝑓 1 𝑥, 𝑝 1 = 1 2 𝐸 1 𝑑 3 𝑝 2 2𝜋 3 2 𝐸 2 𝑑 3 𝑝 3 2𝜋 3 2 𝐸 3 𝑑 3 𝑝 4 2𝜋 3 2 𝐸 4 1 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

Transport Equation for BEC For gas particles: 𝜕 𝑡 + 𝑝 1 𝐸 1 ∙ 𝛻 𝑓 1 𝑔𝑎𝑠 𝑥, 𝑝 1 = 1 2 𝐸 1 𝑑 3 𝑝 2 2𝜋 3 2 𝐸 2 𝑑 3 𝑝 3 2𝜋 3 2 𝐸 3 𝑑 3 𝑝 4 2𝜋 3 2 𝐸 4 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 + 1 2 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

Transport Equation for BEC For condensate particles: 𝜕 𝑡 𝑓 1 𝑐 𝑥, 𝑝 1 = 1 2 𝐸 1 𝑑 3 𝑝 2 2𝜋 3 2 𝐸 2 𝑑 3 𝑝 3 2𝜋 3 2 𝐸 3 𝑑 3 𝑝 4 2𝜋 3 2 𝐸 4 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 1 2𝜋 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

Transport Equation for BEC Whether the condensation occurs? 𝑅 𝑐 𝑔𝑎𝑖𝑛 = 𝑛 𝑐 4𝜋 3 𝑑 𝑝 3 𝑑 𝑝 4 𝑓 3 𝑓 4 1+ 𝑓 2 𝑝 3 𝑝 4 𝐸 3 𝐸 4 𝐸 ℳ 34→12 2 𝑠 𝐸− 𝑝 2 + 𝑚 2 =𝑚 𝐸= 𝐸 3 + 𝐸 4 , 𝑝= 𝑝 3 + 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→12 2 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

Transport Equation for BEC Whether the condensation occurs? 𝐹= ℳ 34→12 2 𝑠 𝑠=0 = ? Case 1: interactions with the isotropic distribution of the collision angle. ℳ 34→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 ℳ 34→12 2 ~ 𝑠 2 𝑡 2 ≈ 𝑠 2 𝑡− 𝑚 𝐷 2 2 , 𝐹=0 at 𝑠=0. ℳ 34→12 2 ~ 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

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) 𝜕 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

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 1 2𝜋 3 2 𝐸 1 𝑑 3 𝑝 2 2𝜋 3 2 𝐸 2 ℳ 34→12 2 2𝜋 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:1408.1313 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

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

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 1 2𝜋 3 2 𝐸 1 𝑑 3 𝑝 2 2𝜋 3 2 𝐸 2 ℳ 34→12 2 2𝜋 7 𝛿 (4) ⋯ 𝒏 𝒄 𝜹 (𝟑) 𝒑 𝟏 1+ 𝑓 2 = 𝜋 2 𝑛 𝑐 (1+ 𝑓 2 ) 1 𝑝 ℳ 34→12 2 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 2 1 𝑝 ℳ 34→12 2 𝑠 × 1 4𝜀 2 𝐸−𝑝 − 1 𝜀 𝜃 𝜀− 𝐸−𝑝 2 Zhe Xu

Recent results ℳ 34→12 2 ≈ 12𝜋 2 𝛼 𝑠 2 𝑠 2 𝑡 𝑡− 𝑚 𝐷 2 , 𝑚 𝐷 2 =16𝜋 𝑁 𝑐 𝛼 𝑠 𝑑 3 𝑝 (2𝜋) 3 1 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

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

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

Recent results Thermlization with BE Condensation Zhe Xu

Recent results Thermlization with BE Condensation Zhe Xu

Recent results Thermlization with BE Condensation perfect agreement ! Zhe Xu

Recent results Thermlization with BE Condensation perfect agreement ! Zhe Xu

Recent results Scaling behaviour of BE condensation BEC completion Zhe Xu

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

Recent results Entropy production before BE condensation Zhe Xu

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

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

Backup Zhe Xu

Transport Equation for BEC 𝑅 𝑐 𝑔𝑎𝑖𝑛 = 𝑛 𝑐 4𝜋 3 𝑑 𝑝 3 𝑑 𝑝 4 𝑓 3 𝑓 4 1+ 𝑓 2 𝑝 3 𝑝 4 𝐸 3 𝐸 4 𝐸 ℳ 34→𝟏2 2 𝑠 𝑠=0 𝑅 𝑐 𝑙𝑜𝑠𝑠 = 𝑛 𝑐 4𝜋 3 𝑑 𝑝 3 𝑑 𝑝 4 1+ 𝑓 3 1+ 𝑓 4 𝑓 2 𝑝 3 𝑝 4 𝐸 3 𝐸 4 𝐸 ℳ 𝟏2→34 2 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

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 ( 𝑛 𝑐 =2.98 𝑓𝑚 −3 ). Zhe Xu

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

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

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

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

Recent results Computation setups: Tests: 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: 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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