THERMALIZATION AND ENTROPY PRODUCTION FROM GLASMA INITIAL CONDITION IN CLASSICAL YANG-MILLS DYNAMICS

Hideaki Iida (Kyoto Univ.) in collab. with

T.Kunihiro, B.Mueller, A.Ohnishi, A.Schaefer, T.T.Takahashi

(Phys. Rev. D88, 094006 (2013))

10 Dec. 2013 NFQCD@YITP

CONTENTS

  Introduction: entropy history  Kolmogorov-Sinaï (KS) entropy  Previous study: Entropy production in classical Yang-Mills

dynamics from random initial condition  Entropy production from realistic initial condition

  Modulated initial condition   Numerical simulation

 Conclusion

Entropy history

Entropy history of a central Au+Au collision at top RHIC energy (from B.Mueller and A.Schaefer, Int. J. of Mod. Phys. E20, 2235-2267 (2011).)

Main part of the entropy production occurs before thermalization … study the entropy production at the initial stage of heavy-ion collision

KOLMOGOROV-SINAÏ ENTROPY We study the time evolution of entropy in classical Yang-Mills dynamics

 Kolmogorov-Sinaï (KS) entropy:

: Lyapunov exponent

Lyapunov exponent: an index of initial-value sensitivity

- Instability ・Weibel instability (Mrowczynski (1993)) ・Nielsen-Olesen instability (Iwazaki(2008), Fujii&Itakura (2008)) … considered to lead early thermalization - Chaoticity ・Lyapunov exponent is related to mixing property of chaos (mentioned later)

|�Xi(t)| � e�it

KS entropy … related to instability & chaoticity

…slightly different from at initial time, respectively

ENTROPY PRODUCTION IN CLASSICAL YANG-MILLS DYNAMICS

  Focusing on entropy production through the instability & chaotic behavior in the CYM system with random initial condition.

Entropy production rate (Kolmogolov-Sinaï entropy)

Distance between two trajectories in phase space

Two quantities

T.Kunihiro, B.Mueller, A.Ohnishi, A.Schaefere, T.T.Takahashi and A.Yamamoto, PRD82, 114015(2010).

PREVIOUS STUDY: ENTROPY PRODUCTION IN CLASSICAL YANG-MILLS DYNAMICS

 Entropy production is investigated in CYM with random initial condition

Time evolution of δX for several initial energy density

Time evolution of spectrum of Lyapunov exponent

T.Kunihiro, B.Mueller, A.Ohnishi, A.Schaefere, T.T.Takahashi and A.Yamamoto, PRD82, 114015(2010).

Chaotic behavior & production of entropy are studied

ENTROPY PRODUCTION FROM REALISTIC INITIAL CONDITION

  High energy collision… high gluon density → gluons are coherent field rather than particles First approximation: classical treatment

  Initial condition: color-glass condensate (CGC), Glasma (L.McLerran and R.Venugopalan 1994; T.Lappi and L.McLerran 2006, …)

Classical Yang-Mills calculation with Glasma initial condition … appropriate for the study of thermalization process

We study thermalization process from a coherent background field

Entropy production from realistic initial condition … initial condition with coherent background field

Fluctuation (noise)

Magnetic field from spatial modulation (neglecting noise)

     Modulated  initial  condition  

(+ fluctuation)

magnetic field

Bx = �2m�

Nz�2 sin

�2nx�

Nx

�cos

�2mz�

Nz

�� 0, By = 0

Bz =2�

Nx�1 cos

�2x�

Nx

�+

2n�

Nx�2 cos

�2nx�

Nx

�sin

�2mz�

Nz

�� 2�

Nx�1 cos

�2x�

Nx

�

�1 � �2

x

zy

→ glasma-like

-11

-10

-9

-8

-7

-6

-5

-4

0 50 100 150 200 250 300

log(

DFF

) or l

og(D

EE)

t

DFF and DEE

DFFDEE

-11

-10

-9

-8

-7

-6

-5

-4

0 50 100 150 200 250 300

log(

DFF

) or l

og(D

EE)

t

DFF and DEE

DFFDEE

-11

-10

-9

-8

-7

-6

-5

-4

0 50 100 150 200 250 300

log(

DFF

) or l

og(D

EE)

t

DFF and DEE

DFFDEE

RESULTS: TIME DEP. OF DISTANCE

only fluctuation

only background field

・Modulation only … No instability & chaotic behavior ・Modulation + (tiny) fluctuation … unstable & chaotic

-25

-24

-23

-22

-21

-20

0 50 100 150 200 250 300

log(

DFF

) or l

og(D

EE)

t

DFF and DEE

DFFDEE

only tiny fluctuation

cf.)

background + fluctuation

-11

-10

-9

-8

-7

-6

-5

-4

0 20 40 60 80 100 120 140 160 180

log(

DFF

)

t

DFF

w=0.00001w=0.0001w=0.0004w=0.0008w=0.0016

DISTANCES FOR VARIOUS INITIAL FLUCTUATION

Onset time of increase of D depends on the ratio of fluctuation to background modulation field

large fluctuation

w: magnitude of fluctuations

INTERMEDIATE LYAPUNOV EXPONENT & KOLMOGOROV-SINAÏ ENTROPY

※ Lyapunov exponent is invariant under the residual gauge trans. in temporal gauge

… averaged Lyapunov exponent in a certain time interval

: Intermediate Lyapunov exponent

… sum of positive ILE cf) Kunihiro, Muller, Schafer, Takahashi, Yamamoto, PRD82, 114015 (2008).

KS entropy … related to mixing property of chaos mixing property … roughly speaking, complexity of classical orbit in phase space in a certain time interval

Orbit in phase space:

Periodic orbit → SKS=0 Complex orbit (dense) → SKS > 0

・　　　　　　be a probability of finding a orbit in the cell chain

then the entropy of the collection of orbit is given by

・Choose a cell at each time slice and call the cell chain

MEANING OF KOLMOGOROV-SINAI ENTROPY

…

t 0 1 2 n-1 n

1 2 3 4

5 …

1 2 3 4

5 …

1 2 3 4

5 …

1 2 3 4

5 …

Time T

・Divide time interval 0-T into n-pieces, and also divide the phase space: a cell has volume

Consider a collection of orbits starting from various points in phase space

・Entropy production rate…Kolmogorov-Sinai entropy SKS:

cell

RESULTS: INTERMEDIATE LYAPUNOV EXP. (V=43)

・Distribution of Lyapunov exp. is stable at large t, and then, significant portion of Lyapunov exponents  keeps positive in a robust way →chaotic & entropy production

・Instability spreads to many modes already at t=40.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 50 100 150 200

t=0t=40t=50

0

0.05

0.1

0.15

0.2

0 50 100 150 200

t=70t=100t=150

distribution of positive intermediate Lyapunov exponent

INTERMEDIATE LYAPUNOV EXPONENT IN LARGER VOLUME (V=83)

・For large volume, qualitative behavior does not change: → indicates that the entropy production occurs in infinite volume limit

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 200 400 600 800 1000 1200 1400 1600

t=0t=40t=50

0

0.05

0.1

0.15

0.2

0 200 400 600 800 1000 1200 1400 1600

t=70t=100t=150

DEPENDENCE OF LYAPUNOV EXPONENT TO VOLUME、KOLMOGOROV SINAÏ ENTROPY

The value of Lyapunov exp. in V=83 is always larger than that in V=43 for each mode → indication of entropy production in infinite volume

V=83

V=43

KS entropy/volume

ε=3.53×10-3

V=43: 0.079 @ t=150 V=83: 0.099 @ t=150

→ entropy production may occur in infinite volume limit

SUMMARY AND CONCLUSIONS

Background magnetic field + (tiny) fluctuation (simple model of realistic initial condition) → instability & chaotic behavior → entropy production in classical Yang-Mills dynamics

・Entropy production proceeds already in the CYM evolution. ・Instability is related to entropy production. … possible scenario of early thermalization ・Fluctuations on top of the background Glasma configulation is crucial for the entropy production.

TIME EVOLUTION OF GLUON COHERENT STATE AND ITS DECOHERENCE ENTROPY IN HEAVY-ION COLLISIONS

in collab. with T.Kunihiro (Kyoto Univ.), A.Ohnishi (YITP) and T.T.Takahashi (Gumma Nat. Coll. of Tech.)

(paper: in preparation)

CONTENTS

 Previous study：Kolmogorov-Sinaï entropy  Decoherence entropy：

time evolution of gluon coherent state from realistic initial condition of heavy-ion collisions and the calculation of decoherence entropy   from Glasma initial condition   from Modulated initial condition

 Conclusions and perspectives

重イオン衝突におけるエントロピー生成

Entropy history of a central Au+Au collision at top RHIC energy (from B.Mueller and A.Schaefer, Int. J. of Mod. Phys. E20, 2235-2267 (2011).)

多くのエントロピーが、熱化の前に生み出されている → 重イオン衝突直後のエントロピー生成に関して研究する

KOLMOGOROV-SINAÏ （KS）ENTROPY

 Entropy defined in Hamilton dynamics

: Lyapunov exp. |�Xi(t)| � e�it

・Lyapunov exp. … Index of instability - Weibel instability（Mrowczynski(1993)) - Nielsen-Olesen instability (Iwazaki(2008), Fujii&Itakura(2008)) ・Existence of many positive Lyapunov exponents in non-linear systems → System is chaotic (mixing property) → production of KS entropy

0

0.05

0.1

0.15

0.2

0 200 400 600 800 1000 1200 1400 1600

t=70t=100t=150

Disribution of Lyapunov exp.

Ref.) H.Iida, T.Kunihiro, B.Müller, A.Schäfer, T.T.Takahashi, arXiv:1304.1807, to appear in PRD.

We calculate ILE from background magnetic fields with fluctuations in CYM system → Distribution of Lyapunov exponents becomes stable and the number of positive Lyapunov exp. is the same order of that of the total DoF. → indication of production of KS entropy

However, calculation of KS is time-consuming…

Number of Lyapunov exp. = DoF of the system → We should evaluate the eigenvalues of Large matrix:

DoF=(V×(Nc2-1)×3×2)

In this study, we study decoherence entropy … entropy created by decoherence

Ref.) W.H.Zurek, Physics Today (1991).

Pure state mixed state

(density matrix) has off-diagonal components is diagonal

Vanishing of quantum coherence due to interactions between a system and a environment

　　 - Decoherence entropy in the initial stage of HIC (cf. H.T.Elze, NPB436, 213 (1995); B.Mueller and A.Schaefer, arXiv:hep-ph/0306309; PRC73, 054905 (2006))

・Assuming gluon fields as coherent states in init. stage of HIC :

・Probability of finding a n-particle state in a coherent state：

・Decoherence entropy is calculated as (Assuming incoherence between n-particle states)

Here, , and we substitute these expectation values by classical fields E, A calculated in CYM.

・Classical treatment of initial stage of HIC （CGC-Glasma） is good → We interpret the classical fields calculated from CYM as (quantum) coherent states. 　　（coherent states：states with minimum uncertainty, good correspondence to classical fields） ・We calculate of decoherence entropy with the assumption that the Fock states in coherent states become decoherent each other completely.

Concrete calculation：

( → )

CALCULATION WITH GLASMA INIT. CONDITION (GIC) Refs)L.McLerran and R.Venugopalan 1994; T.Lappi and L.McLerran 2006, … Romatschke and Venugopalan, PRD74 (2006) 045011. Fukushima and Gelis, Nucl. Phys. A874 (2012) 108-129.

0 20

40 60

80 100

120 0

20

40

60

80

100

120-2

-1.5-1

-0.5 0

0.5 1

1.5 2

’confA0001adj0001imu0001.dat’

-2-1.5-1-0.5 0 0.5 1 1.5 2

0 20

40 60

80 100

120

0 20

40 60

80 100

120

-20-15-10

-5 0 5

10 15

’lambdap0001adj0001.dat’

-20-15-10-5 0 5 10 15

Source

Solving Poisson eq.

・Calculation in nonexpanding geometry (V=203) ・We compare two cases: with and without longitudinal fluctuations（z-axis）

・profile in x-y plane:

… realistic gluon configuration in initial stage of HIC

0

50000

100000

150000

200000

250000

300000

0 50 100 150 200

Sdec: G-Bf20Sdec: G-B20

DECOHERENCE ENTROPY FROM GIC

Without longitudinal fluctuations

With longitudinal fluctuation

・Longitudinal fluctuations makes the entropy ten-times larger

Indicating the importance of longitudinal fluctuations from GIC for thermalization

t

DECOHERENCE ENTROPY WITH VARIOUS LONGITUDINAL FLUCTUATIONS

0

50000

100000

150000

200000

250000

300000

0 20 40 60 80 100 120 140

w=0.00001w=0.0001

w=0.001w=0.01

w=0.1

(w: amplitude of longitudinal fluctuations)

・The onset time of increasing of decoherence entropy depends on longitudinal fluctuations. ・The slopes are roughly the same for various fluctuations.

ISOTROPIZATION & DECOHERENCE ENTROPY

・longitudinal fluctuation → early isotropization ・increase of entropy and isotropization: … isotropization time and the saturation time of decoherence entropy is roughly the same

With longitudinal fluctuations Without longitudinal fluctuations

cf: decoherence entropy

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 50 100 150 200

E2perpE2long

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 50 100 150 200

E2perpE2long

0

50000

100000

150000

200000

250000

300000

0 50 100 150 200

Sdec: G-Bf20Sdec: G-B20

with longitudinal fluctuations

without long. fluctuations

CALCULATION IN MODULATED INIT. COND. (MIC)

Fluctuation (noise)

Magnetic field from spatial modulation (neglecting noise)

(+ fluctuation)

magnetic field

Bx = �2m�

Nz�2 sin

�2nx�

Nx

�cos

�2mz�

Nz

�� 0, By = 0

Bz =2�

Nx�1 cos

�2x�

Nx

�+

2n�

Nx�2 cos

�2nx�

Nx

�sin

�2mz�

Nz

�� 2�

Nx�1 cos

�2x�

Nx

�

�1 � �2

x

zy

→ glasma-like

※The initial condition has the essential features of GIC.

DECOHERENCE ENTROPY FROM MIC

0

50000

100000

150000

200000

250000

300000

0 50 100 150 200

Sdec: M-Bf20Sdec: M-B20

without fluctuations

with fluctuations

・without fluctuations … no increase of entropy ・with fluctuations … entropy increases rapidly around t〜20

Existence of fluctuations in background magnetic fields is crucial for the growth of entropy → fluctuation is important for thermalization … same feature as KS entropy

t

ISOTROPIZATION & DECOHERENCE ENTROPY

・Fluctuations induces isotropization ・Strong correlation between increase of entropy and isotropization

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 50 100 150 200

E2perpE2long

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0 50 100 150 200

E2perpE2long

with fluctuations without fluctuations

0

50000

100000

150000

200000

250000

300000

0 50 100 150 200

Sdec: M-Bf20Sdec: M-B20

cf: decoherence entropy

CONCLUSIONS AND PERSPECTIVES

  Time dependence of gluon coherent states are calculated from CYM, assuming that the classical gluon fields are regarded as coherent states, and decoherence entropy is calculated from the coherent states.

  Longitudinal fluctuations are important for entropy creation from Glasma initial condition

  Longitudinal fluctuations leads early isotropization, and there is a strong correlation b.w. entropy creation and isotropization.

Perspectives： ・Calculation in expanding geometry … suppression of entropy production? ・Quantitative estimation of entropy production

NON-COMPACT FORMALISM IN CYM

  (A,E) formalism is used lather than (U,E) formalism … Gauge inv. is violated in EOM by discretization.  How harmful is the violation? The violation is reflected in the Lyapunov spectrum: 1/3 of Lyapunov exp. are zero, if gauge inv. (Gauss’ law) is kept exactly.

-0.1

-0.05

0

0.05

0.1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000x

mod. init. cond., V-83, t=150

only 2.3% of the sum of positive ILE

cf) Lyapunov spectrum (V=23) Gong, Phys.Rev.D49, 2642 (1994).

Good gauge invariance !

 横軸の単位gμ   decoherence entropyの増加率は\sqrt{gB}?  WeibelとNielsen Olesen？