numerical modelling of bec *

Numerical modelling of BEC **

Numerical modelling of BEC **

Oleg UtyuzhOleg Utyuzh

The Andrzej Sołtan Institute for Nuclear Studies (SINS), Warsaw, Poland

* In collaboration with G.Wilk and Z.Wlodarczyk

22KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh

High-Energy collisions High-Energy collisions

0

0

K K

K K

0K

p

p

p

p


0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

Quantum Correlations (QS)Quantum Correlations (QS)

1 212 1 2( ) ( )p pA x x 1 22 1( ) ( )p px x

x1

x2

p1

p2

12 1 2

2 1 222

( , )( )

( , ), ref

N p pC Q

N p pp p

BE enhancement

2

2 1 2 12 11 2 2( , ) ~ ( , ( , ))N p p A x x x dx


2 1 22 1 2

2 1 2

( , )( { , })

( , )

BE

ref

N p pC Q p p

N p p 2 1 2

2 1 22 1 2

( , )( { , })

( , )

BE

ref

N p pC Q p p

N p p

CorrelationCorrelation functionfunction (1D) – (1D) – sourcesource sizesize

24

2 ( ) 1 ( ) iQxC Q d x x e

x1

x2

p1

p2

R sourcesize

12

( )QR

R

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2( )C Q1~

R

R.Hunbury Brown and Twiss, Nature 178 (1956) 1046G.Goldhaber, S.Goldhaber, W.Lee and A.Pais, Phys.Rev 120 (1960) 300

k r Ψ ( ) e i- kr

2 1 21 ( ), ) )( (ρ x x ρ x ρ x×=


0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2( )C Q

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2( )C Q

2 1 22 1 2

2 1 2

( , )( { , })

( , )

BE

ref

N p pC Q p p

N p p 2 1 2

2 1 22 1 2

( , )( { , })

( , )

BE

ref

N p pC Q p p

N p p

CorrelationCorrelation functionfunction (1D) - chaoticity (1D) - chaoticity

24

2 ( ) 1 ( ) iQxC Q d x x e 12

( )QRλ

x1

x2

p1

p2

12

( )QR

R

chaoticitchaoticityy

• resonancesresonances• finalfinal statestate interactionsinteractions• flowsflows• particlesparticles mismisinindificationdification• momentum resolutionmomentum resolution• ......

1


( )1 1

1( ; ) exp exp

!

N N

i ipi i

i iir rp ix x pN

W. Zajc, Phys. Rev. D35 (1987) 3396

NNππ-particle state-particle state

1r

ir

2r

1p

2p2x

1x

ix

ip


W. Zajc, Phys. Rev. D35 (1987) 3396

2

( ) npp r r d r

3

3*

ii pdp

pd N

..., ,...old iP p

*..., ,...ew in pP

FOR 1,i N

Metropolisalgorithm

ip fixed

*ACCEPT with min 1, new

oi

ld

PProbp

P

NEXT i

speckles

specklespeckless

Numerical symmetrization – (A)Numerical symmetrization – (A)

1

( )1

1exp exp

!

N N

i ipi

ii

ir i p ix pN

r

TIME !!!TIME !!!


1(max)ir C 1(max)ir C

J. Cramer, Univ. of Washigton preprint

(1996 unpublished)

J. Cramer, Univ. of Washigton preprint

(1996 unpublished)

1(max)iC

1( , )Xi iC p p

Numerical symmetrization – (B)Numerical symmetrization – (B)

Monte-Carlo Monte-Carlo rejectionrejection

Monte-Carlo Monte-Carlo rejectionrejection

Xp

1( , )Xi iC p p

1(max) ( 1)!iC i

1PICK UP 0 (max)iX C

SELECT FROM ( )XXp f p

1ACCEPT IF ( , )i iX Xp pC p X

NEXT i

clustersclusters

TIME !!!TIME !!!

10pN

p


PermanentPermanent

1 1 1 1 1

2 2 2 2 2

1 1 1 1 1

1 2 1

1 2 1

1 2 1

1 2 1

1 2 1

i i i i i

N N N N N

N N N N N

i N N

i N N

i N N

i N N

i N N

1 1

2 2

1 2

1 1

H. Merlitz, D. Pelte, Z. Phys. A357 (1997) 175

Numerical symmetrization – (C)Numerical symmetrization – (C)

i i

1 11

1

N N

N N

N N

N N

TIME !!!TIME !!!2.5i j pp p

Fact

orizat

ion

Fact

orizat

ion

Fact

orizat

ion

Fact

orizat

ion

1

( )1

1exp exp

!

N N

i ipi

ii

ir i p ix pN

r

2

2

( )( ,0) ~ exp ( )i

i i ip

P pp iX P p


cos H. Merlitz, D. Pelte, Z. Phys. A357 (1997) 175

2.5i j pP P

2

2

( )( ,0) ~ exp ( )i

i i ip

P pp iX P p

(1) ( )

1( ) ( )... ( )

!i N Np p p

N

3( ,0) ( ,0)exp( )i ip d x x ipx

clusters


(1) 1 ( )

1( ) ( )... ( )

!N Np r r r

N

1

( )1

1exp exp

!

N N

i ipi

ii

ir i p ix pN

r


Problem with numerical symmetrization Problem with numerical symmetrization ……

TIME !!!TIME !!!

Existing ways out:

replace modeling by simulations … (afterburners)

Examples:

• shifting of momenta

• weighting proceduresweighting procedures

Problems:

• changing of initial distributions – changing of physicschanging of initial distributions – changing of physics

• exampleexample O.V.Utyuzh, G.Wilk and Z.Wlodarczyk; Phys. Lett. B522 (2001) 273 andActa Phys. Polon. B33 (2002) 2681.


Numerical symmetrizationNumerical symmetrization

CLUST

ERS

CLUST

ERS

SPECKLES

SPECKLES

SSTTAATTEESS

BUNCHESBUNCHES

CLANSCLANS

CELLSCELLS


1( ; )

1 11

nn

P nn n

1 2

1, ,...,

( ; )

1

i

i

k

n

nn n n i

n

kP n k

n

k

( ; )!

n

nnP n e

n

1( ; )

1

n

n k

n

kn kP n

n n

k

k

2 ( )D k2 ( ) 1

nD n

kk

2 ( 11)D n n

2 ( )D n

EEC’s – A.D. 1996EEC’s – A.D. 1996

M. Biyajima, N. Suzuki, G. Wilk, Z. Wlodarczyk, Phys. Lett. B386 (1996) 297

EElementary lementary EEmitting mitting CCells (ells (EECEEC))


mean number

of - EEC C

Spairs

totpairs

n

N

21 1( 1) ( 1)

2 2Spairs

totpairs

n

N C C n C n n


coshi T im y

MMaximalization of aximalization of IInformation nformation EEntropy (ntropy (MIEMIE))

phasephase spacespace (1D) (1D)

( ) ( )1i i in n

ii

P eZ

( ) ( )lni i

i

n ni i

i n

S P P MIEMIEMIEMIE

constiy

( ), ( )yy y

( )i

i

ni i

i n

n n P ( )i

i

ni i i

i n

E n P ( ) 1i

i

ni

n

P

min

1( )

2iy y i y T. Osada, M. Maruyama and F. Takagi, Phys. Rev. D59 (1999)

014024


MIEMIE - Results - Results


( ) e!

N

BltzP NN

( ) (1 ) NBEP N

( )N i ii

x { , }

1( )

!N i jP i j i

xN

1

( ) e 1iE

kTin E

ssymmetrizationymmetrization**ssymmetrizationymmetrization**

non-identicalnon-identical VSVS identicalidentical BoltzmannBoltzmann VSVS Bose-EinsteinBose-Einstein

QuantumQuantum statisticsstatisticsQuantumQuantum statisticsstatistics

GEOMETRICALGEOMETRICAL

K.Zalewski, Nucl. Phys. B (Proc. Suppl. ) 74 (1999) 65


phasephase spacespace (1D) (1D)

*O. Utyuzh, G. Wilk, Z. Włodarczyk, Acta Phys. Hung. (Heavy Ion Physics) A25 (2006) 83

0

0 eE

kT

0

0 eE

kT

cell formationuntil first

failure( ) (1 ) N

BEP N ( ) (1 ) NBEP N

Quantum Quantum Clan model (1d-QCM)Clan model (1d-QCM)

2

2

( )

2( )cell

E

E E

cellg E e

2

2

( )

2( )cell

E

E E

cellg E e

smearing

particle energyin the cells

EECEEC


Algorithm ...Algorithm ...

PICK UP 0 1RAND

10SELECT FROM ( )f EE

10- /

0ADD particle IF E TP Pe RAND spaceE 10E

21021

1

21SELECT FROM ( ) E

E E

g E eE

20E

1 Nf N P P

1EEC 2EEC

probability of particle cellN

cell formationuntil first

failure

1

PN

P

/

1

1E TN E

e

/

0E TP Pe


Quantum Quantum Clan modelClan model

HadronicSource

Ind

epen

den

t p

rod

uct

ion

Ind

epen

den

t p

rod

uct

ion

( )PAP N

-1

1 [ (1 ) / ]( )

1 !

N jN

Pólya Aepplij

N p pP N e p

j j

Bo

se-E

inst

ein

Bo

se-E

inst

ein

Bo

se-E

inst

ein

Bo

se-E

inst

ein

Bo

se-E

inst

ein

Bo

se-E

inst

ein

1EEC

cellNEEC

iEEC

O. Utyuzh, G. Wilk and Z. Włodarczyk, Acta Phys. Hung.

A25 (2006) 83


Some results …Some results …

( ) ( )cell partP N P n

0 5 10 15 20

100

101

102

103

104

P(N

cell)

Ncell

<Ncell

> = 6.28, N = 1.53

<Ncell

> = 6.30, N = 1.57

0 5 10 15 20 25 30 35 40 45 50 551E-6

1E-5

1E-4

1E-3

0.01

0.1

DELPHI [email protected] GeV <nch

>=20.71, n=6.28

T=3.5 GeV, P0=0.7, =0.3*T; <n

ch>=20.87,

n=6.35

T=3.7 GeV, P0=0.7, =0.1*T; <n

ch>=20.76,

n=6.76

P(n

ch)

nch

0 5 10 15 20 25 3010-1

100

101

102

103

104

105

106

<np> = 1.53,

n = 1.02

<np> = 1.57,

n = 1.07

P(n

p)

np


Results …Results … ( (first application to first application to Simple Cascade Simple Cascade ModelModel))

Results …Results … ( (first application to first application to Simple Cascade Simple Cascade ModelModel))

0.0 0.2 0.4 0.6 0.8 1.00.8

1.0

1.2

1.4

1.6

1.8

P = 0.5C2(Q)

Q [GeV]

0.8

1.0

1.2

1.4

1.6

1.8

2.0

P = 0.23C2(Q)

1 101

10

Fq

Mbin

1

10F

q

2 1 22

2 1 2

( , )

( , )

BEC

ref

N p pC

N p p

1

1

( ) ( 1) ( 1)q M

q m m mqm

MF y n n n q

N


MIE vs MIE vs 1d-1d-QCMQCM

phasephase spacespace (1D) (1D)y-spacey-space

consty

20

2

( )

E

E E

E e

( )

1ii E

ne

phasephase spacespace (1D) (1D)E-spaceE-space


1 11

1

N N

N N

N N

N N

i i

1 1

2 2

1 2

1 1

( )1

1( ) ~ exp

!

n

i ipi

r i p rn

2| ( ) |N pP r

What we are proposing … What we are proposing … symmetrization symmetrization

1 1 2 2 1 2 2 12

1( )

2ip r ip r ip r ip r

np r e e e e

2 1 2 1 2cosP p p r r

2 - particle

approximation

2 - particle

approximation 1122

33

44

55

1EEC

iEEC

cellNEEC


model (3D)model (3D)

p-Spacep-Space x-Spacex-Space

x·x·p-correlationsp-correlations

1+cos(δx δp)

symetrizationsymetrizationplane waves

2626KrakówKraków 2006 2006KrakówKraków 2006 2006Oleg UtyuzhOleg UtyuzhOleg UtyuzhOleg Utyuzh

W T T P0 <nch> σn <npart> <ncell>

45.645.6 3.53.5 0.30.3 1.01.055

0.0.77

10.8310.83 4.294.2922

1.54/1.01.54/1.022

3.23/1.613.23/1.61

91.291.2 3.53.5 0.30.3 1.01.055

0.0.77

20.8820.88 6.376.3700

1.55/1.01.55/1.055

6.31/2.396.31/2.39

182.182.44

3.53.5 0.30.3 1.01.055

0.0.77

41.9741.97 8.988.9855

1.57/1.01.57/1.088

12.60/3.212.60/3.299

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.9

1.0

1.1

1.2

1.3

C2(Q

i)

Qx,z

, [GeV]

Rsphere

= 1.0 fm, psphere

T=3.5 GeV, = 0.3*T GeV, P=0.7*exp(...) W = 46.5 GeV W = 91.2 GeV W = 182.4 GeV

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

C2(Q

inv)

Qinv

[GeV]

Rsphere

= 1.0 fm, psphere

T=3.5 GeV, = 0.3*T GeV, P=0.7*exp(...) W = 46.5 GeV W = 91.2 GeV W = 182.4 GeV

W - dependence


T T P0 <nch> σn <npart> <ncell>

3.13.1 0.30.3 0.930.93 0.70.7 23.3423.34 6.6966.696 1.56/1.01.56/1.044

7.12/2.57.12/2.500

3.53.5 0.30.3 1.051.05 0.70.7 20.8820.88 6.3706.370 1.55/1.01.55/1.055

6.31/2.36.31/2.399

3.93.9 0.30.3 1.171.17 0.70.7 18.8618.86 6.0756.075 1.57/1.01.57/1.077

5.72/2.45.72/2.422

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.9

1.0

1.1

1.2

C2(Q

i)

Qx,z

, [GeV]

Rsphere

= 1.0 fm, psphere

T = 3.1 GeV | T = 3.5 GeV > = 0.3*T GeV, P=0.7*exp(...) T = 3.9 GeV |

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

C2(Q

inv)

Qinv

[GeV]

Rsphere

= 1.0 fm, psphere

T = 3.1 GeV | T = 3.5 GeV > = 0.3*T GeV, P=0.7*exp(...) T = 3.9 GeV |

T - dependence



3.3.55

0.30.3 1.051.05 0.60.6 19.9119.91 5.7145.714 1.41/0.71.41/0.700

6.74/2.46.74/2.466

3.3.55

0.30.3 1.051.05 0.70.7 20.8820.88 6.3706.370 1.55/1.01.55/1.055

6.31/2.36.31/2.399

3.3.55

0.30.3 1.051.05 0.80.8 22.1522.15 7.3017.301 1.79/1.51.79/1.555

5.89/2.25.89/2.266

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.9

1.0

1.1

1.2

C2(Q

i)

Qx,z

, [GeV]

Rsphere

= 1.0 fm, psphere

P0 = 0.6 |

P0 = 0.7 > T = 3.5 GeV, =0.3*T GeV

P0 = 0.8 |

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

C2(Q

inv)

Qinv

[GeV]

Rsphere

= 1.0 fm, psphere

P0 = 0.6 |

P0 = 0.7 > T = 3.5 GeV, =0.3*T GeV

P0 = 0.8 |

P0 - dependence


- dependence

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.9

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

C2(Q

inv)

Qinv

[GeV]

Rsphere

= 1.0 fm, psphere

0 = 0.1 |

0 = 0.3 > T = 3.5 GeV, P=0.7*exp(...)

0 = 0.5 |

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.9

1.0

1.1

1.2

1.3

C2(Q

i)

Qx,z

, [GeV]

Rsphere

= 1.0 fm, psphere

0 = 0.1 |

0 = 0.3 > T = 3.5 GeV, P=0.7*exp(...)

0 = 0.5 |


3.3.55

0.10.1 0.350.35 0.70.7 21.8421.84 6.9276.927 1.57/1.01.57/1.077

6.62/2.46.62/2.444

3.3.55

0.30.3 1.051.05 0.70.7 20.8820.88 6.3706.370 1.55/1.01.55/1.055

6.31/2.36.31/2.399

3.3.55

0.50.5 1.751.75 0.70.7 19.6519.65 5.8165.816 1.56/1.01.56/1.044

5.99/2.25.99/2.299


-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00

1

2

3

N2(U

),C

2(U

)

U [= dP]

pi, p

j - uniform

pi, p

j - uniform

COS(dP*dR) < 2*Rand - 1 ( / )

20

20

sin ( )1

( )

R p

R p

20

20

sin ( )1

( )

R p

R p

How to model numerically How to model numerically COS(…) COS(…) ??

2 1 2 1 2cosP p p r r


-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00

1

2

3

N2(U

),C

2(U

)

U [= dP]

pi, p

j uniform with selection

COS(dP*dR) < 2*Rand - 1 dP from COS(), dP = /dR

|R| < 1.0 fm, |P| < 1.0 GeV

0 0 003

0

3 sin( ) ( ) cos( )1 cos( )

( )

R p R p R pR p

R p

0 0 003

0

3 sin( ) ( ) cos( )1 cos( )

( )

R p R p R pR p

R p

20

20

sin ( )1

( )

R p

R p

20

20

sin ( )1

( )

R p

R p

( ) ( )x X

Xf x dX f X p

x

( ) ( )x X

Xf x dX f X p

x

( ) (... ) ( )x Xf x dXf X p x X δx

2-ways of modeling of 2-ways of modeling of COS(…)COS(…) … …


1( 1)( 2)

2additional links

N N

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.5

1.0

1.5

2.0

C2(U

)

U [= dP]

100 particles 20 particles 4 particles 2 particles theory for 2 particles

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.5

1.0

1.5

2.0

C2(U

)

U [= dP]

100 particles 20 particles 4 particles 2 particles theory for 2 particles theory for 4 particles

1122

33

44

55

2

02 2

0

sin ( )2( )

2 ( )N N R p

C pN R p

2

2 02 2

0

sin ( )( ) 1

( )

R pC p

R p

2

2 02 2

0

sin ( )( ) 1

( )

R pC p

R p

3N

Pairs counting …Pairs counting …

2

02 2

0

sin ( )( ) 1

( )N R p

C pR p

pairs misidentification effect ???pairs misidentification effect ???


-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.5

1.0

1.5

2.0

C2(U

)

U [= dP]

Np = 2

Np = 6, 2-p relations





1122

33

44

55

6611

22

33

44

55

66

NN -particles via 2-particles -particles via 2-particles

1(2)

11

1 cos( )n

n n in ini

P P x

1

(2)1

1

1 cos( )n

n n in ini

P P x


( )1

1( ) ~ exp

!

n

i ipi

r i p rn

!

( ) '( )' 1 1

21 cos

!

n n

i i ii

P p r rn

max

2 11 !( ! 1) !

! 2P n n n

n

1,...,i np p

True True NN -particles -particles


-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.5

1.0

1.5

2.0

C2(U

)

U [= dP]

2 particles 4- (via 2 particles) theory for 2 particles 4- (via 4 particles)

1(2)

11

1 cos( )n

n n in ini

P P x

!

( )( ) '( )

' 1 1

21 cos

!

n nn

n i ii

P r rn

1

( )...(max) !

n

nn p pP n 1

( )...(max) !

n

nn p pP n

1

1( 1)(2) 2

...(max) 2n

n n

n p pP

1

1( 1)(2) 2

...(max) 2n

n n

n p pP

<


-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.5

1.0

1.5

2.0

C2(U

)

U [= dP]

uniform cantor set (s=0.333) cantor set (s=0.111)

ssss

Fractal sourceFractal source


1 10 100

100

101

102

103 s = 0.333 q=2, =0.35109 q=3, =0.73765 q=4, =1.14699 q=5, =1.56974

Fq(M

)

M [= Y/y]1 10 100

100

101

102

103 s = 0.444 q=2, =0.12256 q=3, =0.24102 q=4, =0.39913 q=5, =0.58540

Fq(M

)

M [= Y/y]

-1.0 -0.5 0.0 0.5 1.00

1

2

3

4

5 Cantor set (s=1/3)

(x)

x [fm]

-1.0 -0.5 0.0 0.5 1.00

1

2

3

4

5 Cantor set (s=0.444)

(x)

x [fm]

Fractal sourceFractal source


0.0 0.2 0.4 0.6 0.8 1.00.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

BEC with cells (P=0.7*e-E/3.5) BEC w/o cells

C2(

E)

E [GeV]

BE statistics => cells ?... BE statistics => cells ?...

A. Kisiel et al., Comput. Phys. Commun. 174 (2006) 669


qq qqqqqq

qq qq

qq qqqqqq

qq qq

stimulated

emission

3 ( )d x

particles

bunching

Possible further applications …Possible further applications …


SummarySummary

BEC =BEC = CELLSCELLSGEOMETRIC

DISTRIBUTION

GEOMETRIC

DISTRIBUTION

++


ProblemProblem ofof λλ interpretationsinterpretations Problem of normalization ofProblem of normalization of CC22(Q)(Q) Single-particleSingle-particle spectra modifications

Instead of sInstead of summaryummary … …

resonances

final state interactions

flows

particles misindentification

momentum resolution

...

1

1

1

PHYSICSEECN

0 10 20 30 40 50 60

1E-5

1E-4

1E-3

0.01

0.1

P(n

ch)

nch

BEC Boltzmann


Back-Up SlidesBack-Up Slides


‘If one insists on representing photons by wave packets and demands

an explanation in those terms of the extra fluctuation, such an

explanation can be given. But I shall have to use language which

ought, as a rule, to be used warily. Think, then, of a stream of wave

packets, each about c/ long, in a random sequence. There is a

certain probability that two such trains accidentally overlap. When

this occurs they interfere and one may find (to speak rather loosely)

four photons, or none, or something in between as a result. It is

proper to speak of interference in this situation because the

conditions of the experiment are just such as will ensure that these

photons are in the same quantum state. To such interference one may

ascribe the “abnormal” density fluctuations in any assemblage of

bosons’.E. M. Purcell,

Nature 178 (1956) 1449-1450

Quantum Optics - pQuantum Optics - particles articles bunchings bunchings ……


Quantum Optics - pQuantum Optics - particles articles bunchings bunchings ……

50%

21

3C

BosonsBosons

bunching

correpositi latve ions

FermionsFermions correnegati latve ions

anti-bunching

M. Henny et. al. , Science 284 (1999) 296

0C

0C


- coherent state n

n n 2( ) d

22 21( ) ( )

!

nn n e d P n

n

2

, ,

( )m n m n

n m n m d

21

pure noise

1( )

thermal light

nen

1( )1

n

n

nP n

n

Poisson

transformation

Roy J. Glauber, nucl-th/0604021

Quantum Optics and Heavy Ion PhysicsQuantum Optics and Heavy Ion Physics


0.8

0.9

1.0

1.1

1.2

1.3

1.4

0.0 0.2 0.4 0.6 0.8 1.0

-0.030.000.03

C2(

E)

E [GeV]

0.8

0.9

1.0

1.1

1.2

1.3

1.4

0.0 0.2 0.4 0.6 0.8 1.0-0.030.000.03

C2(

E)

E [GeV]


B.B. Back, et al. (PHOBOS Coll.), Nucl.Phys. A774 (2006) 631-634B.B. Back, et al. (PHOBOS Coll.), Nucl.Phys. A774 (2006) 631-634

B.B. Back, et al. (PHOBOS Coll.), Nucl.Phys. A774 (2006) 631-634B.B. Back, et al. (PHOBOS Coll.), Nucl.Phys. A774 (2006) 631-634


2

1 TBjorken

m dN

R dy

J.D. Bjorken, Phys. Rev. D 27 (1983) 140


“ From these comparisons one can conclude that both MC models reproduce the data well while neither of them is particularly preferred. The perturbative parton shower, on which both MC models are based, seems to play an important role in the origin of the dynamical fluctuations and correlations in e+e− annihilation. The observed differences between the two MC descriptions indicate that the last steps of the hadronization process are not described correctly [2]. Contributions from additional mechanisms to the observed fluctuations and cor-relations are not excluded. “

G.Abbiendi et al., (OPAL Coll.) Eur.Phys.J. C11 (1999) 239-250


J.A.Casado and S.Daté, Phys. Lett. B344 (1995) 441J.A.Casado and S.Daté, Phys. Lett. B344 (1995) 441


( )U x

x

( ) ipxp x e

20

0 2

( )

x

X xipX

p x e



0,0 0,5 1,0 1,5 2,00,9

1,0

1,1

1,2

1,3

C2(Q

i)

Qi [GeV]

Qinv

QE Q

Px

P=0.7*e-E/T, T=3.5 GeV, =0*T

numerical modelling of bec *

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