* Numerical modelling of BEC * Oleg Utyuzh The Andrzej Sołtan Institute for Nuclear Studies (SINS),...

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 0.5 1.0 1.5 2.00.0

Quantum Correlations (QS)Quantum Correlations (QS)

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

12 1 2

2 1 222

( , )( )

( , ), ref

N p pC Q

N p pp p

BE enhancement

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

2 1 22 1 2

( , )( { , })

N p pC Q p p

N p p 2 1 2

2 1 22 1 2

( , )( { , })

N p pC Q p p

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

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

R sourcesize

0.0 0.5 1.0 1.5 2.00.0

2( )C Q1~

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

2( )C Q

0.0 0.5 1.0 1.5 2.00.0

2( )C Q

2 1 22 1 2

( , )( { , })

N p pC Q p p

N p p 2 1 2

2 1 22 1 2

( , )( { , })

N p pC Q p p

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

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

( )QRλ

chaoticitchaoticityy

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

( )1 1

1( ; ) exp exp

i ipi i

i iir rp ix x pN

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

NNππ-particle state-particle state

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

( ) npp r r d r

ii pdp

..., ,...old iP p

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

FOR 1,i N

Metropolisalgorithm

ip fixed

*ACCEPT with min 1, new

PProbp

NEXT i

speckles

specklespeckless

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

1exp exp

ir i p ix pN

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(1996 unpublished)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(1996 unpublished)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

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 !!!

PermanentPermanent

1 1 1 1 1

2 2 2 2 2

1 1 1 1 1

i i i i i

N N N N N

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

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

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

orizat

1exp exp

ir i p ix pN

( )( ,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

( )( ,0) ~ exp ( )i

i i ip

P pp iX P p

(1) ( )

1( ) ( )... ( )

!i N Np p p

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

clusters

(1) 1 ( )

1( ) ( )... ( )

!N Np r r r

1exp exp

ir i p ix pN

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

SPECKLES

SSTTAATTEESS

BUNCHESBUNCHES

CLANSCLANS

CELLSCELLS

1( ; )

P nn n

1, ,...,

nn n n i

kP n k

( ; )!

nnP n e

1( ; )

kn kP n

2 ( )D k2 ( ) 1

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

21 1( 1) ( 1)

2 2Spairs

totpairs

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

( ) ( )lni i

n ni i

S P P MIEMIEMIEMIE

constiy

( ), ( )yy y

n n P ( )i

ni i i

E n P ( ) 1i

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

014024

MIEMIE - Results - Results

( ) e!

BltzP NN

( ) (1 ) NBEP N

( )N i ii

x { , }

!N i jP i j i

( ) 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

cell formationuntil first

failure( ) (1 ) N

BEP N ( ) (1 ) NBEP N

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

2( )cell

cellg E e

2( )cell

cellg E e

smearing

particle energyin the cells

EECEEC

Algorithm ...Algorithm ...

PICK UP 0 1RAND

10SELECT FROM ( )f EE

0ADD particle IF E TP Pe RAND spaceE 10E

21SELECT FROM ( ) E

g E eE

1 Nf N P P

1EEC 2EEC

probability of particle cellN

cell formationuntil first

failure

1E TN E

0E TP Pe

Quantum Quantum Clan modelClan model

HadronicSource

( )PAP N

1 [ (1 ) / ]( )

Pólya Aepplij

N p pP N e p

cellNEEC

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

<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

DELPHI e+e-@91.2 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

0 5 10 15 20 25 3010-1

<np> = 1.53,

n = 1.02

<np> = 1.57,

n = 1.07

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

P = 0.5C2(Q)

Q [GeV]

P = 0.23C2(Q)

2 1 22

N p pC

( ) ( 1) ( 1)q M

q m m mqm

MF y n n n q

MIE vs MIE vs 1d-1d-QCMQCM

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

consty

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

1( ) ~ exp

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

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

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

, [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

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

, [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

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

, [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

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

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

, [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

U [= dP]

j - uniform

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

sin ( )1

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

U [= dP]

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

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

R p R p R pR p

0 0 003

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

R p R p R pR p

sin ( )1

( ) ( )x X

Xf x dX f X p

( ) ( )x X

Xf x dX f X p

( ) (... ) ( )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

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

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

U [= dP]

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

sin ( )2( )

2 ( )N N R p

C pN R p

2 02 2

sin ( )( ) 1

R pC p

2 02 2

sin ( )( ) 1

R pC p

Pairs counting …Pairs counting …

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

U [= dP]

Np = 2

Np = 6, 2-p relations

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

1 cos( )n

n n in ini

1 cos( )n

n n in ini

1( ) ~ exp

r i p rn

( ) '( )' 1 1

21 cos

i i ii

P p r rn

2 11 !( ! 1) !

! 2P 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

U [= dP]

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

1 cos( )n

n n in ini

( )( ) '( )

21 cos

n i ii

P r rn

( )...(max) !

nn p pP n 1

( )...(max) !

nn p pP n

1( 1)(2) 2

...(max) 2n

n p pP

1( 1)(2) 2

...(max) 2n

n p pP

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

U [= dP]

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

Fractal sourceFractal source

1 10 100

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

M [= Y/y]1 10 100

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

M [= Y/y]

-1.0 -0.5 0.0 0.5 1.00

5 Cantor set (s=1/3)

x [fm]

-1.0 -0.5 0.0 0.5 1.00

5 Cantor set (s=0.444)

x [fm]

Fractal sourceFractal source

0.0 0.2 0.4 0.6 0.8 1.00.7

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

E [GeV]

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

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

qq qqqqqq

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

particles misindentification

momentum resolution

PHYSICSEECN

0 10 20 30 40 50 60

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 ……

BosonsBosons

bunching

correpositi latve ions

FermionsFermions correnegati latve ions

anti-bunching

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

- coherent state n

n n 2( ) d

22 21( ) ( )

nn n e d P n

( )m n m n

n m n m d

pure noise

thermal light

Poisson

transformation

Roy J. Glauber, nucl-th/0604021

Quantum Optics and Heavy Ion PhysicsQuantum Optics and Heavy Ion Physics

0.0 0.2 0.4 0.6 0.8 1.0

-0.030.000.03

E [GeV]

0.0 0.2 0.4 0.6 0.8 1.0-0.030.000.03

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

1 TBjorken

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

( ) ipxp x e

X xipX

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

Qi [GeV]

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

* Numerical modelling of BEC * Oleg Utyuzh The Andrzej Sołtan Institute for Nuclear Studies (SINS),...

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