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
33KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
44KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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×=
55KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
66KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
( )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
77KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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 !!!
88KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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(1996 unpublished)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
99KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
1010KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
Numerical symmetrization – (C)Numerical symmetrization – (C)
(1) 1 ( )
1( ) ( )... ( )
!N Np r r r
N
1
( )1
1exp exp
!
N N
i ipi
ii
ir i p ix pN
r
1111KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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.
1212KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
Numerical symmetrizationNumerical symmetrization
CLUST
ERS
CLUST
ERS
SPECKLES
SPECKLES
SSTTAATTEESS
BUNCHESBUNCHES
CLANSCLANS
CELLSCELLS
1313KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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))
1414KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
mean number
of - EEC C
Spairs
totpairs
n
N
21 1( 1) ( 1)
2 2Spairs
totpairs
n
N C C n C n n
1515KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
1616KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
MIEMIE - Results - Results
1717KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
( ) 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
1818KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
1919KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
2020KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
2121KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
2222KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
2323KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
2424KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
2525KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
2727KrakówKraków 2006 2006KrakówKraków 2006 2006Oleg UtyuzhOleg UtyuzhOleg UtyuzhOleg Utyuzh
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
2828KrakówKraków 2006 2006KrakówKraków 2006 2006Oleg UtyuzhOleg UtyuzhOleg UtyuzhOleg Utyuzh
T T P0 <nch> σn <npart> <ncell>
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
2929KrakówKraków 2006 2006KrakówKraków 2006 2006Oleg UtyuzhOleg UtyuzhOleg UtyuzhOleg Utyuzh
- 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 |
T T P0 <nch> σn <npart> <ncell>
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
3030KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
-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
3131KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
-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(…) … …
3232KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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 ???
3333KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
-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
Np = 6, 3-p relations
Np = 6, 4-p relations
Np = 6, 5-p relations
Np = 6, 6-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
3434KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
( )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
3535KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
-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
<
3636KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
-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
3737KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
3838KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
3939KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
qq qqqqqq
qq qq
qq qqqqqq
qq qq
stimulated
emission
3 ( )d x
particles
bunching
Possible further applications …Possible further applications …
4040KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
SummarySummary
BEC =BEC = CELLSCELLSGEOMETRIC
DISTRIBUTION
GEOMETRIC
DISTRIBUTION
++
4141KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
4242KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
Back-Up SlidesBack-Up Slides
4343KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
‘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 ……
4444KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
4545KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
- 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
4646KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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]
4747KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
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
4848KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
2
1 TBjorken
m dN
R dy
J.D. Bjorken, Phys. Rev. D 27 (1983) 140
4949KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
“ 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
5050KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
5151KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
5252KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
J.A.Casado and S.Daté, Phys. Lett. B344 (1995) 441J.A.Casado and S.Daté, Phys. Lett. B344 (1995) 441
5353KrakówKraków 2006 2006Oleg UtyuzhOleg Utyuzh
( )U x
x
( ) ipxp x e
20
0 2
( )
x
X xipX
p x e
Numerical symmetrization – (C)Numerical symmetrization – (C)
5454KrakówKraków 2006 2006KrakówKraków 2006 2006Oleg UtyuzhOleg UtyuzhOleg UtyuzhOleg Utyuzh
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
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