hadron multiplicity distribution in non extensive statistics
DESCRIPTION
Hadron Multiplicity Distribution in Non Extensive Statistics. Carlos E. Aguiar Takeshi Kodama UFRJ. Non Extensive Statistics. Tsallis entropy:. Non extensivity:. q-biased probabilities:. q-biased averages:. Tsallis Distribution. Variational principle:. Probability distribution:. - PowerPoint PPT PresentationTRANSCRIPT
Hadron Multiplicity Distribution in
Non Extensive Statistics
Carlos E. Aguiar
Takeshi Kodama
UFRJ
Non Extensive Statistics
1q
p1S a
qa
a
qa
qa
a p
pp~
aa
a
aa
a
Np~N
Ep~E
Tsallis entropy:
q-biasedprobabilities:
Non extensivity:
)B(S)A(S)1q()B(S)A(S)BA(S
q-biasedaverages:
Tsallis Distribution
0)p(NES aa00
qaaq
qa )NE(exp
Z
1p~
)1q/(1q ]x)1q(1[)x(exp
a
qaaqq )NE(expZ
S)1q(1
)NE)(1q(/1/1T 00
Temperature:
Variational principle:
Probability distribution:
“Partition function”:
Momentum Distribution
q22
q3mpexp
pd
dn
+ p - X
q = 1T = 0.136 G eV
q = 1.037T = 0.117 G eV
0.0 0.5 1.0 1.5 2.0 2.5m t (G eV)
0.0001
0.001
0.01
0.1
1
10
100
1000
dN/d
p t 2
(G
eV -
2 )
NA22250GeV/c
q = 1.031T = 0.119 G eV
q = 1T = 0.135 G eV
0.0 0.5 1.0 1.5 2.0 2.5m t (G eV)
0.0001
0.001
0.01
0.1
1
10
100
1000
dN/d
p t 2
(G
eV -
2 )
p p - X NA22250GeV/c
q = 1.040T = 0.116 G eV
q = 1T = 0.136 G eV
K+ p - X
0.0 0.5 1.0 1.5 2.0 2.5m t (G eV)
0.0001
0.001
0.01
0.1
1
10
100
1000
dN/d
p t 2
(G
eV-2
)
NA22250GeV/c
3.16nnD
3.8n222
12.0n
nD
k
12
2
Multiplicity Distribution
(NA22) GeV/c 250
particles chargedp
Deviationfrom Poisson
0 10 20 30n
0.0001
0.001
0.01
0.1
1
10
n (
mb)
C harged Partic le M ultip lic ity D istributionN A22 -p 250 G eV/c
Po isson(Boltzm ann)
8.16nnD
9.7n222
14.0n
nD
k
12
2
Multiplicity Distribution
(NA22) GeV/c 250
particles chargedpp
Deviationfrom Poisson
0 10 20 30n
0.0001
0.001
0.01
0.1
1
10
n (
mb)
C harged Partic le M ultip lic ity D istributionN A22 p-p 250 G eV/c
Po isson(Boltzm ann)
0.16nnD
2.8n222
12.0n
nD
k
12
2
Multiplicity Distribution
(NA22) GeV/c 250
particles chargedpK
Deviationfrom Poisson
0 10 20 30n
0.0001
0.001
0.01
0.1
1
10
n (
mb)
C harged Partic le M ultip lic ity D istributionN A22 K -p 250 G eV/c
Po isson(Boltzm ann)
7.112nnD
1.21n222
21.0n
nD
k
12
2
Multiplicity Distribution
(UA5)GeV 200s
particles chargedpp
0 10 20 30 40 50 60n
0.0001
0.001
0.01
0.1
1
Pn
C harged Partic le M ultip lic ity D istribution
U A5 s1/2 = 200 G eV
Po isson(Boltzm ann)
Deviationfrom Poisson
Multiplicity Distribution
(DELPHI)GeV 90s
particles chargedee
4.41nnD
2.21n222
045.0n
nD
k
12
2
Deviation
from Poisson
0 10 20 30 40 50 60n
0.0001
0.001
0.01
0.1
1
10
100
Pn
(%)
C harged Partic le M ultip lic ity D istributionDelphi 90 G eV
Po isson(Boltzm ann)
(NA35) GeV/A 200 particles negative(central) SS
Multiplicity Distribution
7.25nnD
8.20n222
011.0n
nD
k
12
2
0 10 20 30 40n
0.0001
0.001
0.01
0.1
Pn
N egative Partic le M ultip lic ity D istributionN A35 S+S (centra l) 200 G eV/A
Po isson(Boltzm ann)
Deviationfrom Poisson
Negative-Binomial Distribution
nk
kn
)km(
km
!n
)1nk()1k(k)n(P
kn )1t(
k
m1t)n(P)t(F
k/mmDmn
22
generatingfunction:
average andvariance:
k = - Nbinomial
distribution
N/mmD
mn22
N
)1t(N
m1)t(F
k = Poisson
distribution
)1t(mexp)t(F
mDn 2
Multiplicity Distributionin Tsallis Statistics
N
)N(q
N )N(a
q)N(aq
a
qaaqq
Z
)NE(exp
)NE(expZ
q
)N(q
Z
Z)N(P
q
N
N)N(q
N
N Z
tZt)N(P)t(F
Integral Representation for q > 1
)Axexp()x(Gdx)A(exp0
)1q/(1,)xexp(x)(
)x(Gq
maximum at x = 1 , width = [q(q-1)]1/2
0 0.5 1 1.5 2x
G q
Integral Representationof the Partition Function
)V,,x(Z)x(GdxZ x
0
)V,t,x(Z)x(GdxtZ)t(F̂ x
0
qN
N)N(q
Gibbs-Boltzmann)V,,(Z e
)1(F̂
)t(F̂)t(F
Relativistic Ideal Gas
)m(K2
mg)(
]V)(exp[)V,,(Z
22
2
q32Z
g)0(
No ideal Tsallis gas for q > 1
N particles:
NN V)(!N
1)V,N,(Z
qZN3
11q
Relativistic Van der Waals Gas
W(x) = Lambert function:
Number of particles < V / v
qZV3
v1q
])(v[W
v
Vexp)V,,(Z
v = “hard-core volume”
x)]x(Wexp[)x(W
)xln()x(W
2xx)0x(W
(q-1) << 1 and v/V << 1
First Order Correctionsto Ideal Gas
)m(K
)m(Km3
V)(n
2
1
0
220
022
0
0
tnV
v
]tn2/t)n[()1q(
tn)t(F̂ln
)0(
20000
2
20000
nV
v4)1n2(n)1q(nD
nV
v2)1n(n)1q(nn
V
v2)1q(
k
1 2
Tsallis and Van der Waals Corrections
Deviation from Poisson:
)0(
02
00002
02
0000
nnV
v4)1n2(n)1q(nD
n2
1n
V
v2)1n(n)1q(nn
0
2
n2V
v2)1q(
k
1
Tsallis - Van der Waals - Bose - Einstein Corrections
Deviation from Poisson:
)0( )1(
)m(K
)m2(K
2
2
Multiple Fireballs
<n> Nfb <n>
k Nfb k
Nfb