Multiparticle production Multiparticle production processes from the processes from the
Information Theory point of Information Theory point of
viewview
O.UtyuzhO.Utyuzh
The Andrzej Sołtan Institute for Nuclear StudiesThe Andrzej Sołtan Institute for Nuclear Studies (SINS) (SINS), , Warsaw, PolandWarsaw, Poland
WhyWhy and and WhenWhen of information theory in multiparticle production of information theory in multiparticle production
Model 1Model 1
Model 3Model 3
Model 2 Model 2
Information contained Information contained in datain data
Which Which model is correct?model is correct?
Model 1Model 1
Model 3Model 3
Model 2 Model 2
Which model tell us Which model tell us TRUTHTRUTHabout dataabout data
Which model tell us Which model tell us TRUTHTRUTHabout dataabout data
lni ii
S p p
- To quantify this problem one uses notion of information
- and resorts to information theory
- based on Shannon information entropy
- where denotes probability distribution of quantity of interest{ }ip
This probability distribution must satisfy the following:This probability distribution must satisfy the following:
1ii
p
( )i k i ki
p R x R
exp ( )i k k ii
p R x
- to be normalized to unity:
- to reproduce results of experiment:
- to maximize (under the above constraints) information entropy
lni ii
S p p
k- with uniquely given by the experimental constraint equations
Model 1Model 1
Model 3Model 3
Model 2 Model 2
TRUTH TRUTH is hereis hereTRUTH TRUTH is hereis here
( ) exp ( )i i k k ik
p x R x
Common partCommon part
noticenotice
If some new data occur and they turn out to disagree with
( ) exp ( )i i k k ik
p x R x
it means that there is more information which must be accounted for :
• either by some new λ= λk+1
• or by recognizing that system in
nonextensive and needs a new form of exp(...) → expq(...)
• or both ...........
Some examples (multiplicity)Some examples (multiplicity)
chnknowledge of only +
- fact that particles are distinguishable
- fact that particles are nondistinguishable
most probable distribution
geometrical (Bose-Einstein)
Poissonian
- fact that particles are coming from k independent, equally strongly emitting sources Negative Binomial
- second moment 2n Gaussian
Rapidity distributionRapidity distribution
3
21 1( )N
TN
dN Edf y
N dy N d pd p
( ) ln ( )
M
M
Y
N N
Y
S dy f y f y
( ) 1N
YM
YM
dy f y
( ) cosh( ) ( )N T N
Y YM M
Y YM M
Mdy E f y dy y f y
N
• we are looking for
• by maximizing
• under conditions
G.Wilk, Z.Włodarczyk, Phys. Rev. 43 (1991) 794
1
( ) exp coshN Tf y yZ
( , , ) exp coshT T
YM
YM
Z Z M N dy y
1/ 22
2
' 4ln 1 1
2 'T
MT
MY
M
As most probable distribution we get
wherewhere
andand T= (W;N, )
0-YMYM
-
+
-
= 0
= 0
1( ) exp coshN Tf y y
Z
Fact: In multiparticle production processes many observables follow simple exponential form:
( ) exp[ ]x
f x
“thermodynamics” (i.e, T )
Another point of view ...Another point of view ...
NN-particle system-particle system
Heat bathHeat bath
(N-1)–particle sub-system
observed particle
Heat bathHeat bathhh TT
(N-1)–particle sub-system
observed particle
( ) exp( )hh
xf x
T
L.Van Hove, Z.Phys. C21 (1985) 93, Z.Phys. C27 (1985) 135.
( ) exp ( )hq h q
q
xf x
T ( ) exp( )h
h
xf x
T
Heat bathHeat bathHeat bathHeat bathhh TTqq
(N-1)–particle sub-system
observed particle
1
1exp ( ) [1 (1 ) ] qh hq
q q
x xq
T T where
nonextensivity
1
1
qi
iq
pS
q
( ) ( ) ( ) ( 1) ( ) ( )q q q q qS A B S A S B q S A S B
qi
i qi
i
pp
p
i ii
i ii
E pE
N pN
Nonextensive (Tsallis) entropy:
is nonextensive because of
q-biased probabilities:
q-biased averages:
AnotherAnother origin of origin of qq: fluctuations present in the system...: fluctuations present in the system...
Fluctuations of temperature:
0 0( ) ( )T T E T a E E
1
10
0 0 0
ln[ ( )] ( ) 1 (1 )( )
qE EdEd P E P E q
T a E E T
ln[ ( )] ( ) expdE E
d P E P ET T
with 1,
V
aC
then the equation on probability P(E) that a system A interacting with the heat bath A’ with temperature T has energy E changes in the following way
1 :q a
M.P.Almeida, Physica A325 (2003) 426
Summarizing:Summarizing: ‘extensive’‘extensive’ ‘nonextensive’‘nonextensive’
0
expx
L
0
exp expq q
x xL
2 21 1
1 1 21
q
where q measures amound of
fluctuations
and
<….> denotes averaging over
(Gamma) distribution in (1/λ)
G.Wilk, Z.Włodarczyk, Phys. Rev. Lett. 84 (2000) 2770; Chaos,Solitons and Fractals 13/3 (2001) 581
Summary (known origins of Summary (known origins of nonextensivitynonextensivity))
• existence of long range correlations• memory effect• fractality of the available phase space• intrinsic fluctuations existing in the
system• ..... others ....
Applications: ppApplications: pp
1 1( ) exp coshq
q q Tq
dNp y y
N dy Z
cosh ( )3
m
m
Yq q
T qqY
N NK E dy y p yq qs s
cosh ( )m
m
Yq q
T qY
sdy y p y
N
q
- input: , ,T chs N - fitted parameters: , qq
NUWW PR D67 (2003) 114002
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6 E
cm = 53 GeV
Ecm
= 200 GeV E
cm = 540 GeV
Ecm
= 900 GeV E
cm = 1800 GeV
Ecm
= 20 GeV
dN/dy
y
inelasticityinelasticity
0,0 0,2 0,4 0,6 0,8 1,00
1
2
3
4(K)
K
200 GeV 900 GeV
NUWW PR D67 (2003) 114002
0,0 5,0x102 1,0x103 1,5x103 2,0x103
0,2
0,4
0,6
0,8
1,0
K
s1/2 [GeV]
y
From fits to rapidity distribution data one gets:
(*) - distribution ‘partition temperature’
ch
K sT
N
(*) fluctuatingq T
fluctuatingq chN
Conjecture: should measure amount of fluctuation in
1q( )chP N
It does so, indeed, see Fig. where dataon obtained from fits are superimposedwith fit to data on parameter in Negative Binomial Distribution!
qk
101 102 103 104
0.05
0.10
0.15
0.20
0.25
0.30
0.35
s1/2 [GeV]
q-1 1/k
1
0
exp( ) exp( )( )
! ( )
n k kn n n nP N dn
n k
2
2
( )1 1ch
chch
Nk NN
11q
k
- Experiment:
kn
( ) ( ; ( 1) )ch NB chP N P N k k
- is measure of fluctuations1k
( )(1 ) ( ) 1
k
k n
k nn k
with
P.Carruthers,C.C.Shih, Int. J. Phys. A4 (1989) 5587
0 10 20 30 40 50 60Nch
0.0001
0.001
0.01
0.1
1
P(N
ch)
Kodama et al..
PoissonPoisson
UA5 @200 GeVUA5 @200 GeV
0 1 2 3 4 5 6 7 8 910-610-510-410-310-210-1100101102103104105
x102
x101200 GeV
540 GeV
900 GeV
UA1
pT [GeV]
E d3/dp3
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6 E
cm = 53 GeV
Ecm
= 200 GeV E
cm = 540 GeV
Ecm
= 900 GeV E
cm = 1800 GeV
Ecm
= 20 GeV
dN/dy
y
SS1/21/2 qqLL TTLL=1/=1/ββLL
200200 1.2031.203 12.1212.12
546546 1.2621.262 22.3822.38
900900 1.2911.291 29.4729.47
SS1/21/2 qqTT TTTT=1/=1/ββTT
200200 1.0951.095 0.1340.134
546546 1.1051.105 0.1350.135
900900 1.1101.110 0.1400.140
2 2 2( ) ( ) ( )L TT T T
NUWW Physica A300 (2004) 467
2 2 2 212 2
q T q T T TL L T T L TqT T
0 1 2 30
50
100
150
200
8.8 GeV
12.3 GeV
17.3 GeV NA490-7%
dN /dy
y
Example of use of MaxEnt methodapplied to some NA49 data for π-
production in PbPb collisions (centrality 0-7%):
SS1/21/2 qq KKqq
8.88.8 1.0401.040 0.220.22
12.312.3 1.1641.164 0.300.30
17.317.3 1.2001.200 0.330.33
q=1, two sources of mass M=6.34 GeV located at |y|=0.83
• (blue lines)
this is example of adding new this is example of adding new dynamical assumptiondynamical assumption
• (orange line)
Applications: AAApplications: AA
0 1 2 30
50
100
150
200
8.8 GeV
12.3 GeV
17.3 GeV NA490-7%
dN /dy
y0 2 4 6
0
100
200
300
400
500
600
700
800
19.6 GeV
130 GeV
200 GeV PHOBOS 0-6%
dNch
/d
“tube”symmetric case
asymmetric case
Applications: pAApplications: pA
1(1 )
2N N
2 2( 1)s s m 2 2 22 2 LABs m m p m
- input: , ,T chs N , ,Ts N
with
( , )E P ( , )E P
( , )w v
1 1( , )E P 1 1( , )E P
1 1( , )e p
M
( )i
( , )E P ( , )E P
( , )i iw v
1 1( , )E P 1 1( , )E P
1 1( , )e p
M
1 1( , )E P1 1( , )E P
1
W.Busza, Acta Phys. Polon. B8 (1977) 333,C.Halliwell et al., Phys. Rev. Lett. 39 (1977) 1499
Rapidity distributionRapidity distribution
3
21 1( )N
TN
dN Edf y
N dy N d pd p
( )max
( )min
( ) ln ( )
Y
N N
Y
S dy f y f y
( )max
( )min
( ) 1N
Y
Y
dy f y
( ) ( )max max
( ) ( )min min
( ) cosh( ) ( )N T N
Y Y
Y Y
Wdy E f y dy y f y
N
• we are looking for
• by maximizing
• under conditions
( ) ( )max max
( ) ( )min min
( ) sinh( ) ( )N T N
Y Y
Y Y
Pdy P f y dy y f y
N
0P
1
( ) exp co sinh( )sh TT yf y yZ
( )max
( )min
co
( , , , )
exp cosh shT
T
T
Y
Y
Z Z W N
dy y
P
y
( )2s
W R K
As most probable distribution we get
wherewhere
andand 24
( )2
s mP R K
2 2 2 2( ) R KM W P KR s R K m K s
-4 -2 0 2 40
1
2
3
4
dN/dy
pXe
pAr
pp
y
Kpp
= 0.45 K
pAr = 0.5
KpXe
= 0.6
-4 -2 0 2 40
1
2
3
4
dN/dy
pXe
pAr
pp
y
Kpp
= 0.45 K
pAr = 0.45, R = 0.55
KpXe
= 0.45, R = 0.90
“symmetric” “asymmetric”
( )i
( , )E P ( , )E P
1 1( , )w v
1 1( , )E P 1 1( , )E P
1 1( , )e p
iM
1 1( , )E P
(1) (1)1 1( , )E P ( ) ( )
1 1( , )i iE P
M1M
( , )i ie p ( , )e p
( , )i iw v ( , )w v
1
( ) ( )ii
f y f y
“sequential”
1 2
1( 1)
2N N N
1 22 ( 1) ( 1)N N N 1,2N
can be visualized
where are such that
-4 -2 0 2 40
1
2
3
4
5
6=4
=3
=2 dN/dy
y
K1 = 0.7, R = 0.35
K2 = 0.17, R = 0.26
K3 = 0.09, R = 0.29
K4 = 0.04, R = 0.36
=1
-4 -2 0 2 40
1
2
3
4
5
6=4
=3
=2
=1
dN/dy
y
K = 0.7, R = 0.35 K = 0.63, R = 0.5 K = 0.6, R = 0.77 K = 0.35, R = 0.75
“tube” “sequential”
summarysummary
When treated by means of information theory methods (MaxEnt approach) the resultant formula are formally identical with those obtained by thermodynamical approach but their interpretation is different and they are valid even for systems which cannot be considered to be in thermal equilibrium.
It means that statistical models based on this approach have more general applicability then naively expected.
Usually regarded to signal some “thermal” behaviour they can also be considered as arising because insufficient information which given experiment is providing us with
In many places one observes simple “exponential” or “exponential-like” behaviour of some selected distributions
summary summary
is not enough and, instead, one should use two (... at least...) parameter formula
with q accounting summarily for all factors mentioned above.
exp( )x
T
1
1exp ( ) [1 (1 ) ] qq
x xq
T T
Therefore: Statistical models of all kinds are widely used as source of some quick reference distributions. However, one must be aware of the fact that, because of such (interrelated) factors as:
fluctuations of intensive thermodynamic parameters finite sizes of relevant regions of interaction/hadronization some special features of the „heath bath” involved in a given process the use of only one parameter T in formulas of the type
Back-up SlidesBack-up Slides
0 1x103 2x1030.0
0.2
0.4
0.6
0.8
1.0
UA7
P238
s1/2 [GeV]
K
Kodama et al..
0 10 20 30 40 50 60Nch
0.0001
0.001
0.01
0.1
1
P(N
ch)
PoissonPoisson
UA5 @200 GeVUA5 @200 GeV
0 10 20 30 40Nch
0.0001
0.001
0.01
0.1P
(Nch)
PoissonPoisson
NA35 @200 GeV/ANA35 @200 GeV/A
S-S (central)S-S (central)
Back-up SlidesBack-up Slides
High-Energy collisions …High-Energy collisions …
AA BB
1
2s
1
2s
K s
High-Energy collisions …High-Energy collisions …
AA’’ BB’’
1(1 )
2s K
1(1 )
2s K
High-Energy collisions …High-Energy collisions …
0
0
K K
K K
0K
p
p
p
p
summary summary
is not enough and, instead, one should use two (... at least...) parameter formula
with q accounting summarily for all factors mentioned above.
In general, for small systems, microcanonical approach would be preferred (because in it one effectively accounts for all nonconventional features of the heat bath...) (D.H.E.Gross, LNP 602)
exp( )x
T
1
1exp ( ) [1 (1 ) ] qq
x xq
T T
Therefore: Statistical models of all kinds are widely used as source of some quick reference distributions. However, one must be aware of the fact that, because of such (interrelated) factors as:
fluctuations of intensive thermodynamic parameters finite sizes of relevant regions of interaction/hadronization some special features of the „heath bath” involved in a given process the use of only one parameter T in formulas of the type
-2 0 20
20
40
60
80
100
120
140
160
NA49 plab
= 80 GeV
dN-/dy
y
Example of use of MaxEnt method
applied to some NA49 data for π –
production in PbPb collisions
(centrality 0-7%) - (I) :
(*) the values of parameters used:
q=1.164 and K=0.3
-2 0 20
20
40
60
80
100
120
140
160
180
200
NA49 plab
= 158 GeV
dN-/dy
y
(*) the values of parameters use
(red line): q=1.2 and K=0.33
(*) q=1, two sources of mass M=6.34 GeV located at |y|=0.83
Example of use of MaxEnt method
applied to some NA49 data for π –
production in PbPb collision
(centrality 0-7%) - (I) :
this is example of adding new this is example of adding new dynamical assumptiondynamical assumption
(*) (*) Nonextensivity – its possible origins .... Nonextensivity – its possible origins .... ”thermodynamics””thermodynamics”
T6
T4T2
T3T1
T5
T7
Tk
h
T varies
fluctuations...
T0=<T>, q
q - measure of fluctuations
Heat bathT0, q
T0=<T>
Historical example:
(*) observation of deviation from the expected exponential behaviour(*) successfully intrepreted (*) in terms of cross-section fluctuation:
(*) can be also fitted by:
(*) immediate conjecture: q fluctuations present in the system
Depth distributions of starting pointsof cascades in Pamir lead chamberCosmic ray experiment (WW, NPB (Proc.Suppl.) A75 (1999) 191
(*) WW, PRD50 (1994) 2318
2 2
2 0.2
3.1;)1(1
exp
1
1
qT
qconstdT
dN
Tconst
dT
dN
q
Some comments on T-fluctuations:
(*) Common expectation: slopes of pT distributions information on T
(*) Only true for q=1 case, otherwise it is <T>, |q-1| provides us additional information
(*) Example: |q-1|=0.015 T/T 0.12
(*) Important: these are fluctuations existing in small parts of the hadronic system with respect to the whole system rather than of the event-by-event type for which T/T =0.06/N 0 for large N
Utyuzh et al.. JP G26 (2000)L39
Such fluctuations are potentially very interesting because they provide a direct measure of the totalheat capacity of the system
11)(
2
2
qC
Prediction: C volume of reaction V, therefore q(hadronic)>>q(nuclear)
Rapidity distributions:Rapidity distributions:
qTqq
qTq
q
yqdyZ
yqZdy
dN
11
11
cosh)1(1
cosh)1(11
Features:(*) two parameters: q=1/Tq and q shape and height are strongly correlated(*) in usual application only =1/T - but in reality ()
1/Zq=1 is always used as another independent parameter height and shape are fitted independently(*) in q-approch they are correlated
() T.T.Chou, C.N.Yang, PRL 54 (1985)
510; PRD32 (1985) 1692
Multiplicity DistributionMultiplicity Distributions: (UA5, DELPHI, NA35)s: (UA5, DELPHI, NA35)
Kodama et al..
e+e-
90GeVDelphi
SS(central)200GeV
<n> = 21.1; 21.2; 20.8D2 = <n2>-<n>2 = 112.7; 41.4; 25.7 Deviation from Poisson: 1/k
1/k = [D2-<n>]/<n2> = 0.21; 0.045; 0.011
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
Poisson(Boltzm ann)
UA5200GeV
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 istributionD elphi 90 G eV
Po isson(Boltzm ann)
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)
Recent example from AA -(1) (RWW, APP B35 (2004) 819)
Dependence of the NBD parameter 1/k onthe number of participants for NA49 andPHENIX data
With increasing centralityfluctuations of the multiplicitybecome weaker and the respective multiplicitydistributions approachPoissonian form. ???Perhaps: smaller NW smallervolume of interaction Vsmaller total heat capacity Cgreater q=1+1/C greater1/k = q-1
Recent example from AA – (2) (RWW, APP B35 (2004) 819)
Dependence of the NBD parameter 1/k on the number of participants for NA49 and PHENIX data
In this case it can be shown that:
56.0/)(
/)(
33.0)1(
)1()()1()(1
22
22
2
2
EE
SSR
qR
qRNDqRN
ND
k
( Wróblewski law )
( for p/e=1/3)
q1.59 which apparently (over)saturates the limit imposed by Tsallis statistics: q1.5 . For q=1.5 one has: 0.33 0.28 (in WL) or 1/3 0.23 (in EoS)
pi = exp[ - ∑k• Rk(xi )]
This is distribution which:
(*) tells us ”the truth, the whole truth” about our experiment,
i.e., it reproduces known information
(*) tells us ”nothing but the truth” about our experiment,
i.e., it conveys the least information (= only those which
is given in this experiment, nothing else)
it contains maximum missing information
G.Wilk, Z.Włodarczyk, Phys. Rev. 43 (1991) 794
Example (from Y.-A. Chao, Nucl. Phys. B40 (1972) 475)
Question: what have in common such successful models as:
(*) multi-Regge model
(*) uncorrelated jet model
(*) thermodynamical model
(*) hydrodynamical model
(*) ..................................
Answer: they all share common (explicite or implicite) dynamical
assumptions that:
(*) only part of initial energy of reaction is used for production
of particles ↔ existence of inelasticity of reaction, K~0.5
(*) transverse momenta of produced secondaries are cut-off ↔
dominance of the longitudinal phase-space
( , )E P ( , )E P
( , )w v
1 1( , )E P 1 1( , )E P
1 1( , )e p
M
( )i
( , )E P ( , )E P
( , )i iw v
1 1( , )E P 1 1( , )E P
1 1( , )e p
M
1 1( , )E P1 1( , )E P
1
summarysummary
0-YMYM
-
+
-
= 0
= 0
