Transcript Maximal Entropy approach to particle production

Multiparticle production
processes from the Information
Theory point of view
O.Utyuzh
The Andrzej Sołtan Institute for Nuclear Studies (SINS),
Warsaw, Poland
Why and When of information theory in multiparticle production
Model 2
Information contained
in data
Model 1
Model 3
Which model is correct?
Model 2
Model 1
Model 3
Which model tell us TRUTH
about data
- To quantify this problem one uses notion of information
- and resorts to information theory
- based on Shannon information entropy
S   pi ln pi
i
- where
{ pi }
denotes probability distribution of quantity of interest
This probability distribution must satisfy the following:
p
- to be normalized to unity:
i
1
i
- to reproduce results of experiment:
 p  R (x ) 
i
k
i
i
- to maximize (under the above constraints) information entropy
S   pi ln pi
i
- with
k  uniquely given by the experimental constraint equations


pi  exp  k  Rk ( xi ) 
 i

Rk
Common part
Model 2
Model 1
Model 3
TRUTH is here


pi ( xi )  exp  k  Rk ( xi ) 
 k

notice
If some new data occur and they turn out to disagree with


pi ( xi )  exp  k  Rk ( xi ) 
 k

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)
knowledge of only
nch
+
most probable distribution
- fact that particles are distinguishable
geometrical
(Bose-Einstein)
- fact that particles are nondistinguishable
Poissonian
- fact that particles are coming from k
independent, equally strongly emitting sources
Negative Binomial
- second moment
n2
Gaussian
Rapidity distribution
• we are looking for
1 dN
1
f N ( y) 

N dy  N N

2
d p
T
 Ed 
 d3 p 


YM
• by maximizing
S

dy  f N ( y)  ln  f N ( y)
 YM
• under conditions
YM
YM

Y M
dy  f N ( y)  1

YM
dy  E  f N ( y)  T
Y M
G.Wilk, Z.Włodarczyk, Phys. Rev. 43 (1991) 794

Y M
dy  cosh( y)  f N ( y) 
M
N
As most probable distribution we get
f N ( y) 
1
 exp 
T  cosh  y 

Z
YM
where
Z  Z ( M , N , T ) 

dy  exp 
 T  cosh  y 
Y M

 M'
YM  ln 
2

 T
  4 2 1 /2  

1   1  T2   
M '  
 

and
 = (W;N,T )
1
f N ( y )   exp   T  cosh  y  
Z
  -
  -
  +
 = 0
=0
-YM
0
YM
Another point of view ...
Fact: In multiparticle production processes many
observables follow simple exponential form:
x
f ( x)  exp[ ]

“thermodynamics” (i.e, T )
N-particle system
(N-1)–particle sub-system
observed
particle
Heat bath
(N-1)–particle sub-system
observed
particle
h
T
xh
f ( xh )  exp( )
T
L.Van Hove, Z.Phys. C21 (1985) 93,
Z.Phys. C27 (1985) 135.
Heat bath
(N-1)–particle sub-system
observed
particle
h
Tq
Heat bath
xxhh
f q((xxhh))  exp
exp(q ( )
Tq
where
exp q ( 
1
h 1 q
xh
x
)  [1  (1  q )  ]
Tq
Tq
nonextensivity
Nonextensive (Tsallis) entropy:
Sq 
1   piq
i
q 1
is nonextensive because of
Sq ( A  B)  Sq ( A)  Sq (B)  (q  1)Sq ( A)  Sq ( B)
q-biased probabilities:
pi q
pi 
q
p
 i
i
q-biased averages:
E   pi Ei
i
N   pi N i
i
Another origin of q: fluctuations present in the system...
Fluctuations of temperature:
T  T (E)  T0  a  (E  E0 )
with
a
1
,
CV
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
q  1 a :
dE
 E
d ln[ P( E )] 
 P( E )  exp   
T
 T
1

E  E0 1q
dE
d ln[ P( E )] 
 P( E )  1  (1  q)

T0  a( E  E0 )
T
0


M.P.Almeida, Physica A325 (2003) 426
Summarizing: ‘extensive’  ‘nonextensive’
 x 
L  exp   
 0 
where q measures amound of
fluctuations
and
<….> denotes averaging over
(Gamma) distribution in (1/λ)
 x
 x
Lq  expq     exp   
 
 0 
2
2
1
1

 


q  1   1 
2
1

G.Wilk, Z.Włodarczyk, Phys. Rev. Lett. 84 (2000) 2770; Chaos,Solitons and Fractals 13/3 (2001) 581
Summary (known origins of nonextensivity)
•
•
•
•
existence of long range correlations
memory effect
fractality of the available phase space
intrinsic fluctuations existing in the
system
• ..... others ....
Applications: pp
- input:
s , T , Nch
- fitted parameters:
1 dNq
1
pq ( y) 

exp   q T cosh y 
N dy
Zq
6
dN/dy
Ecm = 53 GeV
Ecm = 200 GeV
Ecm = 540 GeV
Ecm = 900 GeV
Ecm = 1800 GeV
Ecm = 20 GeV
5
4
Ym

 Ym
q
dy  T cosh y  pq ( y) 
q s
N
 q
q, q
3
2
1
N
Kq 
E
s
q
q
N



dy  T cosh y  pq ( y) 
3q
s Ym
Ym
q
0
0 1 2 3 4 5 6 7 8
y
NUWW PR D67 (2003) 114002
inelasticity
NUWW PR D67 (2003) 114002
1,0
4
0,8
3
c(K)
200 GeV
900 GeV
K
0,6
2
0,4
1
0,2
0,0
5,0x10
2
1,0x10
s
1/2
3
[GeV]
1,5x10
3
2,0x10
3
0
0,0
0,2
0,4
0,6
K
0,8
1,0
From fits to rapidity distribution data one gets:
(*)
y - distribution  ‘partition temperature’
K s
T
N ch
(*)
q fluctuating Nch
q  fluctuating T
Conjecture: q  1 should measure amount
of fluctuation in P(N  ch )
It does so, indeed, see Fig. where data
on qobtained from fits are superimposed
with fit to data on parameter kin
Negative Binomial Distribution!
0.35
0.30
0.25
0.20
0.15
0.10
0.05
q-1
1/k
1
10
2
3
10
10
1/2
s
[GeV]
4
10
1
Poisson
UA5 @200 GeV
1  ( N ch )
1


2
k
N ch
N ch
2
-
1
k
0.1
P(Nch)
- Experiment: P( Nch )  PNB ( Nch ;k(k  1))
0.01
0.001
Kodama et al..
0.0001
0
is measure of fluctuations
10
20
30
Nch
(k  n )
k
n n exp(n )  k n k 1 exp( n )


P ( N )   dn

k n

(1

n
)

(
k
)
n
!

(
k
)


1


0

with  
k
n
1
 q1
k
P.Carruthers,C.C.Shih, Int. J. Phys. A4 (1989) 5587
40
50
60
6
dN/dy
Ecm = 53 GeV
Ecm = 200 GeV
Ecm = 540 GeV
Ecm = 900 GeV
Ecm = 1800 GeV
Ecm = 20 GeV
5
4
NUWW Physica A300 (2004) 467
3
2
1
0
0 1 2 3 4 5 6 7 8
q T2  q T2 T2 T2
q  L L 2 T T  L 2 T 1
T
T
y
S1/2
qL
TL=1/βL
200
1.203
12.12
546
1.262
22.38
900
1.291
29.47
5
10 E d3/dp3
4
10
UA1
3
10
2
10
1
10
900 GeV
0
10
-1
10
2
-2
x10
10
-3
1
10 200 GeV
x10
-4
10
-5
10
540 GeV
-6
10
0 1 2 3 4 5 6 7 89
pT [GeV]
S1/2
 (T )   (TL )   (TT )
2
2
2
qT
TT=1/β
T
200
1.095
0.134
546
1.105
0.135
900
1.110
0.140
Applications: AA
200
150
17.3 GeV
NA49
0-7%
Example of use of MaxEnt method
applied to some NA49 data for πproduction in PbPb collisions
(centrality 0-7%):
12.3 GeV
• (blue lines)
8.8 GeV
100
50
• (orange line)
dN /dy
0
0
1 y
2
3
S1/2
q
Kq
8.8
1.040
0.22
12.3
1.164
0.30
17.3
1.200
0.33
q=1, two sources of
mass M=6.34 GeV
located at |y|=0.83
this is example of adding new
dynamical assumption
200
150
17.3 GeV
800
NA49
0-7%
PHOBOS
0-6%
200 GeV
700
600
12.3 GeV
500
8.8 GeV
130 GeV
400
100
300
200
50
100
dN /dy
0
0
1 y
2
3
19.6 GeV
dNch/d
0
0
2

4
6
Applications: pA
( E1,  P1)
( E1 ,  P1 )
(e1 , p1 )
symmetric
case
( E1,  P1)
( E1 ,  P1 )
(e1 , p1 )
“tube”
M
1
(w , v )
( E1 , P1 )

M

asymmetric
case
(wi , vi )
(i )
( E , P )
- input:
( E , P )
( E , P)
s , T , Nch
s , T , N
s  s  (  1)2 m2
with
2
s  2m2  2m pLAB
 m2
1
N  (1   ) N
2
W.Busza, Acta Phys. Polon. B8 (1977) 333,
C.Halliwell et al., Phys. Rev. Lett. 39 (1977) 1499
( E1, P1)
( E , P)
Rapidity distribution
1 dN
1
f N ( y) 

N dy  N N
• we are looking for
Y ( )

2
d p
T
 Ed 
 d3 p 


max
S
• by maximizing

dy  f N ( y)  ln  f N ( y)
 Y ( )
min
( )
Ymax

• under conditions
dy  f N ( y)  1
Y ( )
min
( )
Ymax
P  0

Y ( )
min
( )
Ymax
dy  E  f N ( y)  T

Y ( )
Y ( )
min
W
N
min
( )
Ymax

dy  cosh( y)  f N ( y) 
( )
Ymax
dy  P  f N ( y)  T

Y ( )
min
dy  sinh( y)  f N ( y) 
P
N
s
W  ( R  K ) 
2
and
s  4m2
P  ( R  K ) 
2
R K
M  W2  P 2  KR s  ( R  K )2 m2 
K s
As most probable distribution we get
1
f ( y) 
 exp   T  cosh  y    T  sinh( y)
Z
where
Z  Z (W , P , N , T )
( )
Ymax


Y ( )
min
dy  exp    T  cosh  y    T  cosh  y 
“asymmetric”
“symmetric”
4
Kpp = 0.45
KpAr = 0.5
KpXe = 0.6
pXe
3
4
Kpp = 0.45
KpAr = 0.45, R = 0.55
KpXe = 0.45, R = 0.90
pXe
3
pAr
dN/dy
2
pAr
dN/dy
2
pp
pp
1
1
0
0
-4 -2
0
y
2
4
-4 -2
0
y
2
4
“sequential”
( E1 ,  P1 )
( E1(1) ,  P1(1) )
( E1(i ) , P1(i ) )
(e1 , p1 )
(ei , pi )
M1
(w1 , v1 )
( E1 , P1 )

Mi
( E1,  P1)
(e , p )
M
(wi , vi )
(w , v )
(i )
( E , P)
( E , P )

f ( y)   fi ( y)
i 1
can be visualized
where
N1,2
1
N  N 1  (  1) N 2
2
are such that
2N1  (  1)  N2  (  1)  N
“sequential”
“tube”
6
K = 0.7, R = 0.35
K = 0.63, R = 0.5
K = 0.6, R = 0.77
K = 0.35, R = 0.75
=4
5
4
=3
dN/dy
=1
2
5
=2
0
0
2
4
dN/dy
=1
2
1
0
y
=3
3
1
-4 -2
K1 = 0.7, R = 0.35
K2 = 0.17, R = 0.26
K3 = 0.09, R = 0.29
K4 = 0.04, R = 0.36
=4
4
=2
3
6
-4 -2
0
y
2
4
summary
In many places one observes simple “exponential” or “exponentiallike” behaviour of some selected distributions

Usually regarded to signal some “thermal” behaviour they can also
be considered as arising because insufficient information which
given experiment is providing us with

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.

summary
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
use of only one parameter T in formulas of the type

x
exp( )
T
is not enough and, instead, one should use two (... at least...) parameter
formula
1
x
x 1q
expq ( )  [1  (1  q) ]
T
T
with q accounting summarily for all factors mentioned above.
the
Back-up Slides
1.0
0.8
K
P238
0.6
0.4
0.2
0.0
0
UA7
3
1x10
2x10
1/2
s
[GeV]
3
Kodama et al..
0.1
1
Poisson
0.1
S-S (central)
0.01
P(Nch)
P(Nch)
UA5 @200 GeV
0.001
0.01
0.001
Poisson
NA35 @200 GeV/A
0.0001
0
10
20
Nch
30
40
0.0001
0
10
20
30
Nch
40
50
60
Back-up Slides
High-Energy collisions …
B
A
1
s
2
1
s
2
K s
High-Energy collisions …
A’
1
s  (1  K )
2
B’
1
s  (1  K )
2
High-Energy collisions …


K
0

0


p
K
K
p

p
p
K




0
K



summary
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
use of only one parameter T in formulas of the type

the
x
exp( )
T
is not enough and, instead, one should use two (... at least...) parameter
formula
1
x
x 1q
expq ( )  [1  (1  q) ]
T
T
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)

160
140
Example of use of MaxEnt method
applied to some NA49 data for π –
production in PbPb collisions
(centrality 0-7%) - (I) :
dN /dy
-
120
100
(*) the values of parameters used:
q=1.164 and K=0.3
80
60
40
20
0
NA49 plab= 80 GeV
-2
0
y
2
200
180
dN /dy
-
Example of use of MaxEnt method
applied to some NA49 data for π –
production in PbPb collision
(centrality 0-7%) - (I) :
160
140
120
(*) the values of parameters use
(red line):
q=1.2 and K=0.33
100
80
(*) q=1, two sources of mass M=6.34
GeV located at |y|=0.83
60
40
20
0
NA49 plab= 158 GeV
-2
0
y
2
this is example of adding new
dynamical assumption
(*) Nonextensivity – its possible origins .... ”thermodynamics”
T varies

fluctuations...
T4
T2
T6
T1
h
T7
T3
Heat bath
T0, q
T0=<T>
T0=<T>, q
Tk
q - measure of fluctuations
T5
Historical example:
(*) observation of deviation from the
expected exponential behaviour
(*) successfully intrepreted (*) in terms of
cross-section fluctuation:

  2     2
 0.2
  2
(*) can be also fitted by:
dN
 T
 const  exp    
dT
 
1
dN
T

 const  1  (1  q ) 1  q ; q  1.3
dT


Depth distributions of starting points
of cascades in Pamir lead chamber
Cosmic ray experiment (WW, NPB
(Proc.Suppl.) A75 (1999) 191
(*) WW, PRD50 (1994) 2318
(*) immediate conjecture:
q fluctuations present in the system
Some comments on T-fluctuations:
Utyuzh et al.. JP G26 (2000)L39
(*) 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
Such fluctuations are potentially very interesting
because they provide a direct measure of the total
heat capacity of the system
 2 ( ) 1
   q 1
2 
 
C
Prediction: C  volume of reaction V, therefore q(hadronic)>>q(nuclear)
Rapidity distributions:


1
dN 1
  1  (1  q)   q  T cosh y 1q
dy Z q
Zq
() T.T.Chou, C.N.Yang, PRL 54 (1985)
510; PRD32 (1985) 1692

 dy1  (1  q)   q  T cosh y 
1
1q
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
Charged Particle Multiplicity Distribution
MultiplicityCharged
Distributions:
DELPHI,
NA35)
Particle Multiplicity(UA5,
Distribution
Negative Particle Multiplicity Distribution
UA5 s1/2 = 200 GeV
Delphi
1
90 GeV
NA35 S+S (central)
200 GeV/A
Kodama et al..
100
Poisson
(Boltzmann)
0.1
Poisson
(Boltzmann)
10
Poisson
(Boltzmann)
0.1
0.01
0.1
e+e90GeV
Delphi
0.01
UA5
200GeV
0.001
0.01
Pn
Pn (%)
Pn
1
0.001
0.0001
0.001
0.0001
0
10
20
30
n
40
50
60
0.0001
0
10
20
SS
(central)
200GeV
30
40
50
60
0
10
n
n
<n> =
21.1;
21.2;
D2 = <n2>-<n>2 =
112.7; 41.4;
 Deviation from Poisson: 1/k
20.8
25.7
1/k = [D2-<n>]/<n2> = 0.21;
0.011
0.045;
20
30
40
Recent example from AA -(1) (RWW, APP B35 (2004) 819)
With increasing centrality
fluctuations of the multiplicity
become weaker and the
respective multiplicity
distributions approach
Poissonian form.
???
Perhaps: smaller NW smaller
volume of interaction V
smaller total heat capacity C
greater q=1+1/C  greater
1/k = q-1
Dependence of the NBD parameter 1/k on
the number of participants for NA49 and
PHENIX data
Recent example from AA – (2) (RWW, APP B35 (2004) 819)
In this case it can be shown that:
1 D2 ( N )

 R(q  1)  D( N )  R(q  1)
k  N 2
( Wróblewski law )
R(q  1)  0.33
(S ) 2 / S 2
R
 0.56
(E ) 2 / E 2
Dependence of the NBD parameter 1/k
on the number of participants for NA49
and PHENIX data
( for p/e=1/3)
q1.59 which apparently
(over)saturates the limit
imposed by Tsallis statistics:
q1.5 . 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
( E1,  P1)
( E1 ,  P1 )
( E1,  P1)
( E1 ,  P1 )
(e1 , p1 )
(e1 , p1 )
M
1
(w , v )

( E1 , P1 )
(i )
( E , P )
( E , P)
( E , P )
M

(wi , vi )
( E1, P1)
( E , P)
summary
  -
  -
  +
 = 0
=0
-YM
0
YM