Multiparticle production processes from the Information Theory point of view

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