Multiparticle production processes from the Information Theory point of view

1. Multiparticle production processes from the Information Theory point of view O.Utyuzh The Andrzej Sołtan Institute for Nuclear Studies (SINS), Warsaw, Poland

2. Why and When of information theory in multiparticle production Model 2 Model 1 Model 3 Information contained in data

3. Which model is correct? Model 2 Model 1 Model 3 Which model tell us TRUTH about data

4. - 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

5. - with uniquely given by the experimental constraint equations This probability distribution must satisfy the following: - to be normalized to unity: - to reproduce results of experiment: - to maximize (under the above constraints) information entropy

6. Model 2 Model 1 Common part Model 3 TRUTH is here

7. notice If some new data occur and theyturn out to disagree with it means that there is more informationwhich must be accounted for : • either by some newλ= λk+1 • or by recognizing that system in • nonextensive and needs a newform ofexp(...) → expq(...) • or both ...........

8. Some examples (multiplicity) most probable distribution knowledge of only + - 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 Gaussian

9. G.Wilk, Z.Włodarczyk, Phys. Rev.43 (1991) 794 Rapidity distribution • we are looking for • by maximizing • under conditions

10. As most probable distribution we get where and

11.   -   -   +  = 0  = 0 YM -YM 0

12. Another point of view ... Fact:In multiparticle production processes many observables follow simple exponential form: “thermodynamics” (i.e, T )

13. N-particle system

14. (N-1)–particle sub-system Heat bath observed particle

15. L.Van Hove,Z.Phys.C21 (1985) 93, Z.Phys.C27(1985) 135. (N-1)–particle sub-system Heat bath observed particle T h

16. (N-1)–particle sub-system Heat bath Heat bath observed particle Tq h where nonextensivity

17. Nonextensive (Tsallis) entropy: is nonextensive because of q-biased probabilities: q-biased averages:

18. M.P.Almeida, PhysicaA325 (2003) 426 Another origin ofq: fluctuations present in the system... Fluctuations of temperature: with 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

19. G.Wilk, Z.Włodarczyk, Phys. Rev. Lett.84 (2000) 2770; Chaos,Solitons and Fractals13/3 (2001) 581 Summarizing:‘extensive’‘nonextensive’ where q measures amound of fluctuations and <….> denotes averaging over (Gamma) distribution in (1/λ)

20. 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 ....

21. - input: - fitted parameters: Applications: pp NUWWPRD67 (2003) 114002

22. inelasticity NUWWPRD67 (2003) 114002

23. fluctuating It does so, indeed, see Fig. where data on obtained from fits are superimposed with fit to data on parameter in Negative Binomial Distribution! From fits to rapidity distribution data one gets: (*) - distribution ‘partition temperature’ (*) fluctuating Conjecture: should measure amount of fluctuation in

24. Poisson 1 UA5 @200 GeV 0.1 P(Nch) 0.01 0.001 Kodama et al.. 0.0001 0 10 20 30 40 50 60 Nch P.Carruthers,C.C.Shih,Int.J.Phys.A4 (1989)5587 - Experiment: - is measure of fluctuations with

25. NUWW Physica A300 (2004) 467

26. Applications: AA Example of use of MaxEnt method applied to some NA49 data for π- production in PbPb collisions (centrality 0-7%): • (blue lines) q=1, two sources of massM=6.34 GeV located at |y|=0.83 • (orange line) this is example of adding new dynamical assumption

28. W.Busza, Acta Phys. Polon. B8 (1977) 333, C.Halliwell et al., Phys. Rev. Lett. 39 (1977) 1499 Applications: pA asymmetric case symmetric case “tube” - input: with

29. Rapidity distribution • we are looking for • by maximizing • under conditions

30. and As most probable distribution we get where

31. “asymmetric” “symmetric”

32. “sequential” can be visualized where are such that

33. “sequential” “tube”

34. summary • In many places one observes simple “exponential” or “exponential-like” 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.

35. 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 the use of only one parameter T in formulas of the type is not enough and, instead, one should use two (... at least...)parameter formula with q accounting summarily for all factors mentioned above.

36. Back-up Slides

38. N e g a t i v e P a r t i c l e M u l t i p l i c i t y D i s t r i b u t i o n N A 3 5 S + S ( c e n t r a l ) 2 0 0 G e V / A 0.1 Poisson 1 UA5 @200 GeV S-S (central) 0.1 0.01 P(Nch) 0.01 P(Nch) 0.001 0.001 0.0001 0 10 20 30 40 50 60 Nch 0.0001 Poisson 0 10 20 30 40 NA35 @200 GeV/A Nch Kodama et al..

39. Back-up Slides

40. High-Energy collisions … B A

41. High-Energy collisions … B’ A’

42. High-Energy collisions …

43. 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 the use of only one parameter T in formulas of the type 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)

44. 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

45. Example of use of MaxEnt method applied to some NA49 data for π– production in PbPb collision (centrality 0-7%) - (I) : (*) the values of parameters use (red line): q=1.2 and K=0.33 (*) q=1, two sources of massM=6.34 GeV located at |y|=0.83 this is example of adding new dynamical assumption

46. (*) Nonextensivity – its possible origins .... ”thermodynamics” • Heat bath • T0, q T0=<T> T2 T4 T varies  T6 fluctuations... T1 T3 T5 h T7 T0=<T>, q Tk q - measure of fluctuations

47. 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 points of cascades in Pamir lead chamber Cosmic ray experiment (WW, NPB (Proc.Suppl.) A75 (1999) 191 (*) WW, PRD50 (1994) 2318

48. 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 arepotentially very interesting because they provide a direct measure of the total heat capacity of the system Prediction: C  volume of reaction V, therefore q(hadronic)>>q(nuclear)

49. Rapidity distributions: 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=1is 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

50. Multiplicity Distributions: (UA5, DELPHI, NA35) Kodama et al.. SS (central) 200GeV e+e- 90GeV Delphi UA5 200GeV <n> = 21.1; 21.2; 20.8 D2 = <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