Universality of hadrons production and the Maximum Entropy Principle

1. Universality of hadrons production and the Maximum Entropy Principle A.Rostovtsev ITEP, Moscow May 2004

2. A shape of the inclusive charged particle spectra SppS HERA ds/dydPT2[pb/GeV2] ds/dydPT2[nb/GeV2] gp W=200 GeV pp W=560 GeV PT[GeV] PT[GeV] Differencein colliding particles and energies in production mechanism for high and low PT Similarityin spectrum shape

3. A comparison of inclusive spectra for hadrons The invariant cross sections are taken for one spin and one isospin projections. m – is a nominal hadron mass Differencein type of produced hadrons Similarityin spectrum shape and an absolute normalization

4. HERA photoproduction 1/(2j+1)ds/(dydpT2) [nb/GeV2] } r0 f0 H1 Prelim f2 h p+published M+PT [GeV] A comparison of inclusive spectra for resonances The invariant cross sections are taken for one spin and one isospin projections. M – is a nominal mass of a resonance Differencein a type of produced resonances Similarityin spectrum shape and an absolute normalization

5. 1 dN/dPt ~ n Pt (1 + ) P0 Stochasticity The properties of aproduced hadron at any giveninteraction cannot bepredicted. But statistical properties energy and momentum averages, correlationfunctions, and probability density functions show regular behavior. The hadron production is stochastic. Power law Ubiquity of the Power law

6. Geomagnetic Plasma Sheet Plasma sheet is hot - KeV, (Ions, electrons) Low density – 10 part/cm3 Magnetic field – open system COLLISIONLESS PLASMA

7. 1 Flux ~ E κ+ 1 (1 + ) κθ Polar Aurora, First Observed in 1972 Energy distribution in a collisionless plasma “Kappa distribution”

8. Turbulence Large eddies, formed by fluid flowing around an object, are unstable, and break up into smaller eddies, which in turn break up into still smaller eddies, until the smallest eddies are damped by viscosity into a heat.

9. n3 h = ( ) e v = (en) 1 1 4 4 Measurements of one-dimensional longitudinal velocityspectra 1500 Damping by viscosity at the Kolmogorov scale 30 Re with a velocity

10. Earthquakes Empirical Gutenberg-Richter Law log(Frequency) vs. log(Area)

11. Avalanches and Landslides log(Frequency) vs. log(Area) an inventory of 11000 landslides in CA triggered by earthquake on January 17, 1994(analyses of aerial photographs)

12. Forest fires log(Frequency) vs. log(Area)

13. Solar Flares log(Frequency) vs. log(Time duration)

14. Rains log(Frequency) vs. log(size[mm])

15. Human activity Male earnings Settlement size First pointed out by George KingsleyZipfand Pareto Zipf, 1949: Human Behaviour and thePrincipleof Least Effort .

16. Sexual contacts A number of partners within 12 months a ≈ 2.5 survey of a random sample of 4,781Swedes (18–74 years)

17. Extinction of biological species

18. Internet cite visiting rate the number of visits to a site, the number of pages within a site, the number of links to a page, etc. Distribution of AOL users' visits to various sites on a December day in 1997

19. Observation: distributions have similar form: (… + many others) • Conclusion:These distributions arise because the same stochastic process is at work, and this process can be understood beyond the context of each example

20. Maximum Entropy Principle S = -Spi log (pi) WHO defines a form of statistical distributions? (Exponential, Poisson, Gamma, Gaussian, Power-law, etc.) In 50thE.T.Jaynes has promotedthe Maximum EntropyPrinciple (MEP) The MEP states that the physical observable has a distribution, consistent with given constraints which maximizes the entropy. Shannon-Gibbs entropy:

21. Flat probability distribution dS = - ln(Pi) – 1 = 0 dPi N g = S Pi = 1 i=1 dg dS dPi dPi - a = 0 Pi = exp(-1-a) = 1/N Shannon entropy maximization subject to constraint (normalization) Method of Lagrange Multipliers (a ) - ln(Pi) – 1 - a = 0 All states (1< i < N) have equal probabilities For continuous distribution with a<x<b P(x) = 1/(b-a)

22. Exponential distribution N N e = S Pi Ei = e g = S Pi = 1 i=1 i=1 - a - b = 0 de dg dS dPi dPi dPi Pi = exp(-1-a-bEi) = A exp(-bEi) For continuous distribution (x>0) P(x) = (1 / e )exp(-x / e ) Shannon entropy maximizationsubject to constraints (normalization and mean value) Method of Lagrange Multipliers (a , b) - ln(Pi) – 1 - a - bEi = 0

23. Exponential distribution (examples) E e = N = kT A. Random events with an average density D=1 / e e B. Isolated ideal gas volume Total Energy (E=Se) and number of molecules (N) are conserved log (dN/de) ε

24. Power-law distribution N N e = S Pi ln(xi) = ln(x) g = S Pi = 1 i=1 i=1 - a - b = 0 de dg dS dPi dPi dPi Pi = exp(-1-a-bxi) = A exp(-bxi) Shannon entropy maximizationsubject to constraints (normalization and geometric mean value) Method of Lagrange Multipliers (a , b) - ln(Pi) – 1 - a - bxi = 0 For continuous distribution (x>0) P(x) = (1 / e )exp(-x / e )

25. Power-law distribution (examples) (Sln(ei)) 1 I= N • Incompressible N-dimensional volumes • (Liouville Phase Space Theorem) Geomagnetic collisionless plasma B. Fractals log (dN/de) An average “information” is conserved eiis a size of i-object log(ε)

26. z = 10 20 50 1 10 100 1000 = . . x . . 2 2 2 1/x , (Q + Q )/Q o o Fractal structure of the protons Scaling, self-similarity and power-law behaviorare F2 properties, which also characterize fractal objects Serpinsky carpet D = 1.5849 Proton: 2 scales Generalized expression for unintegrated structure function:

27. Limited applicability of perturbative QCD ZEUS hep-ex/0208023

28. Forx< 0.01 и 0.35 < Q < 120GeV2: c2 /ndf = 0.82 !!! With only4 free parameters

29. Constraint Correlations Exponential Power Law SPiln(e0+ei) SPiei arithmetic mean geometric mean (Sei) (P(e0+ei))1/N 1 N …+eiej+… No • For ei < e0 Power Law transforms into Exponential distribution • Constraints on geometric and arithmetic mean applied together results in GAMMA distribution

30. Concluding remarks Power law distributions are ubiquitous in the Nature Is there any common principle behind the particle production and statistics of sexual contacts  ??? If yes, the Maximum Entropy Principle is a pleasurable candidate for that. If yes, Shannon-Gibbs entropy form is the first to be considered *) If yes, a conservation of a geometric mean of a variable plays an important role. Not understood  even in lively situations. (Brian Hayes, “Follow the money”, American scientist, 2002) *) Leaving non-extensive Tsallis formulation for a conference in Brasil

31. dSei N N SDei = 0 = 0 dt i=1 i=1 N S = 0 i=1 d S = 0 log( ei ) Dei dt e A flap of a butterfly's wings in Brazil sets off a tornado in Texas Energy conservation is an important to make a spectrum exponential: Assume a relative change of energy is zero: This condition describes an open system with a small scale change compensated by a similar relative change at very large scales. Butterfly effect

32. Fractals / Self-similarity (Sln(ei)) 1 I= N Statistical self-similarity means that the degree of complexity repeats at differentscales instead of geometric patterns. In fractals the average “information” is conserved