Universality of hadrons production and the Maximum Entropy

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

HERA gp W=200 Ge. V ds/dyd. PT 2[pb/Ge. V 2] ds/dyd. PT 2[nb/Ge. V 2] A shape of the inclusive charged particle spectra Spp. S pp W=560 Ge. V PT[Ge. V] Difference in colliding particles and energies in production mechanism for high and low PT Similarity in spectrum shape PT[Ge. V]

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 Difference in type of produced hadrons Similarity in spectrum shape and an absolute normalization

The invariant cross sections are taken for one spin and one isospin projections. M – is a nominal mass of a resonance 1/(2 j+1)ds/(dydp. T 2) [nb/Ge. V 2] A comparison of inclusive spectra for resonances HERA photoproduction } r 0 f 2 h H 1 Prelim p+ published Difference in a type of produced resonances M+PT [Ge. V] Similarity in spectrum shape and an absolute normalization

Stochasticity The properties of a produced hadron at any given interaction cannot be predicted. But statistical properties energy and momentum averages, correlation functions, and probability density functions show regular behavior. The hadron production is stochastic. Power law d. N/d. Pt ~ 1 Pt (1 + P ) 0 n Ubiquity of the Power law

Geomagnetic Plasma Sheet Plasma sheet is hot - Ke. V, (Ions, electrons) Low density – 10 part/cm 3 Magnetic field – open system COLLISIONLESS PLASMA

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

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.

1500 30 Re Measurements of one-dimensional longitudinal velocity spectra Damping by viscosity at the Kolmogorov scale n 3 h=(e ) 1 4 with a velocity 1 v = (en) 4

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

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)

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

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

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

Human activity Male earnings Settlement size First pointed out by George Kingsley Zipf and Pareto Zipf, 1949: Human Behaviour and the Principle of Least Effort.

Sexual contacts A number of partners within 12 months a ≈ 2. 5 survey of a random sample of 4, 781 Swedes (18– 74 years)

Extinction of biological species

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

• 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

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

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

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

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

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

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

Fractal structure of the protons Scaling, self-similarity and power-law behavior are F 2 properties, which also characterize fractal objects z = 10 20 50 Serpinsky carpet D = 1. 5849 1 10. x =. . . 1000 Proton: 2 scales 1/x , (Q 2 o + Q 2 )/Q 2 o Generalized expression for unintegrated structure function:

Limited applicability of perturbative QCD ZEUS hep-ex/0208023

For x < 0. 01 и 0. 35 < Q < 120 Ge. V 2 : c 2 /ndf = 0. 82 !!! With only 4 free parameters

Exponential Constraint S P ie i arithmetic mean 1 N Correlations Power Law SPiln(e 0+ei) geometric mean (Sei) (P(e 0+ei))1/N No …+eiej+… • For ei < e 0 Power Law transforms into Exponential distribution • Constraints on geometric and arithmetic mean applied together results in GAMMA distribution

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

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

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