Universality of hadrons production and the Maximum Entropy

Principle

May 2004

ITEP, Moscow

A.Rostovtsev

d/d

ydP

T2 [

pb/G

eV2 ]

d/d

ydP

T2 [

nb/G

eV2 ]

PT[GeV]PT[GeV]

HERA SppS

A shape of the inclusive charged particle spectra

Difference in colliding particles and energies in production mechanism for high and low PT

Similarity in spectrum shape

pW=200 GeV

ppW=560 GeV

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

A comparison of inclusive spectra for resonances

Difference in a type of produced resonances

Similarity in spectrum shape and an absolute normalization

1/(2

j+1)

d/(

dydp

T2 )

[nb

/GeV

2 ]

M+PT [GeV]

}H1 Prelim

HERAphotoproduction

0

f0f2+ published

The invariant cross sections are taken for one spin and one isospin projections.

M – is a nominal mass of a resonance

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.

Stochasticity

Power law

dN/dPt ~

(1 + )Pt

P0

1

n

Ubiquity of the Power law

Geomagnetic Plasma Sheet

Plasma sheet is hot - KeV, (Ions, electrons)Low density – 10 part/cm3Magnetic field – open system COLLISIONLESS PLASMA

Energy distribution in a collisionless plasma

“Kappa distribution”

Polar Aurora,First Observed in 1972

Flux ~

(1 + )Eκθ

1

κ+ 1

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.

Turbulence

Measurements of one-dimensional longitudinal velocity spectra

1500

30

Re

Damping by viscosity at the Kolmogorov scale

1

4

v = ()1

4

with a velocity

Empirical Gutenberg-Richter LawEmpirical Gutenberg-Richter Law

Earthquakes

log(Frequency) vs. log(Area)

Avalanches and LandslidesAvalanches 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)

log(Frequency) vs. log(Time duration)

Solar Flares

Rains

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

Zipf, 1949: Human Behaviour and the Principle of Least Effort .

Human activityHuman activity

Male earnings Settlement size

First pointed out by George Kingsley Zipf and Pareto

Sexual contactsSexual contacts

survey of a random sample of 4,781 Swedes (18–74 years)

A number of partners within 12 months

≈ 2.5

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:

• Conclusion: These distributions arise because the same stochastic process is at work, and this process can be understood beyond the context of each example

(… + many others)

Maximum Entropy Principle

In 50th E.T.Jaynes has promoted the Maximum Entropy Principle (MEP)

The MEP states that the physical observable has adistribution, consistent with given constraints which maximizes the entropy.

WHO defines a form of statistical distributions?(Exponential, Poisson, Gamma, Gaussian, Power-law, etc.)

S = - pi log (pi)Shannon-Gibbs entropy:

Flat probability distribution

dSdPi

= - ln(Pi) – 1 = 0Shannon entropy maximization

subject to constraint (normalization)

dSdPi

dgdPi

- = 0Method of LagrangeMultipliers ()

- ln(Pi) – 1 - = 0

Pi = exp = 1/N

g = Pi = 1i=1

N

For continuous distribution with a<x<b P(x) = 1/(b-a)

All states (1< i < N) have equal probabilities

Exponential distribution

Shannon entropy maximization subject to constraints

(normalization and mean value)

Method of LagrangeMultipliers ()

- ln(Pi) – 1 - - Ei = 0

Pi = exp(1Ei) = A expEi)

g = Pi = 1i=1

N

= Pi Ei = i=1

N

dSdPi

dgdPi

- - = 0ddPi

For continuous distribution (x>0) P(x) = (1 / exp(-x / )

Exponential distribution (examples)

A. Random events with an average density D=1 /

B. Isolated ideal gas volume

Total Energy (E=) and number of molecules (N) are conserved

ε

log

(dN

/d)

E

N = = kT

Power-law distribution

Shannon entropy maximization subject to constraints

(normalization and geometric mean value)

Method of LagrangeMultipliers ()

- ln(Pi) – 1 - - xi = 0

Pi = exp(1xi) = A expxi)

g = Pi = 1i=1

N

dSdPi

dgdPi

- - = 0ddPi

For continuous distribution (x>0) P(x) = (1 / exp(-x / )

= Pi ln(xi) = ln(x)i=1

N

Power-law distribution (examples)

A.Incompressible N-dimensional volumes(Liouville Phase Space Theorem)

B. Fractals

log(ε)

log

(dN

/d)

ii

N

ipx

1

Geomagnetic collisionless plasma

An average “information” is conserved

I = 1

N(ln(i))

i is a size of

i-object

Fractal structure of the protons

Scaling, self-similarity and power-law behavior are F2 properties,which also characterize fractal objects

Serpinsky carpet

... .

z = 10 20 50

1x =

10 100 1000D = 1.5849

Proton: 2 scales

1/x , (Q + Q )/Q 222o 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 GeV2 : /ndf = 0.82 !!!

With only 4 free parameters

Correlations

Constraint

Exponential Power Law

PiiPiln(0+i)

arithmetic mean geometric mean

1

N(i) (i))1/N

No …+ij+…

• For i < 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 *)

*) Leaving non-extensive Tsallis formulation for a conference in Brasil

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)

Energy conservation is an important to make a spectrum exponential: di

dt= 0 i = 0

i=1

N

i=1

N i

= 0d

dt i=1

N

log( i ) = 0

Assume a relative change of energy is zero:

This condition describes an open system with a small scale change compensated by a similarrelative change at very large scales.

A flap of a butterfly's wings in Brazil sets off a tornado in Texas

Butterfly effect

Statistical self-similarity means that the degree of complexity repeats at different scales instead of geometric patterns.

Fractals / Self-similarity

In fractals the average “information” is conserved I =

1

N(ln(i))