Universality of hadrons production and the Maximum Entropy Principle

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Universality of hadrons production and the
Maximum Entropy Principle
A.Rostovtsev
ITEP, Moscow
May 2004
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SppS
HERA
ds/dydPT2pb/GeV2
ds/dydPT2nb/GeV2
gp W200 GeV
pp W560 GeV
PTGeV
PTGeV
Difference in colliding particles and energies
in production mechanism for high
and low PT Similarity in spectrum shape
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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
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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
Difference in a type of produced
resonances Similarity in spectrum shape and an
absolute normalization
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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
Ubiquity of the Power law
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Plasma sheet is hot - KeV, (Ions, electrons) Low
density 10 part/cm3 Magnetic field open
system COLLISIONLESS PLASMA
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Energy distribution in a collisionless plasma
Polar Aurora, First Observed in 1972
Kappa distribution
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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.
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Measurements of one-dimensional longitudinal
velocity spectra
1500
Damping by viscosity at the Kolmogorov scale
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Re
with a velocity
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Empirical Gutenberg-Richter Law
log(Frequency) vs. log(Area)
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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)
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Forest fires
log(Frequency) vs. log(Area)
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log(Frequency) vs. log(Time duration)
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log(Frequency) vs. log(sizemm)
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First pointed out by George Kingsley Zipf and
Pareto
Zipf, 1949 Human Behaviour and the Principle of
Least Effort .
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A number of partners within 12 months
a 2.5
survey of a random sample of 4,781 Swedes (1874
years)
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(No Transcript)
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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
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• 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

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WHO defines a form of statistical
distributions? (Exponential, Poisson, Gamma,
Gaussian, Power-law, etc.)
In 50th 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
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Shannon entropy maximization
subject to constraint (normalization)
Method of Lagrange Multipliers (a )
- ln(Pi) 1 - a 0
All states (1lt i lt N) have equal probabilities
For continuous distribution with altxltb P(x)
1/(b-a)
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Shannon entropy maximization subject to
constraints
(normalization and mean value)
Method of Lagrange Multipliers (a , b)
- ln(Pi) 1 - a - b?Ei 0
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A. Random events with an average density D1 / e
e
B. Isolated ideal gas volume
Total Energy (ESe) and number of molecules (N)
are conserved
log (dN/de)
e
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Shannon entropy maximization subject to
constraints
(normalization and geometric mean
value)
Method of Lagrange Multipliers (a , b)
- ln(Pi) 1 - a - b?xi 0
For continuous distribution (xgt0) P(x) (1 / e
)?exp(-x / e )
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• Incompressible N-dimensional volumes
• (Liouville Phase Space Theorem)

Geomagnetic collisionless plasma
B. Fractals
log (dN/de)
An average information is conserved
ei is a size of i-object
log(e)
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Fractal structure of the protons
Scaling, self-similarity and power-law behavior
are F2 properties, which also characterize
fractal objects
Serpinsky carpet
D 1.5849
Proton 2 scales
Generalized expression for unintegrated structure
function
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Limited applicability of perturbative QCD
ZEUS hep-ex/0208023
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For x lt 0.01 ? 0.35 lt Q lt 120 GeV2 c2
/ndf 0.82 !!!
With only 4 free parameters
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Exponential
Power Law
SPiln(e0ei)
SPiei
arithmetic mean
geometric mean
(Sei)
(P(e0ei))1/N
1
N
eiej
No

• For ei lt e0 Power Law transforms into
  Exponential distribution
• Constraints on geometric and arithmetic mean
  applied together results in GAMMA distribution

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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
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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
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Statistical self-similarity means that the degree
of complexity repeats at different scales instead
of geometric patterns.
In fractals the average information is conserved