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))