Transcript Document

Universality of hadrons production
and the Maximum Entropy Principle
A.Rostovtsev
ITEP, Moscow
May 2004
HERA
gp
W=200 GeV
ds/dydPT2[pb/GeV2]
ds/dydPT2[nb/GeV2]
A shape of the inclusive charged particle spectra
SppS
pp
W=560 GeV
PT[GeV]
Difference in colliding particles and energies
in production mechanism for high and low PT
Similarity in spectrum shape
PT[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
HERA
photoproduction
The invariant cross
sections are taken
for one spin and one
isospin projections.
r0
M – is a nominal
mass of a resonance
f0
f2
h
}
H1 Prelim
p+ published
Difference in a type of produced resonances
M+PT [GeV]
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
dN/dPt ~
1
Pt
(1 + P )
0
n
Ubiquity of the Power law
Geomagnetic Plasma Sheet
Plasma sheet is hot - KeV, (Ions, electrons)
Low density – 10 part/cm3
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
n3
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 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:
S = - S pi log (pi)
Flat probability distribution
Shannon entropy maximization
dS
= - ln(Pi) – 1 = 0
dPi
N
g = S Pi = 1
subject to constraint (normalization)
i=1
dS
Method of Lagrange
Multipliers (a )
dPi
-a
dg
dPi
- 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)
N
e = S Pi Ei = e
i=1
Method of Lagrange
Multipliers (a , b)
dS
-a
dg
de
-b
= 0
dPi
dPi
dPi
- 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 (dN/de)
e
E
= kT
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)
dS
-a
dg
de
-b
= 0
dPi
dPi
dPi
- 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
N
(Liouville Phase Space Theorem)
e =  xi pi
Geomagnetic collisionless plasma
log (dN/de)
B. Fractals
i =1
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 F2 properties,
which also characterize fractal objects
Serpinsky carpet
z = 10
20
50
D = 1.5849
1 10 .
x = . ..
100
1000
Proton: 2 scales
1/x , (Q 2o + Q2 )/Q2o
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 : c2 /ndf = 0.82 !!!
With only 4 free parameters
Exponential
SPiei
Constraint
arithmetic mean
1
N
Correlations
Power Law
SPiln(e0+ei)
geometric mean
(Sei)
((e0+ei))1/N
No
…+eiej+…
• For ei < e0 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:
dSei
Sei = 0
=0
i=1
dt
Assume a relative change of energy is zero:
N
S
i=1
ei
e
d
=0
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))