GENESIS OF THE LOGNORMAL MULTIPLICITY DISTRIBUTION IN THE e+e− COLLISIONS AND OTHER STOCHASTIC PROCESSES

1990 ◽  
Vol 05 (23) ◽  
pp. 1851-1869 ◽  
Author(s):  
R. SZWED ◽  
G. WROCHNA ◽  
A. K. WRÓBLEWSKI
Keyword(s):  
Limit Theorem ◽  
Multiplicity Distribution ◽  
Scaling Function ◽  
Average Multiplicity ◽  
Linear Functions ◽  
Scale Invariant ◽  
The Central Limit Theorem ◽  
The Mean ◽  
Multiplicity Distributions ◽  
The Scaling Function

It has been observed that the e+e− multiplicity distributions exhibit the following properties: the dispersions are linear functions of the mean and the distributions obey the KNO-G scaling with the scaling function of the lognormal shape. In this paper the scale invariant branching is assumed as a mechanism within which all these properties could be derived. It is shown that the lognormal shape of the scaling function can be obtained within proposed mechanism by using the generalization of the Central Limit Theorem. The dependence of the average multiplicity on energy is also derived within the postulated framework. It is also shown that many other phenomena encountered in nature have the similar statistical properties.

NEW AMY AND DELPHI MULTIPLICITY DATA AND THE LOGNORMAL DISTRIBUTION

1991 ◽  
Vol 06 (03) ◽  
pp. 245-257 ◽  
Author(s):  
R. SZWED ◽  
G. WROCHNA ◽  
A.K. WRÓBLEWSKI
Keyword(s):  
Energy Dependence ◽  
Lognormal Distribution ◽  
Scaling Function ◽  
Average Multiplicity ◽  
Linear Functions ◽  
The Mean ◽  
Multiplicity Distributions ◽  
The Scaling Function ◽  
Skewness And Kurtosis

Multiplicity distributions for e+e−→ hadrons recently reported by the AMY and DELPHI collaborations are compared with the data obtained at lower energies. It is proven that the new data obey the KNO-G scaling and the scaling function can be described by the lognormal distribution. The dispersions are linear functions of the mean as for the data measured at lower energies and the standardized moments (such as skewness and kurtosis) are independent of the energy. The energy dependence of the average multiplicity is described by <nch>=β sα−1.

FINITE-RANGE CONTACT PROCESS ON THE MARKET RETURN INTERVALS DISTRIBUTIONS

2010 ◽  
Vol 13 (05) ◽  
pp. 643-657 ◽  
Author(s):  
JUNHUAN ZHANG ◽  
JUN WANG ◽  
JIGUANG SHAO
Keyword(s):  
Stock Price ◽  
Statistical Physics ◽  
Scaling Function ◽  
Contact Process ◽  
Composite Index ◽  
Finite Range ◽  
Price Model ◽  
Simulation Data ◽  
The Mean ◽  
The Scaling Function

Stochastic system is applied to describe and investigate the fluctuations of stock price changes in a stock market, and a stock price model is developed by the finite-range contact process of the statistical physics systems. In this paper, the scaling behaviors of the return intervals for SSE Composite Index (SSE) and the simulation data of the model are investigated and compared. The database is from the index of SSE in the 6-year period for every 5 minutes, and the simulation data is from the finite-range contact model for different values of the range R. For different values of threshold θ, the statistical analysis shows that the probability density function Pθ(τ) of the return intervals τ for both SSE and the simulation data have similar scaling form, that is [Formula: see text] ([Formula: see text] is the mean return interval), where the scaling function h(x) can be approximately fitted by the function h(x) = ωe-a(ln x)γ, and ω, a, γ are three parameters. Further, with different values of R and θ, the statistical comparison of SSE Composite Index and simulation data are given.

Regression to the Mean

2019 ◽  
pp. 203-226
Author(s):  
Steven J. Osterlind
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
General Public ◽  
Industrial Revolution ◽  
Charles Darwin ◽  
International Health ◽  
Regression To The Mean ◽  
The Central Limit Theorem ◽  
The Mean ◽  
Set Up

This chapter focuses on two events that started the transformation to a quantifying worldview for the general public: (1) developments in transportation, especially the invention of the train (meaning people and goods could travel further) and (2) the consequent tremendous economic expansion which led to a full-blown industrial revolution, first in England and then in America. Work by Charles Darwin showed the broadening impact of quantitative thinking on the discipline of sociology. The chapter also discusses the accomplishments of Francis Galton, including his landmark work Hereditary Genius, the invention of Galton’s bean machine (“quincunx”), which demonstrated the central limit theorem, and his Anthropometric Laboratory, which he set up at the International Health Exhibition to measure mental faculties. Galton also discovered the concept of correlation and “reversion to the mean,” evolving the latter into “regression to the mean,” and invented many other statistical concepts, such as quartile, decile, and ogive.

Directed Polymers in Restricted Geometries

1992 ◽  
Vol 290 ◽  
Author(s):  
G. Zumofens ◽  
J. Klafter ◽  
A. Blumen
Keyword(s):  
Scaling Function ◽  
Random Potential ◽  
Potential Fields ◽  
Directed Polymers ◽  
Transverse Displacement ◽  
Fractal Nature ◽  
One Dimensional ◽  
Characteristic Exponents ◽  
The Mean ◽  
The Scaling Function

AbstractWe study numerically directed polymers in random potential fields for one-dimensional and fractal substrates. For fractal substrates the time evolution of the mean transverse fluctuations depends besides on the randomness of the potential also on the fractal nature of the substrate. The two effects enter in a subordinated way, i.e. the corresponding characteristic exponents due to the potential and the substrate combine multiplicatively. For a one-dimensional substrate the propagator P(x, t), the probability distribution of the transverse displacement x(t), follows the scaling form P(x, t) ∼ 〈x2(t)〉-1/2f (ξ), where ξ is the scaling variable ξ = x/〈x2(t)〉1/2. The numerical results support the scaling function f (ξ) ∼ exp (-cξδ) with δ > 2 which indicates an “enhanced” Gaussian behavior. These results are compared with those of a related “toy model”.

scholarly journals A Study of Pion-Nucleus Interactions in terms of Compound Particles

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Tufail Ahmad
Keyword(s):  
Multiplicity Distribution ◽  
Charged Particles ◽  
Mean Value ◽  
Shower Particle ◽  
Particle Multiplicity ◽  
Compound Multiplicity ◽  
The Mean ◽  
Multiplicity Distributions ◽  
Distribution Shifts ◽  
Compound Particle

We report some results on the compound multiplicity distribution at 340 GeV/c π- nucleus interactions. Compound multiplicity distribution is found to depend on the target size and the distribution becomes broader. The peak of the distribution shifts towards higher values of the compound particle multiplicity. Mean compound multiplicity is found to vary linearly with grey, heavy, and shower particle multiplicity. Correlations between different particle multiplicities have been studied in detail. Dispersion of compound multiplicity distributions and its ratio with the mean value is observed to obey a linear relationship with different particle multiplicities except for shower particles where dispersion is almost independent of shower particles. Mean normalized multiplicity has also been studied in terms of created charged particles.

A CONNECTION OF INELASTICITY WITH MULTIPLICITY DISTRIBUTION AT HIGH ENERGIES

1992 ◽  
Vol 07 (26) ◽  
pp. 2401-2406 ◽  
Author(s):  
ISAY GOLYAK
Keyword(s):  
Multiplicity Distribution ◽  
Secondary Particles ◽  
High Energies ◽  
The Mean ◽  
Multiplicity Distributions ◽  
Scaling Invariant

The widely known experimental value of the mean coefficient of the inelasticity <K>~0.5 is calculated by the investigation of a connection of the inelasticity with KNO scaling invariant multiplicity distributions of secondary particles.

The Central Limit Theorem and the Law of Large Numbers for Pair-Connectivity in Bernoulli Trees

1989 ◽  
Vol 3 (4) ◽  
pp. 477-491
Author(s):  
Kyle T. Siegrist ◽  
Ashok T. Amin ◽  
Peter J. Slater
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Law Of Large Numbers ◽  
Network Reliability ◽  
Random Variable ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
The Mean ◽  
The Law

Consider the standard network reliability model in which each edge of a given (n, m)-graph G is deleted, independently of all others, with probability q = 1– p (0 <p < 1). The pair-connectivity random variable X is defined to be the number of connected pairs of vertices that remain in G. The mean of X has been proposed as a measure of reliability for failure-prone communications networks in which the edge deletions correspond to failures of the communications links. We consider deviations from the mean, the law of large numbers, and the central limit theorem for X as n → ∞. Some explicit results are obtained when G is a tree using martingale difference sequences. Stars and paths are treated in detail.

Cumulative hadron production in pA collisions in the framework of z-scaling

2015 ◽  
Vol 39 ◽  
pp. 1560110
Author(s):  
Aparin Alexey ◽  
Tokarev Mikhail
Keyword(s):  
Cross Section ◽  
Particle Production ◽  
Scaling Function ◽  
Inclusive Cross Section ◽  
Hadron Production ◽  
Average Multiplicity ◽  
Self Similarity ◽  
Kinematic Range ◽  
Different Types ◽  
The Scaling Function

Data on cumulative particle production in pA collisions with momentum [Formula: see text] are analyzed in the [Formula: see text]-scaling approach. Scaling function [Formula: see text] is constructed for different types of target nuclei and inclusive particle angle. The function is expressed via respective inclusive cross section and average multiplicity density of charged particles. This regime corresponds to low [Formula: see text] (low transverse momentum [Formula: see text]). Validity of the concept was verified for this data. A-dependence of the scaling function is studied for low-[Formula: see text] region. Self-similarity of hadron production in pA collisions is confirmed over a wide kinematic range.

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