FROM LOGNORMAL DISTRIBUTION TO POWER LAW: A NEW CLASSIFICATION OF THE SIZE DISTRIBUTIONS

2006 ◽  
Vol 17 (10) ◽  
pp. 1429-1436 ◽  
Author(s):  
LUCIEN BENGUIGUI ◽  
EFRAT BLUMENFELD-LIEBERTHAL
Keyword(s):  
Normal Distribution ◽  
Power Law ◽  
Lognormal Distribution ◽  
Size Distributions ◽  
New Classification ◽  
Multiplicative Process ◽  
Random Multiplicative Process ◽  
Log Normal ◽  
Log Normal Distribution

We propose a new classification of the size distributions of entities based on an exponent α defined from the shape of the log–log Rank Size plot. From an inspection of a large number of cases in different fields, one finds three possibilities: α = 1 giving a power law, α > 1 (parabola like curve) and 0 < α < 1 (analogous to a log normal distribution). A fourth possibility that can be defined when α < 0 was never observed. We present a modified version of models based on a random multiplicative process and an introduction of new entities during the growth. We recover all three kinds of distributions and show that the type of a distribution is conditioned by the rate of the introduction of new entities.

scholarly journals Quasi-Static Variation of Power-Law and Log-Normal Distributions of Urban Population

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 908
Author(s):  
Atushi Ishikawa ◽  
Shouji Fujimoto ◽  
Arturo Ramos ◽  
Takayuki Mizuno
Keyword(s):  
Normal Distribution ◽  
Power Law ◽  
Time Variation ◽  
Time Reversal ◽  
Urban Population ◽  
Large Scale ◽  
Time Reversal Symmetry ◽  
Scale Range ◽  
Log Normal ◽  
Log Normal Distribution

We analytically derived and confirmed by empirical data the following three relations from the quasi-time-reversal symmetry, Gibrat’s law, and the non-Gibrat’s property observed in the urban population data of France. The first is the relation between the time variation of the power law and the quasi-time-reversal symmetry in the large-scale range of a system that changes quasi-statically. The second is the relation between the time variation of the log-normal distribution and the quasi-time-reversal symmetry in the mid-scale range. The third is the relation among the parameters of log-normal distribution, non-Gibrat’s property, and quasi-time-reversal symmetry.

Study of the Effect of Pinning Particles on Grain Size Distributions

2013 ◽  
Vol 753 ◽  
pp. 361-366 ◽  
Author(s):  
Han Li ◽  
Suk Bin Lee ◽  
Anthony D. Rollett
Keyword(s):  
Grain Size ◽  
Normal Distribution ◽  
Grain Growth ◽  
Volume Fraction ◽  
Medium Size ◽  
Second Phase ◽  
Size Distributions ◽  
Grain Size Distributions ◽  
Log Normal ◽  
Log Normal Distribution

The present paper studies grain growth in the presence of inert particles by performing large-scale simulations using a parallel Monte Carlo Potts model. The effect of the second phase particles on the grain size distribution (GSD) is analyzed. The GSDs diverge markedly from log-normal distribution for normal grain growth case. For the cases with low volume fraction of particles, we find that the grain size distributions approach log-normal as stagnation takes hold. For the cases with a high volume fraction of particles, however, medium-size grains reach the log-normal distribution but both lower and upper tails diverge noticeably from the log-normal distribution over time.

scholarly journals The Probability Distribution of Worldwide Forest Areas

2021 ◽  
Vol 13 (3) ◽  
pp. 1361
Author(s):  
Rafael González-Val
Keyword(s):  
Probability Distribution ◽  
Normal Distribution ◽  
Power Law ◽  
Power Laws ◽  
Theoretical Models ◽  
Forest Growth ◽  
Plausible Model ◽  
Forest Coverage ◽  
Log Normal ◽  
Log Normal Distribution

This paper analyses the probability distribution of worldwide forest areas. We find moderate support for a Pareto-type distribution (power law) using FAO data from 1990 to 2015. Power laws are common features of many complex systems in nature. A power law is a plausible model for the world probability distribution of forest areas in all examined years, although the log-normal distribution is a plausible alternative model that cannot be rejected. The random growth of forest areas could generate a power law or log-normal distribution. We study the change in forest coverage using parametric and non-parametric methods. We identified a slight convergence of forest areas over the time reviewed; however, random forest area growth cannot be rejected for most of the distribution of forest areas. Therefore, our results give support to theoretical models of stochastic forest growth.

scholarly journals Using a combined power law and log-normal distribution model to simulate particle formation and growth in a mobile aerosol chamber

2016 ◽  
Author(s):  
M. Olin ◽  
T. Anttila ◽  
M. Dal Maso
Keyword(s):  
Normal Distribution ◽  
Power Law ◽  
Particle Formation ◽  
Distribution Model ◽  
Aerosol Formation ◽  
Aerosol Chamber ◽  
Relative Errors ◽  
Log Normal ◽  
Log Normal Distribution ◽  
Particle Formation And Growth

Abstract. We present the combined power law and log-normal distribution (PL+LN) model, a computationally efficient model to be used in simulations where the particle size distribution cannot be accurately represented by log-normal distributions, such as in simulations involving the initial steps of aerosol formation, where new particle formation and growth occur simultaneously, or in the case of inverse modelling. The model was validated against highly accurate sectional models using input parameter values that reflect conditions typical to particle formation occurring in the atmosphere and in vehicle exhaust, and tested in the simulation of a particle formation event performed in a mobile aerosol chamber at Mäkelänkatu street canyon measurement site in Helsinki, Finland. The number, surface area, and mass concentrations in the chamber simulation were conserved with the relative errors lower than 2 % using the PL+LN model, whereas a moment-based log-normal model and sectional models with the same computing time as with the PL+LN model caused relative errors up to 10 % and 135 %, respectively.

scholarly journals Using a combined power law and log-normal distribution model to simulate particle formation and growth in a mobile aerosol chamber

2016 ◽  
Vol 16 (11) ◽  
pp. 7067-7090 ◽  
Author(s):  
Miska Olin ◽  
Tatu Anttila ◽  
Miikka Dal Maso
Keyword(s):  
Normal Distribution ◽  
Power Law ◽  
Particle Formation ◽  
Distribution Model ◽  
Aerosol Formation ◽  
Aerosol Chamber ◽  
Relative Errors ◽  
Log Normal ◽  
Log Normal Distribution ◽  
Particle Formation And Growth

Abstract. We present the combined power law and log-normal distribution (PL+LN) model, a computationally efficient model to be used in simulations where the particle size distribution cannot be accurately represented by log-normal distributions, such as in simulations involving the initial steps of aerosol formation, where new particle formation and growth occur simultaneously, or in the case of inverse modeling. The model was evaluated against highly accurate sectional models using input parameter values that reflect conditions typical to particle formation occurring in the atmosphere and in vehicle exhaust. The model was tested in the simulation of a particle formation event performed in a mobile aerosol chamber at Mäkelänkatu street canyon measurement site in Helsinki, Finland. The number, surface area, and mass concentrations in the chamber simulation were conserved with the relative errors lower than 2 % using the PL+LN model, whereas a moment-based log-normal model and sectional models with the same computing time as with the PL+LN model caused relative errors up to 17 and 79 %, respectively.

scholarly journals A Universal Model for the Log-Normal Distribution of Elasticity in Polymeric Gels and Its Relevance to Mechanical Signature of Biological Tissues

Biology ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 64
Author(s):  
Arnaud Millet
Keyword(s):  
Elastic Modulus ◽  
Normal Distribution ◽  
Biological Tissues ◽  
Force Microscopy ◽  
Polymeric Gels ◽  
Atomic Force ◽  
Clear Explanation ◽  
Log Normal ◽  
Log Normal Distribution

The mechanosensitivity of cells has recently been identified as a process that could greatly influence a cell’s fate. To understand the interaction between cells and their surrounding extracellular matrix, the characterization of the mechanical properties of natural polymeric gels is needed. Atomic force microscopy (AFM) is one of the leading tools used to characterize mechanically biological tissues. It appears that the elasticity (elastic modulus) values obtained by AFM presents a log-normal distribution. Despite its ubiquity, the log-normal distribution concerning the elastic modulus of biological tissues does not have a clear explanation. In this paper, we propose a physical mechanism based on the weak universality of critical exponents in the percolation process leading to gelation. Following this, we discuss the relevance of this model for mechanical signatures of biological tissues.

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