Emergence of skew distributions in controlled growth processes

Abstract

Starting from a master equation, we derive the evolution equation for the size distribution of elements in an evolving system, where each element can grow, divide into two, and produce new elements. We then probe general solutions of the evolution equation, to obtain such skew distributions as power-law, log-normal, and Weibull distributions, depending on the growth or division and production. Specifically, repeated production of elements of uniform size leads to power-law distributions, whereas production of elements with the size distributed according to the current distribution as well as no production of new elements results in log-normal distributions. Finally, division into two, or binary fission, bears Weibull distributions. Numerical simulations are also carried out, confirming the validity of the obtained solutions.

A novel statistical analysis and interpretation of flow cytometry data

“…For CD4 T cells, the lognormal approximation becomes more accurate as time progresses. This is consistent with the existence of an initial transient distribution of autofluorescence which corresponds to the quiescent state of the cells at the start of the experiment; as cells are activated to divide, the lognormal (or at least skew-right) distribution emerges possibly as a result of growth and division processes [26]. For CD8 T cells, the validity of the approximation does not change much in time.…”

“…Thus the notation of Equation (5) explicitly includes the time-dependence of the autofluorescence density function, p ( t , ζ). Because these intracellular molecules are partitioned among daughter cells during cell division, the distribution of autofluorescence can be intuitively considered as a growth and fragmentation process, which is known to produce skew-right density functions such as the lognormal density function [26]. In fact, it has been shown [13] that the distribution of autofluorescence in the population can be well-approximated using a lognormal density function, and thus can be characterized by its mean and its variance.…”

Accessibility Measurement in Transportation Networks and Application to the Seoul Bus System

“…In contrast to the Gaussian distribution, such a Weibull distribution, which is a representative skew distribution, manifests the presence of correlations among time distances in the system. Specifically, it is known that the evolution of a social or natural system according to appropriate growth and division rates gives rise to skew distributions including the Weibull distribution (Choi et al ; Goh et al ). Indeed this approach has been applied successfully to explain the passenger flow distributions in the subway system (Lee et al ).…”

Time evolution of entropy in a growth model: Dependence on the description

“…In this work, we deal with the case that new elements are produced in uniform size x 0 , i.e., g(x, t) = δ(x − x 0 ). The stationary distribution is then given by a power-law function for x > x 0 [19]:…”

“…The general master equation governing the time evolution of the probability allows one to probe the time evolution of the entropy as well. To clarify the difference between the two entropies, we analyze the simple growth model, with no production or with uniform size production [18,19]. The state of each element is specified by its 'size', which can in general take continuous values.…”