Abstract
The scale, inflexion point and maximum point are important scaling parameters for studying growth phenomena with a size following the lognormal function. The width of the size function and its entropy depend on the scale parameter (or the standard deviation) and measure the relative importance of production and dissipation involved in the growth process. The Shannon entropy increases monotonically with the scale parameter, but the slope has a minimum at p 6/6. This value has been used previously to study spreading of spray and epidemical cases. In this paper, this approach of minimizing this entropy slope is discussed in a broader sense and applied to obtain the relationship between the inflexion point and maximum point. It is shown that this relationship is determined by the base of natural logarithm e ' 2.718 and exhibits some geometrical similarity to the minimal surface energy principle. The known data from a number of problems, including the swirling rate of the bathtub vortex, more data of droplet splashing, population growth, distribution of strokes in Chinese language characters and velocity profile of a turbulent jet, are used to assess to what extent the approach of minimizing the entropy slope can be regarded as useful.
Highlights
Growth phenomena or equivalent growth phenomena exist in many natural and technological processes and most often involve competitive production and dissipation mechanisms to make the size grow initially and decay
We first study the variation of Shannon entropy with respect to the scale parameter σ and establish some relations between the locations and sizes of the inflexion point and maximum point of (1)
We consider in this paper the growth process with a size described by the lognormal function
Summary
Growth phenomena or equivalent growth phenomena exist in many natural and technological processes and most often involve competitive production and dissipation mechanisms to make the size grow initially and decay. We try to explore in this paper more properties of the approach used in [9,21], and to test more cases in order to assess whether this approach can be applied in various problems It was claimed in [9,21] that the principle of maximum rate of entropy production is used while obtaining (2). F (tD) are useful in more situations other than those considered in [9,21] These cases include the swirling rate of the bathtub vortex, more data of droplet splashing, population growth, distribution of strokes in Chinese language characters and the velocity profile of a turbulent jet. Some discussion will be provided to open a question about the value of the Karman constant in turbulence flow
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