Scaling Relations of Lognormal Type Growth Process with an Extremal Principle of Entropy

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.

Keywords

• Shannon entropy
• growth process
• scaling relation