The number e^{(1/2)} is the ratio between the time of maximum value and the time of maximum growth rate for restricted growth phenomena?

Abstract

For many natural process of growth, with the growth rate independent of size due to Gibrat law and with the growth process following a log-normal distribution, the ratio between the time (D) for maximum value and the time (L) for maximum growth rate (inflexion point) is then equal to the square root of the base of the natural logarithm (e^{1/2}). On the logarithm scale this ratio becomes one half ((1/2)). It remains an open question, due to lack of complete data for various cases with restricted growth, whether this e^{1/2} ratio can be stated as e^{1/2}-Law. Two established examples already published, one for an epidemic spreading and one for droplet production, support however this ratio. Another example appears to be the height of humain body. For boys the maximum height occurs near 23 years old while the maximum growth rate is at the age near 14, and there ratio is close to e^{1/2}. The main theoretical base to obtain this conclusion is problem independent, provided the growth process is restricted, such as public intervention to control the spreading of communicable epidemics, so that an entropy is associated with the process and the role of dissipation, representing the mechanism of intervention, is maximized. Under this formulation the principle of maximum rate of entropy production is used to make the production process problem independent.