Explanation of Log-Normal Distributions and Power-Law Distributions in Biology and Social Science

In many biological and social systems, where the ideal of heritance and aberrance exist at the same time, normal distribution failed in describing the distributions of variables. In stead, many real data as well as simulations indicate that log-normal distribution and power-law distribution can characterize many properties of this kind of systems. In this paper, the failure of normal distribution is studied and a general phenomenological model including both heritance and aberrance is set up. Using Fokker-Plank equation, it turns out that, log-normal distribution will be favored if the system has an optimum value for the variable we are studying, but power-law distribution will be favored for discrete variables.