Lerch distribution based on maximum nonsymmetric entropy principle: Application to Conway's game of life cellular automaton

Conway's Game of Life (GoL) is a biologically inspired computational model which can approach the behavior of complex natural phenomena such as the evolution of ecological communities and populations. The GoL frequency distribution of events on log-log scale has been proved to satisfy the power-law scaling. In this work, GoL is connected to the entropy concept through the maximum nonsymmetric entropy (MaxNSEnt) principle. In particular, the nonsymmetric entropy is maximized to lead to a general Zipf's law under the special auxiliary information parameters based on Hurwitz-Lerch Zeta function. The Lerch distribution is then generated, where the Zipf, Zipf-Mandelbrot, Good and Zeta distributions are analyzed as particular cases. In addition, the Zeta distribution is linked to the famous golden number. For GoL simulations, the Good distribution presented the best performance in log-log linear regression models for individual cell population, whose exponents were far from the golden number. This result suggests that individual cell population decays slower than a hypothetical slope equal to a (fast decaying) negative golden number. (c) 2021 Elsevier Ltd. All rights reserved.

Automated summary

Conway's Game of Life is a computational model inspired by biology that can simulate complex natural phenomena. By connecting GoL to the concept of entropy through the MaxNSEnt principle, a general Zipf's law is derived, with Lerch distribution and Zipf, Zipf-Mandelbrot, Good, and Zeta distributions as special cases.Additionally, the Zeta distribution is linked to the famous golden number, and the Good distribution shows the best performance in log-log linear regression models for individual cell population in GoL simulations.