Self-organized criticality in the 'Game of Life"

Abstract

THE 'Game of Life'^1,2 is a cellular automaton, that is, a lattice system in which the state of each lattice point is determined by local rules. It simulates, by means of a simple algorithm, the dynamical evolution of a society of living organisms. Despite its simplicity, the complex dynamics of the game are poorly understood. Previous interest in 'Life' has focused on the generation of complexity in local configurations; indeed, the system has been suggested to mimic aspects of the emergence of complexity in nature^1,2. Here we adopt a different approach, by using concepts of statistical mechanics to study the system's long-time and large-scale behaviour. We show that local configurations in the "Game of Life" self-organize into a critical state. Such self-organized criticality provides a general mechanism for the emergence of scale-free structures^3–5, with possible applications to earth-quakes^6,7, cosmology⁸, turbulence⁹, biology and economics¹⁰. By contrast to these previous studies, where a local quantity was conserved, 'Life' has no local conservation laws and therefore represents a new type of universality class for self-organized criticality. This refutes speculations that self-organized criticality is a consequence of local conservation¹¹, and supports its relevance to the natural phenomena above, as these do not involve any locally conserved quantities. The scaling is universal in the sense that the exponents that characterize correlation functions do not depend on details of the local rules.