Phase transitions in a conservative Game of Life

TL;DR

This paper explores a conservative variant of Conway's Game of Life, revealing phase transitions between active and inactive states as dead cell density varies, through simulations and mean-field analysis.

Contribution

It introduces a conservative version of the Game of Life and characterizes its phase transitions using both lattice simulations and mean-field calculations.

Findings

Identifies a discontinuous phase transition from inactive to active phase.

Discovers a re-entrant transition back to inactivity at higher dead cell densities.

Shows differences between lattice and mean-field transition types.

Abstract

We investigate the dynamics of a conservative version of Conway's Game of Life, in which a pair consisting of a dead and a living cell can switch their states following Conway's rules but only by swapping their positions, irrespective of their mutual distance. Our study is based on square-lattice simulations as well as a mean-field calculation. As the density of dead cells is increased, we identify a discontinuous phase transition between an inactive phase, in which the dynamics freezes after a finite time, and an active phase, in which the dynamics persists indefinitely in the thermodynamic limit. Further increasing the density of dead cells leads the system back to an inactive phase via a second transition, which is continuous on the square lattice but discontinuous in the mean-field limit.