The Game of Life at finite temperature

In this paper we propose a cellular array model whose transition function is based on a continuously valued expression that includes a temperature parameter T and two shift parameters E 0 and x 0 . This transition function coincides with the transition function of the Game of Life cellular automaton model when the temperature approaches the limit T → 0 . States in this model are continuously valued for T > 0 , and cell state transitions, though deterministic, are affected by thermal fluctuation. Spectral analysis of the cell state’s time series shows a 1 / f spectrum in the low temperature region, as in the Game of Life, and a Lorentzian spectrum in the high temperature region. Analysis of the average entropy shows a division along approximately the same critical point into a low temperature region indicative of the Game of Life and a high temperature region with different behavior. We show that the model is robust to deviations in the values of parameters E 0 and x 0 , but that the robustness declines with an increase in temperature, up to a critical temperature at which all robustness is lost.