# Thermodynamics in Stochastic Conway’s Game of Life

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction to Classical Conway’s Game of Life (CCGoL)

- 1.
- If a dead cell has exactly 3 neighbors, it comes alive in the next cycle.
- 2.
- If a living cell has 2 or 3 neighbors, it survives in the next cycle.
- 3.
- If a cell has a different number of neighbors than stated above, it will be dead in the next cycle.

## 2. Introduction to Stochastic Classical Conway’s Game of Life (SCCGoL)

**SCCGoL(D)**). One shall consider all possible scenarios (cell configurations) or subsets of it characterized by a given probability that might take place during the next time step as depicted in Figure 5a. This is somewhat similar to the case of Quantum Mechanics, where one evolves from one point in space to another point in space over a large class of trajectories formally recognized as path integral approach. Instead of discrete values of 0 and 1, we introduce cell states that have continuous values between 0 and 1, which are called mass that will be later assigned to Stochastic Conway’s Game of Life in Continuous mode =

**SCCGoL(C)**. Due to the fact that SCCGoLs have different rules from CCGoLs, cells almost never have exactly two or three neighbors. A condition for a given cell to come to life from a dead cell state (creationism of a live cell) is that it has a number of neighbors in a certain range of values. Similar rules apply to a living cell, justifying its live or dead state in the next time iteration. By setting standard intervals of allowed/forbidden numbers of neighboring values, in which the cell is alive/dead, and by adding the additional spontaneous rule probability for the cell to change in the next iteration (probability of changing the state of a cell regardless of the number of neighbors), we are able to create a simulation where the cells almost never die, since it is very difficult from a probabilistic point of view for a given cell to stay alive. If we modify the system dynamics starting from the live cell configuration, it has fixed $1-p$ probability to follow a standard Conway’s Game of Life neighborhood condition for being dead/alive in the already defined intervals during the next time step, and if we select p probability for a cell to change its state during the next time step independently from its neighbors, we can trace various dynamics as we systematically increase the spontaneous probability level p from $0\%$ to $100\%$, resulting in the graph depicted in Figure 6. The algorithm is based on consecutive steps that are shown in Figure 5b.

## 3. Generalization of Stochastic Classical Conway’s Game of Life (SCCGoL) to the Case of $\mathit{N}$ Competing Cellular Automaton Species

## 4. Methodology of Describing SCCGoL Dynamics Using Tools of Classical Statistical Physics

## 5. Numerical Analysis of SCCGoL Dynamics with Methodology of Classical Statistical Physics

**Figure 10.**(

**a**,

**b**) Evolution of diffusion process in cellular automaton system for a limited lattice size of 100 by 100 in SCCGoL(C) (averaged over 1000 trials). The final saturation of mass and entropy (as depicted in Figure 9) implies values approaching thermodynamic equilibrium with characteristic fluctuations of mass and entropy around effective stationary values.

## 6. Numerical Study of One-Species Cellular Automata with Various Boundary Conditions

## 7. Numerical Study of Two-Species Cellular Automata in Perturbative Interaction by Narrow Constriction

## 8. Behavior of One-Species Cellular Automata with Cycle Movement of Barriers Mimicking Thermodynamical Cycle

## 9. Conclusions and Future Perspectives

- 1.
- 2.
- 3.
- 4.
- 5.
- 6.
- Identification of a brief Shannon entropy peak that later minimizes and saturates in SGoL (Figure 9 and Figure 12). Monotonicity in increase of entropy is twice briefly interrupted by a small decline, which is associated with automaton cells “colliding” with the barriers and experiencing a brief propagation slowdown (Figure 14).

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Berto, F.; Tagliabue, J. Cellular Automata. In The Stanford Encyclopedia of Philosophy, Spring 2022 ed.; Zalta, E.N., Ed.; Metaphysics Research Lab, Stanford University: Stanford, CA, USA, 2022. [Google Scholar]
- Wolfram, S. Statistical mechanics of cellular automata. Rev. Mod. Phys.
**1983**, 55, 601. [Google Scholar] [CrossRef] - Gardner, M. Mathematical Games—The fantastic combinations of John Conway’s new solitaire game “life”. Sci. Am.
**1970**, 223, 120–123. [Google Scholar] [CrossRef] - Bandyopadhyay, P.S.; Grunska, N.; Dcruz, D.; Greenwood, M.C. Are Scientific Models of Life Testable? A Lesson from Simpson’s Paradox. Sci
**2021**, 3, 2. [Google Scholar] [CrossRef] - Peitgen, H.O.; Jürgens, H.; Saupe, D. Chaos and Fractals; Springer: Cham, Switzerland, 1983. [Google Scholar]
- Aguilera-Venegas, G.; Galán-García, J.L.; Egea-Guerrero, R.; Galán-García, M.Á.; Rodríguez-Cielos, P.; Padilla-Domínguez, Y.; Galán-Luque, M. A probabilistic extension to Conway’s Game of Life. Adv. Comput. Math.
**2019**, 45, 2111–2121. [Google Scholar] [CrossRef] - Vandevelde, S.; Vennekens, J. ProbLife: A Probabilistic Game of Life. arXiv
**2022**, arXiv:2201.09521. [Google Scholar] - Shannon, C. A Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Shannon, C. A Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 623–656. [Google Scholar] [CrossRef] - Velazquez Abad, L. Principles of classical statistical mechanics: A perspective from the notion of complementarity. Ann. Phys.
**2012**, 327, 1682–1693. [Google Scholar] [CrossRef] - Mishin, Y. Thermodynamic Theory of Equilibrium Fluctuations; Elsevier: Amsterdam, The Netherlands, 2015. [Google Scholar]
- Feynman, R.P. Statistical Mechanics; Westview Press: Boulder, CO, USA, 1972. [Google Scholar]
- Huang, K. Introduction to Statistical Physics; CRC Press: Boca Raton, FL, USA, 2001. [Google Scholar]
- Huang, K. Statistical Mechanics; John Wiley & Sons: New York, NY, USA, 1963. [Google Scholar]
- Baez, J.C.; Pollard, B.S. Quantropy. arXiv
**2013**, arXiv:1311.0813. [Google Scholar] [CrossRef] - Abe, S.; Okuyama, S. Similarity between quantum mechanics and thermodynamics: Entropy, temperature, and Carnot cycle. Phys. Rev. E Stat. Nonlin. Soft Matter Phys.
**2011**, 83, 021121. [Google Scholar] [CrossRef] [PubMed] - Pomorski, K. Equivalence between Classical Epidemic Model and Quantum Tight-Binding Model; Springer: Cham, Switzerland, 2022. [Google Scholar]
- Pomorski, K. Equivalence between finite state stochastic machine, non-dissipative and dissipative tight-binding and Schroedinger model. Math. Comput. Simul.
**2023**, 209, 362–407. [Google Scholar] [CrossRef] - Kotula, D.; Pomorski, K. Thermodynamics of Stochastic Conway Game of Life; ShanghaiAI Lectures. 2022. Available online: https://youtu.be/kLOB9VlF-R4 (accessed on 1 January 2023).
- Flitney, A.P.; Abbott, D. A Semi-quantum Version of the Game of Life. In Advances in Dynamic Games: Applications to Economics, Finance, Optimization, and Stochastic Control; Nowak, A.S., Szajowski, K., Eds.; Birkhäuser Boston: Boston, MA, USA, 2005; pp. 667–679. [Google Scholar]

**Figure 1.**Evolution of a one-dimensional cellular automaton in successive cycles with left side partial logical negation rule (if the state of nearest left cell is alive, then the state of a given cellular automaton will change to its opposite).

**Figure 2.**Evolution of various topologies of cellular automaton structures over time with deterministic rules of CCGoL. Two dynamically unstable structures and two structures can be identified that have dynamical stability over time.

**Figure 3.**Evolution of the “glider” configuration (

**a**–

**e**) of cellular automata propagating over time in deterministic CCGoL.

**Figure 4.**Evolution of the “toad” configuration of cellular automata acting as an oscillator in successive cycles in CCGoL, where numbers correspond to number of living neighbors of the given cell. The blue color of digits means that in the next cycle the given cell will be dead, while in case of the green ones, that in the next cycle the given cell will be alive.

**Figure 5.**Computation procedure for Stochastic Classical Conway’s Game of Life (SCCGoL). (

**a**) Possible scenarios during one time step in Stochastic Classical Conway’s Game of Life (D/C) with Discrete $\{0,1\}$ or Continuous lattice point values. (

**b**) Flowchart of SCCGoL(D/C) simulation with predefined initial structure, probability level and maximum number of simulation cycles (time indices).

**Figure 6.**Schematic view of initial conditions in SCCGoL(D) for different lattice sizes ((

**a**) N = 10 and (

**b**) N = 20 correspondingly); (

**c**–

**f**) Dependence of automata population average cycle lifetime (over 1000 trials) on probability of spontaneous change of cell state from alive to dead and conversely, with preservation of standard rules in Conway’s Game of Life.

**Figure 7.**(

**a**) Initial mass distribution of one species of cellular automata using a two-dimensional isotropic Gaussian function. (

**b**) Example of initial distribution of three cellular automaton tribes with boundary existence between each separate tribe and a rule that one geometrical place on the lattice is occupied by the species of cellular automata with dominant mass.

**Figure 8.**Evolution of map distribution of four cellular automaton populations of different species with 100 by 100 lattice size (

**a**,

**b**). One can notice the tendency of each species to occupy the maximum possible territory at the cost of other species in four-species SCCGoL(C) = SCCGoL(C)4S, which can be understood as a weak antagonistic relation.

**Figure 9.**Evolution of mass and entropy in successive cycles in an SCCGoL(C) manifesting maximization with saturation and approaching stationary thermodynamic equilibrium with initial conditions depicted in Figure 10.

**Figure 11.**Dynamics of thermodynamic parameters (mass, entropy, and temperature) with simulation time in SCCGoL(C) (lattice size 100 by 100), also given in Figure 9. Thermodynamic equilibrium is accompanied with a final distribution of negative temperature (starting from positive temperature distribution, in accordance with Formula (3)), as an experimentally observed necessary criteria for final thermodynamic stability. One can spot various similarities between the statistical behavior of SCCGoL(C) and physical systems described by classical thermodynamics (maximization and saturation of entropy, decay of temperature gradients and final thermalization, uniform distribution of mass and energy). (

**a**) Mass at t = 4. (

**b**) Mass at t = 26. (

**c**) Mass at t = 92. (

**d**) Entropy at t = 4. (

**e**) Entropy at t = 26. (

**f**) Entropy at t = 92. (

**g**) $T(x,y,t=4)$. (

**h**) $T(x,y,t=26)$. (

**i**) $T(x,y,t=92)$. (

**j**) $\frac{dm}{dt}$ with time. (

**k**) $\frac{dS}{dt}$ with time. (

**l**) Temperature with time.

**Figure 12.**Diffusion process in the

**SCCGoL(C)**p20L100b100 system of two weakly interconnected chambers (

**a**,

**b**) connected by means of two small holes in the barrier (lattice size 100 by 100, 20% is p probability of spontaneous change of cell state from alive to dead and conversely, with preservation of standard rules in Conway’s Game of Life, as also given in Figure 6). Two stages of diffusion can be noticed in the mass and entropy dynamics (

**c**), corresponding to two consecutive processes: full diffusion in the left chamber leading to full diffusion in the right chamber. More details of the evolution of the space dependence of the thermodynamic parameters over time are given by Figure 13.

**Figure 13.**Dynamics of thermodynamic parameters with simulation time in

**SCCGoL(C)**p20L100b100 with two chambers weakly connected by two small holes in a barrier, as initially depicted in Figure 12. (

**a**) Mass at t = 5. (

**b**) Mass at t = 70. (

**c**) Mass at t = 100. (

**d**) Entropy at t = 5. (

**e**) Entropy at t = 70. (

**f**) Entropy at t = 100. (

**g**) $T(x,y,t=5)$. (

**h**) $T(x,y,t=70)$. (

**i**) $T(x,y,t=100)$. (

**j**) $\frac{dm}{dt}$ with time. (

**k**) $\frac{dS}{dt}$ with time. (

**l**) Temperature with time.

**Figure 14.**Dynamics of diffusion for a system

**SCCGoL(C)**p20L100b100 with two barriers, with two small holes in each barrier (generalization of situation from Figure 13) creating three weakly interconnected chambers perturbed by mutual interactions mediated by holes (

**a**,

**b**). Monotonicity in increase of entropy (

**c**) is briefly interrupted twice by a small decline, which is associated with automaton cells “colliding” with barriers and experiencing a short lasting slowdown in propagation. Details on the evolution of space dependent thermodynamic parameters over time are given in Figure 15.

**Figure 15.**Space dependent dynamics of thermodynamic parameters with simulation time in

**SCCGoL(C)**p20L100b100 for three chambers weakly connected by four small holes, also depicted in Figure 14. (

**a**) Mass at t = 30. (

**b**) Mass at t = 70. (

**c**) Mass at t = 110. (

**d**) Entropy at t = 30. (

**e**) Entropy at t = 70. (

**f**) Entropy at t = 110. (

**g**) $T(x,y,t=30)$. (

**h**) $T(x,y,t=70)$. (

**i**) $T(x,y,t=110)$. (

**j**) $\frac{dm}{dt}$ with time. (

**k**) $\frac{dS}{dt}$ with time. (

**l**) Temperature with time.

**Figure 16.**Diffusion process in a system with two cellular automata tribes weakly interacting with each other via a small hole in a double barrier; (

**a**) presents the initial configuration, (

**b**) shows the state of long thermodynamic equilibrium, in which the two cellular automata tribes coexist in two different geometrical domains, effectively geographically separated. In both cases, the mass (

**c**) and entropy (

**d**) of the tribes are saturated, with the tendency to oscillate manifesting system thermodynamic equilibrium. (

**e**) Mass at t = 3. (

**f**) Mass at t = 34. (

**g**) Mass at t = 200.

**Figure 17.**Dynamics of thermodynamic variables for two competing cellular automaton tribes (the first tribe is on the left and the second tribe is on the right). (

**a**) $\frac{dm}{dt}$ over time for first and second automaton tribes. (

**b**) $\frac{dS}{dt}$ over time for first and second automaton tribes. (

**c**) Temperature over time for first and second automaton tribes.

**Figure 18.**Diffusion process in the SCCGoL(C) system of one-species cellular automata with moving sinusoidal two barriers (

**a**,

**b**) and quasi-periodic functions of mass and entropy from the 100th cycle (

**c**). More details of the evolution of the space dependence of the thermodynamic parameters over time are given by Figure 19.

**Figure 19.**Space dependent dynamics of thermodynamic parameters with simulation time in SCCGoL(C) for one-species cellular automata with moving sinusoidal two barriers, also depicted in Figure 18. (

**a**) Mass at t = 60. (

**b**) Mass at t = 150. (

**c**) Mass at t = 305. (

**d**) Entropy at t = 60. (

**e**) Entropy at t = 150. (

**f**) Entropy at t = 305. (

**g**) $T(x,y,t=60)$. (

**h**) $T(x,y,t=150)$. (

**i**) $T(x,y,t=305)$. (

**j**) $\frac{dm}{dt}$ with time. (

**k**) $\frac{dS}{dt}$ with time. (

**l**) Temperature with time.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pomorski, K.; Kotula, D.
Thermodynamics in Stochastic Conway’s Game of Life. *Condens. Matter* **2023**, *8*, 47.
https://doi.org/10.3390/condmat8020047

**AMA Style**

Pomorski K, Kotula D.
Thermodynamics in Stochastic Conway’s Game of Life. *Condensed Matter*. 2023; 8(2):47.
https://doi.org/10.3390/condmat8020047

**Chicago/Turabian Style**

Pomorski, Krzysztof, and Dariusz Kotula.
2023. "Thermodynamics in Stochastic Conway’s Game of Life" *Condensed Matter* 8, no. 2: 47.
https://doi.org/10.3390/condmat8020047