Towards quantization Conway Game of Life

TL;DR

This paper explores a quantum-inspired, dissipative version of Conway's Game of Life using complex non-Hermitian Hamiltonians, revealing thermodynamic behaviors and non-conservation phenomena.

Contribution

It introduces a novel quantum mechanical framework for cellular automata with complex energies and dissipation, linking thermodynamics and quantum concepts.

Findings

System exhibits negative temperatures at equilibrium.

Mass, energy, and temperature diffuse consistent with thermodynamic laws.

Complex-valued mass mimics wave-function behavior.

Abstract

Classical stochastic Conway Game of Life is expressed by the dissipative Schr\"odinger equation and dissipative tight-binding model. This is conducted at the prize of usage of time dependent anomalous non-Hermitian Hamiltonians as with occurrence of complex value potential that do not preserve the normalization of wave-function and thus allows for mimicking creationism or annihilationism of cellular automaton. Simply saying time-dependent complex value eigenenergies are similar to complex values of resonant frequencies in electromagnetic resonant cavities reflecting presence of dissipation that reflects energy leaving the system or being pumped into the system. At the same time various aspects of thermodynamics were observed in cellular automata that can be later reformulated by quantum mechanical pictures. The usage of Shannon entropy and mass equivalence to energy points definition of cellular automata temperature. Contrary to intuitive statement the system dynamical equilibrium is always reflected by negative temperatures. Diffusion of mass, energy and temperature as well as phase of proposed wave function is reported and can be directly linked with second thermodynamics law approximately valid for the system, where neither mass nor energy is conserved. The concept of complex-valued mass mimics wave-function behavior. Equivalence an anomalous second Fick law and dissipative Schr\"odinger equation is given. Dissipative Conway Game of Life tight-binding Hamiltonian is given using phenomenological justification.