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The spin system of the $2d$ Ising model having a hexagonal-lattice issimulated using non-deterministic Cellular Automata. The method to implementthis program is outlined and our results show a good approximation to the exactanalytic solution. The phase transition in $2d$ Ising model is studied with a$40\times40$ hexagonal-lattice with five different boundary conditions (bcs)i.e., adiabatic, periodic, reflexive, fixed $+1$ and fixed $-1$ with randomorientation of spins as initial conditions in the absence of an externalapplied magnetic field. The critical temperature below which the spontaneousmagnetization appears as well as other physical quantities such as themagnetization, energy, specific heat, susceptibility and entropy with each ofthe bcs are calculated. The phase transition occurs around $T^H_c$ = 1.5 whichapproximates well with the result obtained from exact analytic solution byWannier and Houtappel. We compare the behavior of magnetisation per cell forfive different types of bcs by calculating the number of points close to theline of zero magnetization for $T>T^H_c$. We find that the periodic, adiabaticand reflexive bcs give closer approximation to the value of $T^H_c$ than fixed$+1$ and fixed $-1$ bcs with all three initial conditions for lattice size lessthan $50\times50$. However, for lattice size between $50\times50$ and$200\times200$, fixed $+1$ bc and fixed $-1$ bc give closer approximation tothe $T^H_c$ with initial conditions in which all spins are in downconfiguration and all spins are in up configuration respectively.

A comparative study of $2d$ Ising model at different boundary conditions using non-deterministic Hexagonal Cellular Automata