A comparative study of 2d Ising model at different boundary conditions using Cellular Automata

Abstract

Using Cellular Automata, we simulate spin systems corresponding to [Formula: see text] Ising model with various kinds of boundary conditions (bcs). The appearance of spontaneous magnetization in the absence of magnetic field is studied with a [Formula: see text] square lattice with five different bcs, i.e. periodic, adiabatic, reflexive, fixed ([Formula: see text] or [Formula: see text]) bcs with three initial conditions (all spins up, all spins down and random orientation of spins). In the context of [Formula: see text] Ising model, we have calculated the magnetization, energy, specific heat, susceptibility and entropy with each of the bcs and observed that the phase transition occurs around [Formula: see text] as obtained by Onsager. We compare the behavior of magnetization versus temperature for different types of bcs by calculating the number of points close to the line of zero magnetization after [Formula: see text] at various lattice sizes. We observe that the periodic, adiabatic and reflexive bcs give closer approximation to the value of [Formula: see text] than fixed [Formula: see text] and fixed [Formula: see text] bcs with all three initial conditions for lattice sizes less than [Formula: see text]. However, for lattice size between [Formula: see text] and [Formula: see text], fixed [Formula: see text] bc and fixed [Formula: see text] bc give closer approximation to the [Formula: see text] with initial conditions all spin down configuration and all spin up configuration, respectively.