A solution of 3d-Ising Model with boundary conditions was studied. Because of the lack of 3d-Pfaffian we used the symmetry of a cubic lattice to obtain the final results. We obtained 23% upper value of the transition point compared with the one which was given by Monte Carlo Method for a cubic lattice.
The Ising model (and also the Ising model in a transverse field) has been used extensively to investigate ferroelectric phase transitions. Numerical Monte Carlo simulations of phase transitions in three-dimensional Ising systems with variable size (2 ≤ L ≤ 100) have been performed. The susceptibility maximum and peak temperature Tc 3D(L,Lc) for free and periodic boundary conditions have been determinated and have been observed to be well fitted by simple expressions as a function of L. For L → ∞, as expected [Tc 3D]fbc = [Tc 3D]pbc being fbc = free boundary conditions and pbc = periodic boundary conditions. Scaling plots MLβ/v vs |ϵ|Ll/v for larger L with both boundary conditions were obteined and compared. The scaling function for our data up to L ≈ 100 was examined closely and an excellent fit at T → Tc was obtenied.
A Monte Carlo method is applied to simulate the static critical behavior of a cubic-lattice 3D Ising model for systems with quenched disorder. Numerical results are presented for the spin concentrations of p = 1.0, 0.95, 0.9, 0.8, 0.6 on L × L × L lattices with L = 20–60 under periodic boundary conditions. The critical temperature is determined by the Binder cumulant method. A finite-size scaling technique is used to calculate the static critical exponents α, β, γ, and ν (for specific heat, susceptibility, magnetization, and correlation length, respectively) in the range of p under study. Universality classes of critical behavior are discussed for three-dimensional diluted systems.
In this work, Markov Chain-Monte Carlo technique was used to study the phase transition in two and three dimensional Ising Model (IM) in a square and cubic lattice. The study of temperature dependence of average magnetization and specific heat in different magnetic fields has been carried out in the 3x3 and 3x3x3 lattice with periodic boundary. Critical temperature point kBTc / J for 2D and 3D Ising Model has been observed at around 2.2 and 4.3 respectively at zero field. Our work satisfies Onsager’s critical value in 2D IM. The simulation suggests bifurcation in average magnetization below critical temperature Tc. Temperature plays the role of increasing randomness of spins. We found that Ising Model in small lattice size still retains interesting features like spontaneous magnetization and symmetry breaking below Tc at B = 0. At a non-zero field, the likelihood of spins to prefer certain alignment depends on the direction of the external field and magnitude of magnetization depends on magnitude of field ±B. Specific heat Cv, which gives us fluctuation of energy at particular temperature, has been found to attain maximum value at critical regions.