A new graphical method is developed to calculate the critical temperature of 2-and 3-dimensional Ising models as well as that of the 2-dimensional Potts models. This method is based on the transfer matrix method and using the limited lattice for the calculation. The reduced internal energy per site has been accurately calculated for different 2-D Ising and Potts models using different size-limited lattices. All calculated energies intersect at a single point when plotted versus the reduced temperature. The reduced temperature at the intersection is 0.4407, 0.2746, and 0.6585 for the square, triangular, and honeycombs Ising lattices and 1.0050, 0.6309, and 1.4848 for the square, triangular, and honeycombs Potts lattices, respectively. These values are exactly the same as the critical temperatures reported in the literature, except for the honeycomb Potts lattice. For the two-dimensional Ising model, we have shown that the existence of such an intersection point is due to the duality relation. The method is then extended to the simple cubic Ising model, in which the intersection point is found to be dependent on the lattice sizes. We have found a linear relation between the lattice size and the intersection point. This relation is used to obtain the critical temperature of the unlimited simple cubic lattice. The obtained result, 0.221 (2), is in a good agreement with the accurate value of 0.22165 reported by others.

A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of anisotropic two-layer Ising ferromagnet, on strips of r wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest neighbor couplings K x , K y (where K x and K y are the nearest neighbor interactions within each layer in the x and y directions, respectively) and with inter-layer coupling K z , using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted on a power series in terms of the lattice sizes.The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained, are in good agreement with the accurate values reported by others.

Abstract. Random walk simulation of the Levy flight shows a linear relation between the mean square displacement and time. We have analyzed different aspects of this linearity. It is shown that the restriction of jump length to a maximum value (l m ) affects the diffusion coefficient, even though it remains constant for l m greater than 1464. So, this factor has no effect on the linearity. In addition, it is shown that the number of samples does not affect the results. We have demonstrated that the relation between the mean square displacement and time remains linear in a continuous space, while continuous variables just reduce the diffusion coefficient. The results are also implied that the movement of a levy flight particle is similar to the case the particle moves in each time step with an average length of jumping . Finally, it is shown that the non-linear relation of the Levy flight will be satisfied if we use time average instead of ensemble average. The difference between time average and ensemble average results points that the Levy distribution may be a non-ergodic distribution.

A numerical method based on the transfer matrix method is developed to calculate the critical temperature of two-layer Ising ferromagnet with a weak interlayer coupling. The reduced internal energy per site has been accurately calculated for symmetric ferromagnetic case, with the nearest neighbor coupling K 1 = K 2 = K (where K 1 and K 2 are the nearest neighbor interaction in the first and second layers, respectively) with inter layer coupling J. The critical temperature as a function of the inter-layer coupling,is obtained for very weak inter-layer interactions,. Also a different function is given for the case of the strong inter-layer interactions ( 1 > ξ ). The importance of these relations is due to the fact that there is no well tabulated data for the critical points versus J/K. We find the value of the shift exponent γ φ = is 1.74 for the system with the same intra-layer interaction and 0.5 for the system with different intra-layer interactions. PACS numbers: 05.50.+q;64.60.-i Keywords: Ising Model, Critical Temperature / 1 c c) 0 ( ) ( J T J T −In particular, these theories predict that when the coupling K is the same in ach layer, then γ φ = , where γ is the critical exponent describing divergence of susceptibility upon approaching the critical point. We have estimated the shift exponent by using the data in Table 1 and a short program, which is written in Mathematica:

The critical points of the 3-states two-layer Potts model on square lattice for different interlayer couplings (K x =K y ≠K z ) are calculated with high precision using probabilistic cellular automata with Glauber algorithm, where K x and K y are the nearest-neighbor interactions within each layer in the x and y directions, respectively and K z is the