# Study of Quasi-Static Magnetization with the Random-Field Ising Model
## 4 Results and Discussion  
Numerical simulations using RFIM were carried out to verify the Gaussian pseudo-random number generator and spin flip procedure. Subsequently, the dynamics of the magnetization process of 2D and 3D structures was examined. Model verification included both the discrepancies in the average value of the generated pseudo-random set \(h_{\mathrm{i}}\) according to dependencies 3 and 4 and the correlation of the spin flip process \(G\) (\(x\), \(R\)) depending on the set of pseudo-random parameter values using the correlation functions (Equations (8) and (9)). Analyzing the changes in the average value of the \(h_{\mathrm{i}}\) parameters against the size of the model space \(N\) for various values of the disorder \(R\) (Figure 1), one can assume that these changes are negligibly small in comparison with the \(h_{\mathrm{i}}\) values for the model space comprising above \(N=10^{4}\) spins. However, if the value \(N\) falls below \(10^{4}\), one should either consider the non-equilibrium magnetization state resulting from the non-zero mean \(h_{\mathrm{i}}\) or reduce the discussed problem by increasing the size of the model space so that the number of spins would exceed \(10^{4}\). Thus, the reduced impact of the non-equilibrium initial state occurs in simulations of two-dimensional spaces (\(D=2\)) for the values of the parameter \(L>100\), and \(L>22\) in the case of the three-dimensional model (\(D=3\)). Another tested parameter of the model was the correlation of spin flips in areas of the model where avalanche processes occurred. The effect of avalanches, which introduce discontinuous changes in the magnetization of the model space, was determined on the basis of Equation (9). It can therefore be assumed that the discrete states of magnetization, recorded in the course of simulation, also strongly depend on the correlation of spin flips. The value of spin flip correlation can be estimated by determining the distance \(x\) (avalanche size) defined as the number of sites of inverted spins relative to the spin starting the avalanche process. Based on the normalized number of classes, range width, and class limits, the histogram of correlated inversions was determined in accordance with the relationships found in Equations (8) and 9). A graph of the spin flip correlation function was created by accumulating number of flipped spins in class intervals for a given distance x of correlated spin interactions (Figure 2) The high variability of the spin flip correlation function indicated that avalanches were nucleatedfrequently [13; 30], but their propagation was often quenched by fluctuations of random fields \(h_{\rm i}\). The size of avalanches depended on the value of disorder introduced to the model space. Test results of the developed pseudo-random number generator (PRNG) showed that the numerical performance of the generator did not introduce significant limitations associated with simulation time. The implementation efficiency of the PRNG algorithm at \(10^{7}\) numbers per second was sufficient for the RFIM model, because the simulation did not require the generation of pseudo-random numbers ad hoc. Considering the limitations related to the mean value of the \(h_{\rm i}\) set, it should be stated that the model space with \(10^{6}\) spins guarantees the correct convergence of computer simulations  
The impact of both the disorder and the range of variation of the disorder parameter \(R\) on magnetization were examined. The magnetizing process depending on the disorder parameter \(R\) was depicted in graphs of meta-stable magnetization changes d\(M\)/d\(H\) (Figures 4 and 5) and hysteresis loops (Figure 6). Figure 4: Dependence of dynamic changes in magnetization d\(M\)/d\(H\) of the model on the external field \(H\) for given values of disorder \(R\) (\(D=2\), \(L=5000\), \(\Delta H=0.01\)). Figure 3: An example of the distribution of random fields \(h_{\mathrm{i}}\) (\(R=1.00\)) in the 2D model space (200 by 200 spins).