Theory, Application, and Implementation of Monte Carlo Method in Science and Technology


Figure 5.

Figure 6.

149

Comparison of the adding probabilities for the seven-bridge system.

Application of Local Information Entropy in Cluster Monte Carlo Algorithms

DOI: http://dx.doi.org/10.5772/intechopen.88627

Reverse magnetization curves simulated using the classical Padd and modified P<sup>0</sup>

add adding probability.

#### Table 2.

Parameters of the analyzed systems and simulation procedure.

#### Figure 4.

Comparison of the adding probabilities for the one-bridge system.

successes when the procedure starts from the same cluster "seed." Figures 4 and 5 show the comparison for the one- and seven-bridge systems, respectively. The cluster seed is placed in the center, and the white color is attributed to 100 clusterbuilding successes per 100 trials.

As shown, using the classical Wolff algorithm, all spins in the system belong to the same cluster independently on their magnetic properties. This means that during simulations, a possibility of closing the clusters inside the hard or soft phase is not available. The modification of adding probability causes that the clusterbuilding procedure can distinguish the hard and soft phase with relatively high

Application of Local Information Entropy in Cluster Monte Carlo Algorithms DOI: http://dx.doi.org/10.5772/intechopen.88627

Figure 5. Comparison of the adding probabilities for the seven-bridge system.

Figure 6. Reverse magnetization curves simulated using the classical Padd and modified P<sup>0</sup> add adding probability.

successes when the procedure starts from the same cluster "seed." Figures 4 and 5 show the comparison for the one- and seven-bridge systems, respectively. The cluster seed is placed in the center, and the white color is attributed to 100 cluster-

Parameter Soft phase Hard phase Exchange coupling J 1.5e–2 eV 1.5e–2 eV Anisotropy constant K 0 1e–3 eV

Theory, Application, and Implementation of Monte Carlo Method in Science and Technology

Dipolar constant D 1.8e–7 eV Thermal energy kBT 1e–4, 1 e–5, 1e–6 eV α parameter Random value from 0 to 25

Cluster analysis probability Pcl 10<sup>3</sup> Range of the entropy Srange 3 nodes Spin change angle θ π/100 System size (in one direction) n 50 Number of iteration in one MC step 3n<sup>3</sup> = 3,75,000 Number of MC steps for magnetization averaging 400

Parameters of the analyzed systems and simulation procedure.

Simulation procedure

As shown, using the classical Wolff algorithm, all spins in the system belong to the same cluster independently on their magnetic properties. This means that during simulations, a possibility of closing the clusters inside the hard or soft phase is not available. The modification of adding probability causes that the clusterbuilding procedure can distinguish the hard and soft phase with relatively high

building successes per 100 trials.

Comparison of the adding probabilities for the one-bridge system.

Table 2.

Figure 4.

probability. However, with lower probability, the cluster contains whole spins in the system (like in the Wolff algorithm) which is related to a random value of the α parameter. The modification of adding probability causes that the configurations necessary for modeling of real magnetization processes are available during MC iterations. In other words, if magnetization processes require separate behavior of the hard and soft phase, the simulation procedure will test such a possibility.

As a final test, the so-called reverse magnetization curves were simulated using the classical as well as modified adding probability. Initially, the system was saturated in the field direction (all spins are directed up), and then the field was switched off. During calculations the field was increased in the opposite direction. Magnetization, determined as average spin value in the field direction, as a function of the external magnetic field for all examined cases is shown in Figure 6.

The difference between Padd and P<sup>0</sup> add appeared in all studied examples. Note that for the one-bridge system (relatively low interactions between the hard and

soft phases), the two-step reverse magnetization curve is expected. The first step is attributed to the spin flip of the soft phase, while the second one related to spin flip of the hard phase. For the seven-bridge system, the reverse magnetization is different due to stronger interphase coupling. However, at temperatures kBT above 1e–6 eV, this process consists of a subsequent change of the "soft" spins and next, coherent flip of the "hard" spins. Such a scenario is fully confirmed by the spin configurations depicting the reverse magnetization process for the seven-bridge

Reverse magnetization curve and energy of the system in the fields related to the spin flip of the soft phase.

Application of Local Information Entropy in Cluster Monte Carlo Algorithms

DOI: http://dx.doi.org/10.5772/intechopen.88627

Regarding the fact that in some cases also for Padd the two-step behavior occurs, the main question is which curve is physically correct. For this reason, it is worth to analyze the curves in fields when the soft phase changed spin direction using P<sup>0</sup>

the cluster-building procedure. It is known that thermal equilibrium is related to a minimum of the free energy of the system. Comparing the spin configurations for the Wolff and our modified algorithm, one can state that entropy (as a thermody-

to a decreasing of the free energy. Therefore, this approach is correct for which the energy of the system is lower. Figure 8 shows a focus of the interesting region as well as the energy of the system for the two tested algorithms. In all cases the

It is clear that the introduced modification of adding probability results in more effective finding of the free energy minimum and, therefore, produces physically

As it was shown, in some conditions (low temperature and/or strong exchange

coupling) the classical CMC algorithm can produce incorrect results in the

add results in "faster" (in lower fields) minimization of the system

namic function) is higher for the second one, i.e., applying P<sup>0</sup>

energy than the procedure based on the classical adding probability.

reliable results which allows modeling multiphase magnetic systems.

add in

add, which contributes

system at kBT = 1e–4 eV (see Figure 7).

application of P<sup>0</sup>

Figure 8.

5. Conclusions

Figure 7.

Spin configurations depicting the reverse magnetization process for the seven-bridge system at kB T = 1e‒4 eV. Red and blue colors indicate the soft and hard phase, respectively.

Application of Local Information Entropy in Cluster Monte Carlo Algorithms DOI: http://dx.doi.org/10.5772/intechopen.88627

Figure 8. Reverse magnetization curve and energy of the system in the fields related to the spin flip of the soft phase.

soft phases), the two-step reverse magnetization curve is expected. The first step is attributed to the spin flip of the soft phase, while the second one related to spin flip of the hard phase. For the seven-bridge system, the reverse magnetization is different due to stronger interphase coupling. However, at temperatures kBT above 1e–6 eV, this process consists of a subsequent change of the "soft" spins and next, coherent flip of the "hard" spins. Such a scenario is fully confirmed by the spin configurations depicting the reverse magnetization process for the seven-bridge system at kBT = 1e–4 eV (see Figure 7).

Regarding the fact that in some cases also for Padd the two-step behavior occurs, the main question is which curve is physically correct. For this reason, it is worth to analyze the curves in fields when the soft phase changed spin direction using P<sup>0</sup> add in the cluster-building procedure. It is known that thermal equilibrium is related to a minimum of the free energy of the system. Comparing the spin configurations for the Wolff and our modified algorithm, one can state that entropy (as a thermodynamic function) is higher for the second one, i.e., applying P<sup>0</sup> add, which contributes to a decreasing of the free energy. Therefore, this approach is correct for which the energy of the system is lower. Figure 8 shows a focus of the interesting region as well as the energy of the system for the two tested algorithms. In all cases the application of P<sup>0</sup> add results in "faster" (in lower fields) minimization of the system energy than the procedure based on the classical adding probability.

It is clear that the introduced modification of adding probability results in more effective finding of the free energy minimum and, therefore, produces physically reliable results which allows modeling multiphase magnetic systems.
