1. Introduction

The problem of simulation magnetization processes of multiphase magnetic materials seems to be important, regarding the tendency for applications of highefficient permanent magnets with reduced or without rare earth elements. Recently, we reported unique hard magnetic properties of Tb-Fe-B-Nb bulk alloys, i.e., coercive field over 7 T at room temperature, attributed to a specific microstructure of dendrite-like Tb2Fe14B grains [1, 2]. In this system, Tb and Fe magnetic moments are coupled antiferromagnetically, which is responsible for relatively low magnetic remanence (μ0M<sup>r</sup> ≈ 0.3 T) and in consequence |BH|max (about 13 kJ/m<sup>3</sup> ). However, the Fe-Nb-B-Tb bulk alloys can be considered as a material with extremely high resistance to the external magnetic field and can be a source of magnetic anisotropy in powders as well as bulk spring-exchange composites containing magnetically soft and ultra-high coercive phases. For this reason, the ability of simulating of such systems is very useful in the process of examining and designing of spring-exchange composites in the pre-lab phase.

Monte Carlo (MC) algorithms are now widely used to clarify various physical phenomena, as well as to investigate their potential application in modern technology. Among many simulation methods, the Metropolis MC (MMC) approach [3, 4] is especially attractive in statistical physics for the determination of system physical quantities in thermodynamic equilibrium. The MMC algorithm realizes an ergodic stochastic process, ensuring the fulfillment of the detailed balance condition.

One of the bright examples of the application of the MMC algorithm is the Ising model of spins located on the nodes of some lattice. Indeed, using this method one can study a course of magnetic ordering and its dependence of temperature and details of interactions between the spins.

spins, we get a total of 2100 states—the number that makes the summation occur-

Application of Local Information Entropy in Cluster Monte Carlo Algorithms

f g<sup>α</sup> <sup>F</sup><sup>α</sup> exp ð Þ �βE<sup>α</sup> <sup>p</sup>�<sup>1</sup> <sup>P</sup> <sup>α</sup> f g<sup>α</sup> exp ð Þ �βE<sup>α</sup> <sup>p</sup>�<sup>1</sup>

In order to estimate the average ´F, a nonuniform sampling of the system states can be applied. If f gα denotes a set of indices of M system states selected with the probability pα, the equilibrium value of the observable F can be modeled [13] by the

α

where probabilities p<sup>α</sup> were selected to be equal exp ð Þ �βE<sup>α</sup> =Z. All we need is a method that generates a set of system states with the Boltzmann probabilities pα. Because the exact value of the partition function Z is unknown, the generation of the states is usually carried out by the ergodic Markov process. This process produces a proper chain of states under the assumption that transition probability Wαβ (from α to β state) is independent of the states preceding α. Moreover, It is also assumed that the detailed balance condition, pβWβα ¼ pαWαβ, is satisfied when the

The transition probability Wαβ can be considered as a product of the selection probability gαβ and the acceptance ratio (probability)Aαβ. In general, the selection probabilities can be chosen to a large extent freely, e.g., they can be symmetrical gαβ ¼ gβα [12]. In that case the acceptance probabilities satisfy the equation:

Aαβ=Aβα ¼ exp �β E<sup>β</sup> � E<sup>α</sup>

E ¼ �J

new spin configuration can be accepted or rejected with an acceptance ratio

As an example, let us consider Ising model of N interacting spins placed at the nodes of a two-dimensional regular. The energy of the system is, then, given by the

> X N

i6¼j¼1

where si ¼ �1 describes the spin state at the ith lattice node and J refers to the exchange integral. In order to determine the physical properties of our magnetic system, the Metropolis algorithm can be employed. It relies on the particular choice of both the selection probability and the acceptance ratio. Having some configuration of spins, the next one is obtained by the flip of a single spin (single-spin-flip algorithm) [12, 13]. This procedure results in uniform distribution of the selection probabilities, i.e., each new state participates in simulations with probability gαβ ¼ 1=N. Then, the

� � � � (for <sup>E</sup><sup>β</sup> � <sup>E</sup><sup>α</sup> <sup>&</sup>gt;0) and <sup>A</sup>αβ <sup>¼</sup> 1 (for all other cases). Although the Metropolis algorithm can be applied to a variety of physical problems, when applied to magnetic systems, it has disadvantage that relies on a very rapid increase of the correlation time as well as correlation length near the critical point. As a result the system contains domains of the same oriented spins and therefore becomes configurationally frozen. This unexpected behavior (critical slowing down) of the Metropolis algorithm is the reason for the difficulties in the generation of statistically independent spin configurations that are needed for the calculation of the estimator FS. The solution of the critical slowing down problem was proposed by Swendsen-Wang [7] and later by Wolff [8]. The approach developed by Wolf (cluster-flipping

¼ 1 N X f gα

Fα, (2)

� � � � (3)

sisj (4)

ring in Eq. (1) impossible.

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

FS ¼

system is in a state of equilibrium [12–15].

P

estimator:

formula

Aαβ ¼ exp �β E<sup>β</sup> � E<sup>α</sup>

141

The MMC method utilizes the single-spin-flip procedure to change the spin configuration; however, in many cases (e.g., simulations of magnetization processes) a more effective algorithm is needed. The simplest approach relies on the generation of a cluster of uniformly oriented spins and their subsequent flip to reach new state of the system. The main question is how to determine the cluster and how to establish a rule of its acceptance, simultaneously satisfying the detailed balance condition. Classical approaches, based on the Kasteleyn-Fortuin theorem [5, 6], were proposed by Swendsen and Wang (SW) [7] as well as by Wolff [8] who assumed a specific cluster-building procedure controlled by the so-called adding probability. It is known that the cluster Monte Carlo methods (CMC) are very efficient in the analysis of critical phenomena, e.g., transformation from ferromagnetic to paramagnetic phase [9, 10]. In contrast, their application for studying magnetization processes of systems far below the Curie point produces artificial results, which can be demonstrated for the systems containing magnetically different phases (e.g., hard and soft) as well as geometrical irregularities. The clusterbuilding algorithms implemented within SW and Wolf approaches are steered by the exchange interactions and the system temperature, but they are not sensitive to other features, potentially affecting the clusterization of spins.

In order to broaden the applications of the CMC methods to simulations of real magnetic composites, we proposed a new method based on some modification of the SW/Wolff adding probability and a particular Metropolis-like algorithm, ensuring the principle of detailed balance [11]. The idea is based on the fact that some kind of regions of the system, characterized by a local disorder of selected system property, constitutes natural barriers for the extension of clusters. In the case of magnetic multiphase composites, spatial distribution of the magnetic anisotropy can be considered as the property affecting the cluster formation.

In the chapter, the disorder-based CMC algorithm is introduced and discussed in a context of classical CMC methods. We show that the new simulation procedure is efficient leading to physically reliable results, especially for multiphase magnetic composites.
