## 1. Introduction

Real magnets and Ising models have provided a rich and productive field for the interaction between theory and experiment over the past 86 years (Ising, 1925). In order to identify the real magnets with a simple microscopic Hamiltonian, one needs to understand the behaviour of individual magnetic ions in crystalline environment (Wolf, 2000). Spin–1/2 Ising model and its variants such as Blume-Capel, Blume-Emery-Griffiths and mixed spin models were regarded as theoretical simplifications, designed to model the essential aspects of cooperative systems without detailed correspondence to specific materials. The similarities and differences between theoretical Ising models and a number of real magnetic materials were widely reviewed by many authors. The early experiments were focused on identifying Ising-like materials and characterizing the parameters of the microscopic Hamiltonian. Various approximate calculations were then compared with thermodynamic mesurements. Although both the theoretical and experimental studies concerning Ising-like systems have concentrated on static properties, very little has been said about its dynamic characteristics.

Lyakhimets (Lyakhimets, 1992) has used a phenomenological description to study the magnetic dissipation in crystalline magnets with induced magnetic anisotropy. In his study, the components of the second-order tensor which describes the induced anisotropy of the magnet were taken as thermodynamic variables and the nonequilibrium linear Onsager thermodynamics was formulated for the system. Such an approach reflects all symmetry characteristics of the relaxation problem. The relaxation parameters and their angular denpendencies were formulated for spin waves and moving domain walls with the help of the dissipation function. The implications of nonequilibrium thermodynamics were also considered for magnetic insulators, including paramagnets, uniform and nonuniform ferromagnets (Saslow & Rivkin, 2008). Their work was concentrated on two topics in the damping of insulating ferromagnets, both studied with the methods of irreversible thermodynamics: (a) damping in uniform ferromagnets, where two forms of phenomenological damping were commonly employed, (b) damping in non-uniform insulating ferromagnets, which become relavent for non-monodomain nanomagnets. Using the essential idea behind nonequilibrium thermodynamics, the long time dynamics of these systems close to equilibrium was well defined by a set of linear kinetic equations for the magnetization of insulating paramagnets (and for ferromagnets). The dissipative properties of these equations were characterized by a matrix of rate coefficients in the linear relationship of fluxes to appropriate thermodynamic forces.

Investigation of the relaxation dynamics of magnetic order in Ising magnets under the effect of oscillating fields is now an active research area in which one can threat the sound propagation as well as magnetic relaxation. In most classes of magnets, a very important role is played by the order parameter relaxation time and it is crucial parameter determining the sound dynamics as well as dynamic susceptibility. As a phenomenological theory, nonequilibrium thermodynamics deals with approach of systems toward steady states and examines relaxation phenomena during the approach to equilibrium. The theory also encompasses detailed studies of the stability of systems far from equilibrium, including oscillating systems. In this context, the notion of nonequilibrium phase transitions is gaining importance as a unifying theoretical concept.

In this article, we will focus on a general theory of Ising magnets based on nonequilibrium thermodynamic. The basics of nonequilibrium thermodynamics is reviewed and the time-reversal signature of thermodynamic variables with their sources and fluxes are discussed in Section 2. Section 3 then considers Ising spin models describing statics of ferromagnetic and antiferromagnetic orders in magnets. Section 4 contains a detailed description of the kinetic model based on coupled linear equations of motion for the order parameter(s). The effect of the relaxation process on critical dynamics of sound propagation and dynamic response magnetization is investigated in Section 5. Comparison with experiments is made and reasons for formulating a phenomenological theory of relaxation problem are given in Section 6. Finally, the open questions and future prospects in this field are outlined.

## 2. Basics of nonequilibrium thermodynamics

Nonequilibrium thermodynamics (NT), a scientific discipline of 20 th century, was invented in an effort to rationalize the behavior of irreversible processes. The NT is a vast field of scientific endeavour with roots in physics and chemistry. It was developed in the wake of the great success of certain symmetry relations, known as Onsager reciprocal relations in the phenomenological laws. These symmetry relations between irreversible phenomena have found a wide field of application in all branches of the physical science and engineering, and more recently in a number of interdisciplinary fields, including environmental research and, most notably, the biological sciences. Above applications can be classified according to their tensorial character. First one has scalar phenomena. These include chemical reactions and structural relaxation phenomena. Onsager relations are of help in this case, in solving the set of ordinary differential equations which describe the simultaneous relaxation of a great number of variables. Second group of phenomena is formed by vectorial processes, such as diffusion, heat conduction and their cross effects (e.g. thermal diffusion). Viscous phenomena and theory of sound propagation have been consistently developed within the framework of nonequilibrium thermodynamics.

Before introducing the notion of nonequilibrium thermodynamics we shall first summarize briefly the linear and nonlinear laws between thermodynamic fluxes and forces. A key concept when describing an irreversible process is the macroscopic state parameter of an adiabatically isolated system. These parameters are denoted by

It is known empirically that the irreversible flows, time derivatives of deviations (

where the quantities

It is well known that the entropy of an isolated system reaches its maximum value at equilibrium: so that any fluctuation of the thermodynamic parameters results with a decrease in the entropy. In response to such a fluctuation, entropy-producing irreversible process spontaneously drive the system back to equilibrium. Consequently, the state of equilibrium is stable to any perturbation that reduces the entropy. In contrast, one can state that if the fluctuations are groving, the system is not in equilibrium. The fluctuations in temperature, volume, magnetization, kuadrupole moment, etc. are quantified by their magnitude such as

In this expansion, the second term represents the *first-order* terms containing
*second-order* terms containing
*first-order* terms vanishes wheares the leading contribution to the increment of the entroıpy originates from the *second-order* term

The thermodynamic forces in Eqs. (2) are the intensive variables conjugate to the variables

where

Clearly the first term in Eq. (5) is zero as the fluxes vanish when the thermodynamic forces are zero. The term which is linear in the forces is evidently derivable, at least formally, from the equilibrium properties of the system as the functional derivative of the fluxes with respect to the forces computed at equilibrium,

where the coefficients defined by

Here the coefficients

In the nonlinear thermodynamic theory, a nonlinear generalization of Onsager’s reciprocal relations was obtained using statistical methods (Hurley & Garrod, 1982). Later, the same generalization was also proved with pure macroscopic methods (Verhas, 1983). The proof of the generalization is based on mathematical facts. None of these generalizations are of general validity. The principle of macroscopic reversibility proposed by Meixner gives a good insight to the structure of the Onsager-Casimir reciprocal relations and says that the entropy production density in invariant under time inversion if it is quadratic function of independent variables. Demanding its validity to higher order leads to conflict only with the rules of the chemical reactions (Meixner, 1972).

## 3. Ising model and equilibrium properties based on the mean field approximation

In this section, we consider the Ising model on a regular lattice where each interior site has the same number of nearest-neighbour sites. This is called the coordination number of the lattice and will be denoted by

with

where
*et al.,* 1971). Recently, there have been many theoretical studies of mixed spin Ising systems. These are of interest because they have less translational symmetry than their single-spin counterparts since they consist of two interpenetrating inequivalent sublattices. The latter property is very important to study a certain type of ferrimagnetism, namely molecular-based magnetic materials which are of current interest (Kaneyoshi & Nakamura, 1998).

For sake of the brevity, here we will focus on the equilibrium properties of the

where

and we define the critical temperature

From Eq. (10), it is seen that the

To find the magnetization we solve Eq. (12) for

Now, using the definition

one obtains the following expression for the susceptibility

Among the physical systems which undergo phase transitions, the most interesting class is the ferromagnet-paramagnet transtions in simple magnets. The free energy in such systems is nonanalytical function of its arguments. This is a manifestation of very strong fluctuations of quantity called order parameter. Phase transformations in ferromagnets are the continuous phase transitions which show no latent heat, seen in
Figure 2
. On the other hand, many physical quantities such as specific heat and static susceptibility diverge to infinity or tend to zero when approaching the critical temperature

## 4. Thermodynamic description of the kinetic model

In this section, a molecular-field approximation for the magnetic Gibbs free-energy production is used and a generalized force and a current are defined within the irreversible thermodynamics. Then the kinetic equation for the magnetization is obtained within linear response theory. Finally, the temperature dependence of the relaxation time in the neighborhood of the phase-transition points is derived by solving the kinetic equation of the magnetization. For a simple kinetic model of Ising magnets, we first define the time-dependent long-range order parameter

where

where

In Eq. (19), the coefficients are called as Gibbs production coefficients:

The rate (or kinetic) equation is obtained using Eqs. (18)-(25) in the relaxation equation (Eq. (17)):

In order to find the relaxation time (

Assuming a solution of the form

Using Eq. (20) yields

The behaviour of the relaxation time near the phase-transition points can be derived analytically from the critical exponents. It is a well-known fact that various thermodynamic functions represents singular behavior as one approaches the critical point. Therefore, it is convenient to introduce an expansion parameter, which is a measure of the distance from the critical point (

In the vicinity of the second-order transition the magnetization vanishes at

The critical exponent for the function

This description is valid for all values of

The behavior of the relaxation time *τ* as a function of temperature is given in Figure 4. One can see from
Figure 4
that
*et al*., 1989). According to mean-field calculations, the dynamic critical exponent of the Ising model is
*et al.* (Halperin *et al.*, 1974) found the critical-point singularity of the linear dynamic response of various models. The linear response theory, however, describes the reaction of a system to an infinitesimal external disturbance, while in experiments and computer simulations it is often much easier to deal with nonlinear-response situations, since it is much easier to investigate the response of the system to finite changes in the thermodynamic variables. A natural question is whether the critical-point singularity of the linear and nonlinear responses is the same. The answer is yes for ergodic systems, which reach equilibrium independently of the initial conditions (Racz, 1976). The assumption that the initial and intermediate stages of the relaxation do not affect the divergence of the relaxation time (motivated by the observation that the critical fluctuations appear only very close to equilibrium) led to the expectation that in ergodic systems
*et al*. (Koch et al., 1996) presented field-theoretic arguments by making use of the Langevin equation for the one-component field

For the ferromagnetic interaction, a short range order parameter as well as the long range order is introduced (Tanaka et al., 1962; Barry, 1966) while there are two long range sublattice magnetic orders and a short range order in the Ising antiferromagnets (Barry & Harrington, 1971). Similarly the number of thermodynamic variables (order parameters) also increases when the higher order interactions are considered (Erdem & Keskin, 2001; Gülpınar et al., 2007; Canko & Keskin, 2010). For a general formulation of Ising spin kinetics with a multiple number of spin orderings (

(34) |

where the coefficients are defined as

Then a set of linear rate equations may be written in terms of a matrix of phenomenological coefficients which satisfy the Onsager relation (Onsager, 1931):

where the generalized forces are

The matrix equation given by Eq. (36) can be written in component form using Eq. (37), namely a set of

where the matrixes are defined by

(39) |

Since the phenomenological coefficients

## 5. Critical behaviours of sound propagation and dynamic magnetic response

In this section, we will discuss the effect of the relaxation process on critical dynamics of sound propagation and dynamic response magnetization for the Ising magnets with single order parameter (

In order to obtain the critical sound propagation of an Ising system we focus on the case in which the lattice is stimulated by the sound wave of frequency

Solving Eq. (40) for

The response in the pressure

then

Finally, the derivative of the pressure with respect to volume gives

Here

From the real and imaginary parts of Eq. (45) one obtains the velocity of sound and attenuation coefficient for a single relaxational process as

where

In the presence of many thermodynamic variables for more complex Ising-type magnets, there exist more than one relaxational process with relaxational times (

Similarly, theoretical investigation of dynamic magnetic response of the Ising systems has been the subject of interest for quite a long time. In 1966, Barry has studied spin–1/2 Ising ferromagnet by a method combining statistical theory of phase transitions and irreversible thermodynamics (Barry, 1966). Using the same method, Barry and Harrington has focused on the theory of relaxation phenomena in an Ising antiferromagnet and obtained the temperature and frequency dependencies of the magnetic dispersion and absorption factor in the neighborhood of the Neel transition temperature (Barry & Harrington, 1971). Erdem investigated dynamic magnetic response of the spin–1 Ising system with dipolar and quadrupolar orders (Erdem, 2008). In this study, expressions for the real and imaginary parts of the complex susceptibility were found using the same phenomenological approach proposed by Barry. Erdem has also obtained the frequency dependence of the complex susceptibility for the same system (Erdem, 2009). In Ising spin systems mentioned above, there exist two or three relaxing quantities which cause two or three relaxation contributions to the dynamic magnetic susceptibility. Therefore, as in the sound dynamics case, a general formulation (section 4) is followed for the derivation of susceptibility expressions. In the following, we use, for simplicity, the theory of relaxation with a single characteristic time to obtain an explicit form of complex susceptibility.

If the spin system descibed by Eq. (8) is stimulated by a time dependent magnetic field

If this equation is substituted into the kinetic equation Eq. (17) we find following form:

Solving Eq. (49) for

Eq. (50) is needed to calculate the complex initial susceptibility

where

where

Finally the magnetic dispersion and absorbtion factors become

In Figures 7 and 8 we illustrate the temperature variations of the magnetic dispersion and absorption factor in the low frequency limit

Finally the high frequency behavior (

## 6. Comparison of theory with experiments

The diverging behavior of the relaxation time and corresponding slowing down of the dynamics of a system in the neighborhood of phase transitions has been a subject of experimental research for quite a long time. In 1958, Chase (Chase, 1958) reported that liquid helium exhibits a temperature dependence of the relaxation time consistent with the scaling relation

It is well known fact that measurements of sound propagation are considered useful in investigating the dynamics of magnetic phase transitions and therefore many experimental and theoretical studies have been carried out. Various aspects of ultrasonic attenuation in magnetic insulators (Lüthi & Pollina, 1969; Moran & Lüthi, 1971) and in magnetic metals (Lüthi et al., 1970; Maekawa & Tachiki, 1978) have been studied. In these works, the transtion temperature was associated with the experimentally determined peaks whose maximum shift towards the lower temperatures as the sound frequency increases. Similarly, acoustic studies, especially those of dispersion, have also been made on several magnetic systems such as transition metals (Golding & Barmatz, 1969), ferromagnetic insulators (Bennett, 1969) and antiferromagnetic semiconductors (Walter, 1967). It was found that the critical changes in sound velocity show a uniform behaviour for all substances studied, namely, a frequency-independent and weak temperature-dependent effect. It was also found that, in the ordered phase, the minima of the sound velocity shifted to lower temperatures with increasing frequency (Moran & Lüthi, 1971).

Dynamic response of a spin system to a time-varying magnetic field is an important subject to probe all magnetic systems. It is also called AC or dynamic suceptibility for the magnetization. The dynamic susceptibility is commonly used to determine the electrical properties of superconductors (Kılıç et al, 2004) and magnetic properties of some spin systems such as spin glasses (Körtzler & Eiselt, 1979), cobal-based alloys (Durin et al., 1991), molecule-based magnets (Girtu, 2002), magnetic fluids (Fannin et al., 2005) and nanoparticles (Van Raap et al., 2005). The dynamic magnetic response of these materials and the development of methods for its modification are important for their potential applications. For example, cores made of cobalt-based alloys in low signal detectors of gravitational physics contribute as a noise source with a spectral density proportional to the ac susceptibility of the alloy. The knowledgement of dynamic susceptibility for nanocomposite particles is very important for the design of magneto-optical devices.

## 7. Conclusion

In this chapter, we have discussed a simple kinetic formulation of Ising magnets based on nonequilibrium thermodynamics. We start with the simplest relaxation equation of the irreversible thermodynamics with a characteristic time and mention a general formulation based on the research results in the literature for some well known dynamic problems with more than one relaxational processes. Recent theoretical findings provide a more precise description for the experimental acoustic studies and magnetic relaxation measurements in real magnets.

The kinetic formulation with single relaxation process and its generalization for more coupled irrevesible phenomena strongly depend on a statistical equilibrium description of free energy and its properties near the phase transition. The effective field theories of equilibrium statistical mechanics, such as the molecular mean-field approximation is used as this century-old description of free energy. However, because of its limitations, such as neglecting fluctuation correlations near the critical point and low temperature quantum excitatitions, these theories are invaluable tools in studies of magnetic phase transitions. To improve the methodology and results of mean-field analysis of order parameter relaxation, the equilibrium free energy should be obtained using more a reliable theory including correlations. This was recently given on the Bethe lattice using some recursion relations. The first major application of Bethe-type free energy for the relaxation process was on dipolar and quadrupolar interactions to study sound attenuation problem (Albayrak & Cengiz, 2011).

Bethe lattice treatment of phenomenological relaxation problem mentioned above has also some limitations. It predicts a transition temperature higher than that of a bravais lattice. Also, predicting the critical exponents is not reliable. Therefore, one must consider the relaxation problem on the real lattices using more reliable equilibrium theories to get a much clear relaxation picture. In particular, renormalization group theory of relaxational sound dynamics and dynamic response would be of importance in future.