# Study of Quasi-Static Magnetization with the Random-Field Ising Model
## 1 Introduction  
The concept of a cellular automaton was introduced by [PERSON] and [PERSON] [1; 2]. At present, cellular automata are used as a scientific tool in studies of complex dynamic systems, microstructures, and many other applications [3; 4; 5; 6; 7]. Cellular automaton (CA) is an example of how simple rules and local interactions can lead to very diverse and complicated behaviors [8; 9]. Usually, cellular automata are simple and regular grids. Each node of the grid is in a defined discrete state. The states of the nodes are updated synchronously at discrete moments and the state of each node at the next moment is a function of the state of its neighbors in the current moment. The Ising model, being a magnetic system based on statistical mechanics, is a magnificific example of a cellular automaton that was developed much earlier than the aforementioned cellular automaton concept [10; 11]. The model is widely used in computer physics for the analysis of magnetic hysteresis [12; 13] phase transitions [14; 15], dynamics of magnetization [16; 17], Barkhausen noise [13; 15; 17; 18], and energy loss in magnetic materials [19], as well as for defect and inhomogeneity studies in magnetic structures [20; 21; 22; 23; 24; 25]. Magnetic modeling using the interactions of so-called \"spins\" arranged in a lattice of regular unit cells is the most widely used approach. A single spin can be interpreted as a magnetic moment of an atom in a crystal lattice, crystal grain, magnetic domain, or quantized amorphous space [26]. Modeling of spontaneous magnetization processes including hysteresis, displacements of domain walls, and the Barkhausen effect are some of the common applications of the Ising model [26; 27; 28]. The vast majority of works use the random-field Ising model (RFIM) based on the Hamiltonian as the energy function with an additional term representing the random molecular magnetic field assigned to each spin [13; 14; 29; 30]. However, the RFIM does not explicitly take into account the time scale. Therefore, the simulation results are qualitatively and quantitatively consistent only for time-independent processes such as the primary magnetization curve or the static magnetic hysteresis loop [28]. Studies of hysteresis in soft magnetic materials have always been a part of scientific work. Various magnetic, physical, structural, thermodynamic, and many other parameters are identified on the basis of the shape and area of the hysteresis loop. Basically, loops can be classified as static or dynamic [12; 31], but the magnetizing field must be variable over time to obtain both hysteresis loops. The static loop is interesting because it illustrates the magnetizing process without time-dependent effects. In practice, the change in material magnetization is quite complex and there is no strict interpretation when the process can be considered static, even when an extremely low frequency sweep rate is used [6; 32]. Therefore, this approach shall only be considered as a quasi-static approximation of a boundary static process [32]. Many works on testing magnetic materials arbitrarily assume, without any explanation, that the frequency sweep rate is low enough that, e.g., eddy currents may be neglected. The real limitation of the static condition results very often from the technical limitations of the measuring instruments. In this case, one cannot assess how large the discrepancy of such an approximation is. In this context, the paper provides a contribution in the field of slowly driven magnetic systems through the identification of the quasi-static limiting of magnetization processes in magnetic materials. [PERSON] et al. [19] showed the dependency of field sweep rate in the RFIM on the coercive field of magnetic thin films and pointed out a direct link between the theory of loss separation for bulk materials and a simple magnetic system with disorder.