The multicritical behaviors and thermodynamic properties of disordered magnetic systems have been a focus of investigations in condensed-matter physics because their properties are richer and more complex than those of their pure and non-disordered counterparts [[1], [2], [3], [4], [5]]. In particular, the magnetic properties of these systems have been examined in the presence of random single-ion anisotropies [6,7], random magnetic fields [8,9], and disorders in the couplings between spins [10,11]. The impact of random field effects on magnetic systems has been the subject of experimental investigations [12]. A random crystal field was introduced to observe the critical behavior of ³He-⁴He mixtures in aerogel, which was modeled using the spin-1 Blume-Capel (BC) system. In this model, the crystal fields are assumed to have positive or negative values. The positive crystal field is defined as a bulk field that controls the concentration of ³He atoms. The negative crystal field is correspondingly defined as the field at the pore-grain interface [13,14]. The development of a superfluid ³He-⁴He solution has also been achieved through experimental studies [15,16].

As demonstrated in the relevant literature, the equilibrium properties of the BC model have been the subject of study in the presence of a random crystal field by various authors [[17], [18], [19], [20], [21]]. Recent studies have revealed new phase diagrams, which include a partially ferromagnetic phase. The experimental interest in this model has arisen from its potential to describe cooperative physical systems, including, but not limited to, superconducting films [22], dysprosium aluminum garnet [23], liquid crystals [24], metallic alloys [25], magnetic thin films [26], and others. The BC model has been extensively investigated in theoretical studies by a variety of methods, including mean-field theory (MFT) [17,18,[27], [28], [29]], renormalization group calculations (RGT) [6,14,30,31], Monte Carlo simulations [[32], [33], [34], [35]], effective field theory [36], the cluster variation approach [37], the replica method [5], and the pair approximation approach [38]. Randomness significantly affects the behaviors above, as shown in the example; the phase diagrams present rich behavior in these studies under quenched randomness. Quenched randomness can cause changes to the critical behavior of a system. For instance, it has been shown to cause first-order transitions to be replaced by continuous transitions in two dimensions and to lower the tricritical temperature in higher dimensions [7,8,39,40]. The equilibrium phase diagrams of the spin-1 BC model under a bimodal crystal field have been investigated using the mean-field approximation (MFA) by Bocarra et al. [17], the RGT by Branco et al. [14], and El Bouziani et al. [31]. The equilibrium behavior of the mean-field spin-1 Blume-Capel model with another form of random crystal-field distributions corresponding to quenched diluted single-ion anisotropy has been studied extensively by Benyoussef et al. [6]. Moreover, studies on dynamic phase diagrams [41,42] and the application of relaxation theory to various systems [43,44] have drawn significant attention to this model [43,44]. The behavior of the relaxation time () and the dynamics of the system in the neighborhood of phase transitions have been the subject of experimental research for quite a considerable time [45,46]. To the best of our knowledge, the relaxation dynamics of the spin-1 BC model with quenched random crystal field distributed according to a symmetric, two-valued probability distribution defined by two delta functions have not yet been studied. Our main goal in the present work is to investigate the effect of quenched disorder on the relaxation processes that occur under the application of a small field for quite a short time. To this end, the kinetic equation for the magnetization () is derived within the framework of linear response theory. Subsequently, the temperature () and the crystal field () dependencies of the relaxation time in the neighborhood of the phase-transition points are obtained by solving the kinetic equation for the magnetization. The focal point of this study is the behavior of the relaxation time, which characterizes the magnetization relaxation process in proximity to the critical, isolated critical, multicritical, and first-order transition points. The findings presented here constitute a novel contribution to the field, owing to the observation of successive first-order phase transitions, which occur only under strong disorder.

The structure of this work is delineated as follows: Section 2 provides a concise exposition of the model and details its equilibrium properties under the MFA. The third section offers a detailed exposition of the derivation of the kinetic equation and the relaxation time. Section 4 provides the results of the present study, focusing on the MFA phase diagrams and relaxation behavior of the system in the vicinity of critical, multicritical, and first-order points. Finally, the last section provides a summary and conclusion.