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In this paper, we consider translation-invariant Gibbs measures (TIGM) for the Blume-Capel model in the case of a graph of type wand on a Cayley tree of order k ≥ 2. An exact critical value θcr = k+1√ kk(k−1)/2k is found such that for θ ≥ θcr there is a single TIGM, and for θ < θcr there are exactly three TIGMs in the case of a graph of type wand for the model under consideration. [ABSTRACT FROM AUTHOR]

In this paper, we study the periodic Gibbs measures for the HC–Blume–Capel model in the case of a "wand" on a Cayley tree of order two. It is proved that in this case, for there exist exactly three periodic Gibbs measures that are translation-invariant; and for there exist exactly three periodic Gibbs measures, one of which is translation-invariant, two others are periodic (not translation-invariant) with period two. In addition, the problem of the (non) extremes of these measures has been studied. [ABSTRACT FROM AUTHOR]

The work is devoted to the theory of Gibbs measures, the so-called gradient Gibbs measure. In this paper, we study the HC-Blume-Capel model with a countable set Z of spin values and an interaction strength J Ⲉ R of nearest neighbors in the case of a "generalized wand" on a Cayley tree of order k = 2. An exact critical value θcr = 1/3√2 is found such that exactly three translation-invariant gradient Gibbs measures exist for 0 < θ < θcr, and for θ > θcr there are exactly three G(2) k -periodic gradient Gibbs measures (one that is translation-invariant and two not translation-invariant) in the cases of "generalized wand" for the model in question. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove connections among the asymptotic behavior of the magnetization, the structure of the phase transitions, and a class of polynomials that we call the Ginzburg–Landau polynomials. The model under study is a mean-field version of a lattice spin model due to Blume and Capel. It is defined by a probability distribution that depends on the parameters β and K, which represent, respectively, the inverse temperature and the interaction strength. Our main focus is on the asymptotic behavior of the magnetization m( β n , K n ) for appropriate sequences ( β n , K n ) that converge to a second-order point or to the tricritical point of the model and that lie inside various subsets of the phase-coexistence region. The main result states that as ( β n , K n ) converges to one of these points ( β, K), $m(\beta_{n},K_{n})\sim \bar{x}|\beta -\beta_{n}|^{\gamma}\rightarrow 0$ . In this formula γ is a positive constant, and $\bar{x}$ is the unique positive, global minimum point of a certain polynomial g. We call g the Ginzburg–Landau polynomial because of its close connection with the Ginzburg–Landau phenomenology of critical phenomena. For each sequence the structure of the set of global minimum points of the associated Ginzburg–Landau polynomial mirrors the structure of the set of global minimum points of the free-energy functional in the region through which ( β n , K n ) passes and thus reflects the phase-transition structure of the model in that region. This paper makes rigorous the predictions of the Ginzburg–Landau phenomenology of critical phenomena and the tricritical scaling theory for the mean-field Blume–Capel model. [ABSTRACT FROM AUTHOR]

We present four results on the fluctuations of the spin per site around the thermodynamic magnetization in the mean-field Blume-Capel model. Our first two results refine the main theorem in a previous paper (Ellis et al. in Ann. Appl. Probab. 20:2118-2161, ), in which the first rigorous confirmation of the statistical mechanical theory of finite-size scaling for a mean-field model is given. Our first main result studies the asymptotics of the centered, finite-size magnetization, giving its precise rate of convergence to 0 along parameter sequences lying in the phase-coexistence region and converging sufficiently slowly to either a second-order point or the tricritical point of the model. A simple inequality yields our second main result, which generalizes the main theorem in Ellis et al. (Ann. Appl. Probab. 20:2118-2161, ) by giving an upper bound on the rate of convergence to 0 of the absolute value of the difference between the finite-size magnetization and the thermodynamic magnetization. These first two results have direct relevance to the theory of finite-size scaling. They are consequences of our third main result. This is a new conditional limit theorem for the spin per site, where the conditioning allows us to focus on a neighborhood of the pure states having positive thermodynamic magnetization. Our fourth main result is a conditional central limit theorem showing that the fluctuations of the spin per site are Gaussian in a neighborhood of the pure states having positive thermodynamic magnetization. [ABSTRACT FROM AUTHOR]

This paper concerns the microscopic dynamical description of competing metastable states. We study, at infinite volume and very low temperature, metastability and nucleation for kinetic Blume–Capel model: a ferromagnetic lattice model with spins taking three possible values: −1, 0, 1. In a previous paper ([MO]) we considered a simplified, irreversible, nucleation-growth model; in the present paper we analyze the full Blume–Capel model. We choose a region U of the thermodynamic parameters such that, everywhere in U: −\b{1} (all minuses) corresponds to the highest (in energy) metastable state, \b{0} (all zeroes) corresponds to an intermediate metastable state and +\b{1} (all pluses) corresponds to the stable state. We start from −\b{1} and look at a local observable. Like in [MO], we find that, when crossing a special line in U, there is a change in the mechanism of transition towards the stable state +\b{1}. We pass from a situation: 1. where the intermediate phase \b{0} is really observable before the final transition, with a permanence in \b{0} typically much longer than the first hitting time to \b{0}; to the situation: 2. where \b{0} is not observable since the typical permanence in \b{0} is much shorter than the first hitting time to \b{0} and, moreover, large growing 0-droplets are almost full of +1 in their interior so that there are only relatively thin layers of zeroes between +1 and −1. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Blume-Capel thin film model for mixed spins σ = 3/2 and S = 2 placed on a square lattice with four-spin interactions. In fact, we study the effect of the coupling exchange interactions in the presence of both an external and a crystal field. The ground state phase diagrams are studied for this mixed system, and the stable phases have been established. On the other hand, the magnetic properties, at non-null temperatures are investigated using the Monte Carlo simulations. In order to complete this study, we present the thermal behaviors of the hysteresis loops. The dependencies of the coercitif magnetic field on the crystal field, and/or the four-spin interaction couplings, are also discussed. [ABSTRACT FROM AUTHOR]

After showing that it can be parametrized as the elementary coordination game of evolutionary game theory, this paper analyzes the Blume–Capel model using the tensor renormalization group method introduced by Michael Levin and Cody P. Nave. The results obtained along the cross sections defined by the coordination game parametrization expand and corroborate earlier findings regarding the location, order, and critical properties of the order–disorder phase transitions in both the zero-field and non-zero-field versions of the Blume–Capel model. • The Blume–Capel model and the elementary coordination game are exactly equivalent. • The crystal field is temperature-dependent along the corresponding cross sections. • These cross sections are studied here using the tensor renormalization group method. • Phase transition results corroborate earlier findings, confirming the equivalence. • Both the zero- and non-zero-field cases are considered. [ABSTRACT FROM AUTHOR]

In this paper, we study the magnetic properties of ferrimagnetic mixed spins with integer σ = 2 and half-integer S = 7 / 2 in a Blume–Capel model, using Monte Carlo simulations. The considered Hamiltonian includes the first nearest-neighbors and the exchange coupling interactions on the two sub-lattices. The effect of these coupling exchange interactions, in the presence of both the external magnetic field and the crystal field, are studied. The magnetizations and the corresponding susceptibilities are presented and discussed. Finally, we have interpreted the behaviors of the magnetic hysteresis of this model. [ABSTRACT FROM AUTHOR]

Phase diagrams of the mixed spin- and Blume–Capel model on the Bethe lattice in the distribution with probabilities and for and and the adjustment parameter are obtained on the and planes for given values of single-ion anisotropy by varying in the range and in the range . The phase diagrams are constructed by studying the thermal variations of the order parameters in terms of exact recursion relations, and the probability distribution is then implemented into the model. The model presents first- and second-order phase transitions in addition to the tricritical and bicritical points for the coordination number corresponding to the honeycomb lattice. [ABSTRACT FROM AUTHOR]

In this paper, the equilibrium properties of spin-1 Blume-Capel model are studied by using constant-coupling approximation. The formulation is based on developed by Obokata and Oguchi method, where the dependence upon the thermodynamic variables is determined by a set of two-couple nonlinear algebraic equations. The temperature dependence of the order parameters is examined to characterize the nature (continuous or discontinuous) of the phase transitions and to obtain the metastable and unstable branches. For the system, the effect of the uniaxial anisotropy parameter to phase transitions and stable, metastable and unstable states is discussed on the simple cubic lattice with the coordination number = 6. [ABSTRACT FROM AUTHOR]

The effects of bimodal random crystal field and trimodal magnetic field distributions on the order parameters, i.e., magnetization and quadrupole moments, susceptibility and hysteresis properties of the spin-1 Blume-Capel model are investigated. The lowest approximation of the cluster variation method was used. For the distributions of crystal field and magnetic field, two nodes and three nodes distributions were chosen, respectively, with the probabilities q and p. Exploiting the variation of the order parameters and susceptibility as functions of temperature and crystal field, it was found that they exhibit either continuous or discontinuous phase transitions, i.e., first- or second-order, respectively. The model also presents the re-entrant behaviour for some values of our system parameters. Under the constraint of magnetic field and specific values of the system parameters, two, four and six hysteresis loops have been observed. [ABSTRACT FROM AUTHOR]

We derive probabilistic limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume–Emery–Griffiths model [ Phys. Rev. A 4 (1971) 1071–1077]. These probabilistic limit theorems consist of scaling limits for the total spin and moderate deviation principles (MDPs) for the total spin. The model under study is defined by a probability distribution that depends on the parameters n, β, and K, which represent, respectively, the number of spins, the inverse temperature, and the interaction strength. The intricate structure of the phase transitions is revealed by the existence of 18 scaling limits and 18 MDPs for the total spin. These limit results are obtained as (β, K) converges along appropriate sequences (βn, kn) to points belonging to various subsets of the phase diagram, which include a curve of second-order points and a tricritical point. The forms of the limiting densities in the scaling limits and of the rate functions in the MDPs reflect the influence of one or more sets that lie in neighborhoods of the critical points and the tricritical point. Of all the scaling limits, the structure of those near the tricritical point is by far the most complex, exhibiting new types of critical behavior when observed in a limit-theorem phase diagram in the space of the two parameters that parametrize the scaling limits. [ABSTRACT FROM AUTHOR]

We study the antiferromagnetic (AF) spin-1 Blume-Capel (BC) model on a recursive lattice called twofold Cayley tree. Both sublattice magnetizations of the Ising spins are exactly calculated with the aim to obtain phase diagrams and thermal variations of the sublattice magnetizations. The finite-temperature phase diagrams exhibit a small reentrant region if the twofold Cayley tree with a sufficiently high coordination number q > 5 is considered. The results are also compared to obtained by Bethe lattice consideration in recursive approach. [ABSTRACT FROM AUTHOR]

In this study, we investigate the metastable behavior of Metropolis-type Glauber dynamics associated with the Blume–Capel model with zero chemical potential and zero external field at very low temperatures. The corresponding analyses for the same model with zero chemical potential and positive small external field were performed in Cirillo and Nardi (J Stat Phys 150:1080–1114, 2013) and Landim and Lemire (J Stat Phys 164:346–376, 2016). We obtain both large deviation-type and potential-theoretic results on the metastable behavior in our setting. To this end, we perform highly thorough investigation on the energy landscape, where it is revealed that no critical configurations exist and alternatively a massive flat plateau of saddle configurations resides therein. [ABSTRACT FROM AUTHOR]

In this article, we look at DNA as a configuration of the Blume-Capel model and insert it in the path of the Cayley tree. To study the thermodynamic properties of the DNA model, we describe the corresponding translational-invariant Gibbs measure (TIGM) of the model on the Cayley tree. It is shown that, for k ≥ 2, for any temperature T > 0 there is a unique TIGM. Using these results, we study the distributions of the Holliday junctions DNA. For very high and very low temperatures, we give stationary distributions and typical configurations of Holliday junctions. [ABSTRACT FROM AUTHOR]

A new approximation technique is developed so as to study the quantum ferromagnetic spin-1 Blume-Capel model in the presence of a transverse crystal field in the square lattice. Our proposal consists of approaching the spin system by considering islands of finite clusters whose frontiers are surrounded by noninteracting spins that are treated by the effective-field theory. The resulting phase diagram is qualitatively correct, in contrast to most effective-field treatments, in which the first-order line exhibits spurious behavior by not being perpendicular to the anisotropy axis at low-temperatures. The effect of the transverse anisotropy is also verified by the presence of quantum phase transitions. The possibility of using larger sizes constitutes an advantage to other approaches where the implementation of larger sizes is computationally costly. [ABSTRACT FROM AUTHOR]

We consider the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field evolving on a large but finite torus. We obtain sharp estimates for the transition time, we characterize the set of critical configurations, and we prove the metastable behavior of the dynamics as the temperature vanishes. [ABSTRACT FROM AUTHOR]

By using the simplified effective-field theory based on Glauber-type stochastic dynamics, namely the dynamic simplified effective-field theory, the dynamic phase diagrams of a two-dimensional mixed spin-1 and spin-3/2 Blume–Capel model are studied in an oscillating external magnetic field. The dynamic equations are derived for two interpenetrating square lattices. The time variations of average magnetizations and the temperature dependence of the dynamic magnetizations are examined, and the dynamic phase diagrams are presented in the two different planes. The dynamic phase diagrams illustrate several ordered phases, the coexistence phase region and critical points as well as a re-entrant behavior depending on the interaction parameters. Finally, the discussion and comparison of the dynamic phase diagrams are given briefly. [ABSTRACT FROM AUTHOR]

The two-dimensional Blume-Capel model with free surfaces where a surface field $$H_1$$ acts and the 'crystal field' (controlling the density of the vacancies) takes a value $$D _s$$ different from the value $$D$$ in the bulk, is studied by Monte Carlo methods. Using a recently developed finite size scaling method that studies thin films in a $$L \times M$$ geometry with antisymmetric surface fields $$(H_L=-H_1)$$ and keeps a generalized aspect ratio $$c = L^2/M$$ constant, surface phase diagrams are computed for several typical choices of the parameters. It is shown that both second order and first order wetting transitions occur, separated by tricritical wetting behavior. The special role of vacancies near the surface is investigated in detail. [ABSTRACT FROM AUTHOR]

We derive rates of convergence for limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume-Emery-Griffith model. The theorems consist of scaling limits for the total spin. The model depends on the inverse temperature $$\beta $$ and the interaction strength $$K$$ . The rates of convergence results are obtained as $$(\beta ,K)$$ converges along appropriate sequences $$(\beta _n,K_n)$$ to points belonging to various subsets of the phase diagram which include a curve of second-order points and a tricritical point. We apply Stein's method for normal and non-normal approximation avoiding the use of transforms and supplying bounds, such as those of Berry-Esseen quality, on approximation error. [ABSTRACT FROM AUTHOR]

The study of systems with multiple (not necessarily degenerate) metastable states presents subtle difficulties from the mathematical point of view related to the variational problem that has to be solved in these cases. We prove sufficient conditions to identify multiple metastable states. Since this analysis typically involves non-trivial technical issues, we give different conditions that can be chosen appropriately depending on the specific model under study. We show how these results can be used to attack the problem of multiple metastable states via the use of the modern approaches to metastability. We finally apply these general results to the Blume-Capel model for a particular choice of the parameters for which the model happens to have two multiple not degenerate in energy metastable states. We estimate in probability the time for the transition from the metastable states to the stable state. Moreover we identify the set of critical configurations that represent the minimal gate for the transition. [ABSTRACT FROM AUTHOR]

The equilibrium properties of the spin-1 Blume-Capel model are studied by the lowest approximation of the cluster-variation method which is identical to the mean-field approximation. The lattice is divided into two sublattices A and B, and different bilinear interaction parameters JAB and JBA between the nearest-neighbor spins are assumed. The temperature variations of the order-parameters are studied, therefore, the phase diagrams are obtained on the (JAB, T) planes for given values of JBA and crystal fields D. It was found that the model yields both second- and first-order phase transitions at higher positive D values and the lines of which combine at tricritical points. Negative values of D lead only second-order phase transitions. The phase diagrams are symmetric under the exchange of JAB with JBA as expected. [ABSTRACT FROM AUTHOR]

The spin-1 Blume-Capel model is studied on a Bethe lattice which is divided into two sublattices A and B. Alternatingly changing bilinear exchange interactions, JAB and JBA, between the sublattices, i.e., between the nearest-neighbor shell spins, are assumed. The phase diagrams of the model are studied on the (JAB, T) planes for given values of JBA, crystal fields D and the coordination numbers q = 3, 4 and 6. It was found that the model either displays only second-order phase transition lines at higher crystal field values or second- and first-order phase transitions lines combined at tricritical points at lower negative crystal fields. It was also found that the tricritical points move to higher temperatures and to higher values of JAB as the crystal field becomes more negative. [ABSTRACT FROM AUTHOR]

Using mean-field theory, we have studied the effect of quantum transverse field on the magnetic properties and multilayer transitions in the spin-3/2 Blume-Capel model with RKKY interaction. The system is formed by two magnetic multilayer materials, of different thicknesses, separated by a non magnetic spacer of thickness M. It is found that there exist critical transverse fields above which the multilayer transitions occur. This critical value decreases when increasing the thickness of the nonmagnetic spacer M. Phase diagram is established in different plane of space parameter. [ABSTRACT FROM AUTHOR]

We investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model, one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two-dimensional parameter space there exists a curve at which the model undergoes a second-order, continuous phase transition, a curve where the model undergoes a first-order, discontinuous phase transition, and a tricritical point which separates the two curves. We determine the interface between the regions of slow and rapid mixing. In order to completely determine the region of rapid mixing, we employ a novel extension of the path coupling method, successfully proving rapid mixing even in the absence of contraction between neighboring states. [ABSTRACT FROM AUTHOR]

A mean-field approximation is developed for a decorated ferrimagnetic Ising model, in which the two magnetic atoms A and B have spins σ=1/2 and S=1, respectively. In this system, the exchange interaction between nearest-neighbors of atom B is taken into account. Some interesting phenomena, such as the appearance of three types of phase diagrams and the existence of one and two compensation points are found. Phase diagrams and temperature dependence of the magnetizations of the system are investigated in detail. [ABSTRACT FROM AUTHOR]

The so-called diluted-random-cluster model may be viewed as a random-cluster representation of the Blume–Capel model. It has three parameters, a vertex parameter a, an edge parameter p, and a cluster weighting factor q. Stochastic comparisons of measures are developed for the ‘vertex marginal’ when q ∊ [1,2], and the ‘edge marginal’ when q ∊ [1,∞). Taken in conjunction with arguments used earlier for the random-cluster model, these permit a rigorous study of part of the phase diagram of the Blume–Capel model. [ABSTRACT FROM AUTHOR]

We consider a class of spin systems on ℤ d with vector valued spins ( S x ) that interact via the pair-potentials J x, y S x ⋅ S y . The interactions are generally spread-out in the sense that the J x, y 's exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions d≥3, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions d = 1,2 for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique “state,” then in any sequence of translation-invariant Gibbs states various observables converge to their mean-field values and the states themselves converge to a product measure. [ABSTRACT FROM AUTHOR]

We prove existence of the surface tension in the low temperature 2D Blume–Capel model and verify the Ornstein–Zernike asymptotics of the corresponding finite-volume interface partition function. [ABSTRACT FROM AUTHOR]