Peter T. Otto

Analysis of phase transitions in the mean-field Blume–Emery–Griffiths model

In this paper we give a complete analysis of the phase transitions in the mean-field Blume–Emery–Griffiths lattice-spin model with respect to the canonical ensemble, showing both a second-order, continuous phase transition and a first-order, discontinuous phase transition for appropriate values of the thermodynamic parameters that define the model. These phase transitions are analyzed both in terms of the empirical measure and the spin per site by studying bifurcation phenomena of the corresponding sets of canonical equilibrium macrostates, which are defined via large deviation principles. Analogous phase transitions with respect to the microcanonical ensemble are also studied via a combination of rigorous analysis and numerical calculations. Finally, probabilistic limit theorems for appropriately scaled values of the total spin are proved with respect to the canonical ensemble. These limit theorems include both central-limit-type theorems, when the thermodynamic parameters are not equal to critical values, and noncentral-limit-type theorems, when these parameters equal critical values.

Asymptotic behavior of the magnetization near critical and tricritical points via Ginzburg-Landau polynomials

The purpose of this paper is to prove unexpected 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 meanfield version of an important lattice-spin model due to Blume and Capel. It is defined by 1 ar X iv :0 80 3. 01 78 v2 [ co nd -m at .s ta tm ec h] 9 M ay 2 00 8 Ellis, Machta, and Otto: Asymptotics for the Magnetization 2 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(βn,Kn) for appropriate sequences (βn,Kn) 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 (βn,Kn) converges to one of these points (β,K), m(βn,Kn) ∼ x̄|β − βn| → 0. In this formula γ is a positive constant, and x̄ 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. This polynomial arises as a limit of appropriately scaled free-energy functionals, the global minimum points of which define the phase-transition structure of the model. In the asymptotic formula m(βn,Kn) ∼ x̄|β − βn| , both γ and x̄ depend on the sequence (βn,Kn). Six examples of such sequences are considered, each leading to a different asymptotic behavior of m(βn,Kn). Our approach to studying the asymptotic behavior of the magnetization has three advantages. First, for each sequence (βn,Kn) under study, the structure of the global minimum points of the associated Ginzburg-Landau polynomial mirrors the structure of the global minimum points of the free-energy functional in the region through which (βn,Kn) passes and thus reflects the phase-transition structure of the model in that region. In this way the properties of the Ginzburg-Landau polynomials make rigorous the predictions of the Ginzburg-Landau phenomenology of critical phenomena. Second, we use these properties to discover new features of the first-order curve in a neighborhood of the tricritical point. Third, the predictions of the heuristic scaling theory of the tricritical point are made rigorous by the asymptotic formula m(βn,Kn) ∼ x̄|β − βn| , which is the main result in the paper. American Mathematical Society 2000 Subject Classifications. Primary 82B20

Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality

The main focus of this paper is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finite-size magnetization. This is done by comparing the asymptotic behaviors of these two quantities along parameter sequences converging to either a second-order point or the tricritical point in the meanfield Blume–Capel model. We show that the thermodynamic magnetization and the finite-size magnetization are asymptotic when the

Mixing Times for the Mean-Field Blume-Capel Model via Aggregate Path Coupling

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.

Multiple critical behavior of probabilistic limit theorems in the neighborhood of a tricritical point

AbstractsWe 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. A4 (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.

RAPID MIXING OF GLAUBER DYNAMICS OF GIBBS ENSEMBLES VIA AGGREGATE PATH COUPLING AND LARGE DEVIATIONS METHODS

In this paper, we present a novel extension to the classical path coupling method to statistical mechanical models which we refer to as aggregate path coupling. In conjunction with large deviations estimates, we use this aggregate path coupling method to prove rapid mixing of Glauber dynamics for a large class of statistical mechanical models, including models that exhibit discontinuous phase transitions which have traditionally been more difficult to analyze rigorously. The parameter region for rapid mixing for the generalized Curie–Weiss–Potts model is derived as a new application of the aggregate path coupling method.

The aggregate path coupling method for the Potts model on bipartite graph

In this paper, we derive the large deviations principle for the Potts model on the complete bipartite graph

Path Coupling and Aggregate Path Coupling

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Large Deviations and Equilibrium Macrostate Phase Transitions

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Aggregate Path Coupling: One-Dimensional Theory

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