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Register a new userGibbs-non-Gibbs properties for n-vector lattice and mean-field models
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We review some recent developments in the study of Gibbs and non-Gibbs properties of transformed n-vector lattice and mean-field models under various transformations. Also, some new results for the loss and recovery of the Gibbs property of planar rotor models during stochastic time evolution are presented.
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We extend the notion of Gibbsianness for mean-field systems to the set-up of general (possibly continuous) local state spaces. We investigate the Gibbs properties of systems arising from an initial mean-field Gibbs measure by application of given local transition kernels. This generalizes previous case-studies made for spins taking finitely many values to the first step in direction to a general theory, containing the following parts: (1) A formula for the limiting conditional probability distributions of the transformed system. It holds both in the Gibbs and non-Gibbs regime and invokes a minimization problem for a constrained rate-function. (2) A criterion for Gibbsianness of the transformed system for initial Lipschitz-Hamiltonians involving concentration properties of the transition kernels. (3) A continuity estimate for the single-site conditional distributions of the transformed system. While (2) and (3) have provable lattice-counterparts, the characterization of (1) is stronger in mean-field. As applications we show short-time Gibbsianness of rotator mean-field models on the (q-1)-dimensional sphere under diffusive time-evolution and the preservation of Gibbsianness under local coarse-graining of the initial local spin space.
Models of quantum and classical particles on the d-dimensional cubic lattice with pair interparticle interactions are considered. The classical model is obtained from the corresponding quantum one when the reduced physical mass of the particle tends to infinity. For these models, it is proposed to define the convergence of the Euclidean Gibbs states, when the reduced mass tends to infinity, by the weak convergence of the corresponding Gibbs specifications, determined by conditional Gibbs measures. In fact it is proved that all conditional Gibbs measures of the quantum model weakly converge to the conditional Gibbs measures of the classical model. A similar convergence of the periodic Gibbs measures and, as a result, of the order parameters, for such models with the pair interactions possessing the translation invariance, has also been proven.
We discuss spin models on complete graphs in the mean-field (infinite-vertex) limit, especially the classical XY model, the Toy model of the Higgs sector, and related generalizations. We present a number of results coming from the theory of large deviations and Steins method, in particular, Cramer and Sanov-type results, limit theorems with rates of convergence, and phase transition behavior for these models.
Gibbs Phase Rule describes the nature of phase boundaries on phase diagrams, and is a foundational principle in materials thermodynamics. In Gibbs original derivation, he stipulates that the Phase Rule applies only to simple systems--defined to be homogeneous, isotropic, uncharged, and large enough that surface effects can be neglected; and not acted upon by electric, magnetic or gravitational fields. Modern functional materials; spanning nanomaterials, multiferrorics, materials for energy storage and conversion, colloidal crystals, etc.; are decidedly non-simple, leveraging various additional forms of thermodynamic work to achieve their functionality. Here, we extend Gibbs original arguments on phase coexistence to derive a generalized Phase Rule, based in the combinatorial geometry of high-dimensional convex polytopes. The generalized Phase Rule offers a conceptual and mathematical framework to interpret equilibrium and phase coexistence in advanced modern materials.
We consider the problem of approximate sampling from the finite volume Gibbs measure with a general pair interaction. We exhibit a parallel dynamics (Probabilistic Cellular Automaton) which efficiently implements the sampling. In this dynamics the product measure that gives the new configuration in each site contains a term that tends to favour the original value of each spin. This is the main ingredient that allows to prove that the stationary distribution of the PCA is close in total variation to the Gibbs measure. The presence of the parameter that drives the inertial term mentioned above gives the possibility to control the degree of parallelism of the numerical implementation of the dynamics.
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