Uncountable Set Of Spin Values Research Articles

Gibbs measures of models with uncountable set of spin values on lattice systems

Gibbs measures of models with uncountable set of spin values on lattice systems

New Condition on Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

New Condition on Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

Nontranslation invariant Gibbs measures for models with uncountable set of spin values on a Cayley tree

We consider models with nearest-neighbour interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. It is known that the “splitting Gibbs measures” of the model can be described by solutions of a nonlinear integral equation. Recently, by solving this integral equation some periodic (in particular translation invariant) splitting Gibbs measures were found. In this paper we give three constructions of new sets of nontranslation invariant splitting Gibbs measures. Our constructions are based on known solutions of the integral equation (1.5) .

Four competing interactions for models with an uncountable set of spin values on a Cayley tree

We consider models with four competing interactions (external field, nearest neighbor, second neighbor, and three neighbors) and an uncountable set [0, 1] of spin values on the Cayley tree of order two. We reduce the problem of describing the splitting Gibbs measures of the model to the problem of analyzing solutions of a nonlinear integral equation and study some particular cases for Ising and Potts models. We also show that periodic Gibbs measures for the given models either are translation invariant or have the period two. We present examples where periodic Gibbs measures with the period two are not unique.

A model with uncountable set of spin values on a Cayley tree: phase transitions

In this paper we consider a model with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order two. This model depends on two parameters \(n\in \mathbb N\) and \(\theta \in [0,1)\). We prove that if \( 0 \le \theta \le \frac{2n+3}{2(2n+1)}\), then for the model there exists a unique translational-invariant Gibbs measure; If \(\frac{2n+3}{2(2n+1)}< \theta <1\), then there are three translational-invariant Gibbs measures (i.e. phase transition occurs).

Periodic Gibbs measures for models with uncountable set of spin values on a Cayley tree

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. We show that periodic Gibbs measures are either translation-invariant or periodic with period two. We describe two-periodic Gibbs measures of the model. For k = 1 we show that there is no any periodic Gibbs measure. In case k ≥ 2 we get a sufficient condition on Hamiltonian of the model with uncountable set of spin values under which the model has no periodic Gibbs measure. We construct several models which have at least two periodic Gibbs measures.

Phase transitions for a model with uncountable set of spin values on a Cayley tree

In this paper we consider a model with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 2. To study translation-invariant Gibbs measures of the model we drive an nonlinear functional equation. For k = 2 and 3 under some conditions on parameters of the model we prove non-uniqueness of translation-invariant Gibbs measures (i.e., there are phase transitions).

Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\geq 1$. It is known that the "splitting Gibbs measures" of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\geq 2$ we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure.

Non-uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree

In this paper we construct several models with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order k≥2. We prove that each of the constructed model has at least two translational-invariant Gibbs measures.

On Models with Uncountable Set of Spin Values on a Cayley Tree: Integral Equations

We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ⩾ 1. We reduce the problem of describing the “splitting Gibbs measures” of the model to the description of the solutions of some nonlinear integral equation. For k = 1 we show that the integral equation has a unique solution. In case k ⩾ 2 some models (with the set [0, 1] of spin values) which have a unique splitting Gibbs measure are constructed. Also for the Potts model with uncountable set of spin values it is proven that there is unique splitting Gibbs measure.