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On Some Nonlinear Dynamic Systems Arising in Statistical Mechanics
Research Output | Dar Al-hikmah Library | IIUM Only

For Potts model on a Cayley tree we produce recursive relations that provide us the numerically exact phase diagram of the model. Each phase is characterized by a particular attractor and the phase diagram is obtained by following the evolution and detecting the qualitative changements of these attractors. These changements can be either continuous

Multidimensional Mapping for Potts Model with Competing Interactions
Research Output | Dar Al-hikmah Library | IIUM Only

We study the phase diagram for Potts model on a Cayley tree of arbitrary order k with competing interactions: nearest-neighbor interactions J1, prolonged next-nearest-neighbor interactions Jp and one level binary neighbors interaction J0. To perform this study, an iterative scheme similar to that appearing in real space renormalization group framew

Limit Behavior of Dynamic Systems Corresponding to Lattice Models with Competing Prolonged and one-Level Binary Interactions
Research Output | Dar Al-hikmah Library | IIUM Only

We study the phase diagram of the Ising model on a Cayley tree with competing prolonged next-nearest neighbour Jpand one-level next-nearest neighbour interactions J. Vannimenus proved that the phase diagram of Ising model with competing neareast-neighbour interaction J1and prolonged next-nearest neighbour interactions Jpcontains a modulated phase,

The Miracle in the Iron and the Ising Model of the Ferromagnet
Research Output | Dar Al-hikmah Library | IIUM Only

It is well known that the Qur’an has many miraculous attributes proving that it is revelation from All�h. In this paper we consider one of them, namely, the miracle in the Iron. One of the main and very useful property of the Iron is to be magnet. In 1925 Ising wrote his doctoral thesis on a model now called the Ising model. He tried to explain

On Extreme Gibbs Measures of the Vannimenus Model
Research Output | Dar Al-hikmah Library | IIUM Only

One of the main problems of statistical physics is that of describing all Gibbs measures corresponding to a given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probability measures. The purpose of this paper is to investigate extreme Gibbs measures of the Vannimenus model.On extreme Gibbs m

Phase Diagrams of an Ising System with Competing Binary, Prolonged Ternary and next-Nearest Interactions on a Cayley Tree
Research Output | Dar Al-hikmah Library | IIUM Only

In this paper we consider the Ising model with spin values in Φ = {−1, 1}, the relevant Hamiltonian with competing binary nearest-neighbor, prolonged ternary and prolonged next-nearest neighbors interactions has the form H(σ) = −Jp �� >x,y< σ(x)σ(y) − Jt >x�,¯y,z< σ(x)σ(y)σ(z)− J1 � <x,y>σ(x)σ(y), w

Strange Attractor in the Potts Model on a Cayley Tree in the Presence of Competing Interactions
Research Output | Dar Al-hikmah Library | IIUM Only

For Potts model on a Cayley tree we produce recursive relations that provide us the numerically exact phase diagram of the model. Each phase is characterized by a particular attractor and the phase diagram is obtained by following the evolution and detecting the qualitative changements of these attractors. These changements can be either continuous

Potts Model with Competing Interactions on a Cayley Tree of k-Th Order
Research Output | Dar Al-hikmah Library | IIUM Only

We study the phase diagram for Potts model on a Cayley tree of arbitrary order k with competing nearest-neighbor interactions J1, prolonged next-nearest-neighbor interactions Jp and one level k-tuple neighbors interaction J0. To perform this study, an iterative scheme similar to that appearing in real space renormalization group frameworks is estab

Phase Transition in Ferromagnetic Lattice Models on a Cayley Tree with non-Uniform External Magnetic Fields
Research Output | Dar Al-hikmah Library | IIUM Only

In this article we study the phase transition phenomenon for the Ising and Potts models under the action of a non-uniform external magnetic field hx, x V, where V is the set of vertices of the Cayley tree. We show that both models on the Cayley tree with –J < inf {hx: x V} < sup{ hx: x V} < J, have a first-order phase transitionPhase

Lattice Models of Statistical Mechanics and their Applications
Research Output | Dar Al-hikmah Library | IIUM Only

Lattice models are caricatures invented to illuminate various aspects of elementary statistical mechanics, especially the phenomena of phase transitions and spon- taneously broken symmetry. This paper is an attempt to present the basic ideas of the subject and its application to a wider audience.Lattice Models of Statistical Mechanics and their App

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