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

It is well known that the Qur’an has many miraculous attributes proving that it is revelation from God. 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, us

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

From three-state Potts model we produce the recursion relations that provide us numerically exact phase diagrams of the Potts model. Each phase is characterized by a pafticular attractor and the phase diagram is obtained by following the evolution and detecting the qualitative changements of these attractors. Ihese changements can either be continu

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

We study the phase diagrams for the Potts model on the Bethe lattice with competing nearest-neighbor interactions J1 and ternary interactions Jp and Jt: At vanishing temperature, the phase diagram is fully determined for all values and signs of Jp=J1 and Jt=J1; in particular, we show that the phase diagrams contain ferromagnetic, paramagnetic and &

We study an Ising model on a general Cayley tree with competing interaction of next-nearest-neighbour of two types; prolonged and one-level k-tuple interactions. Vannimenus covers a study of Ising model on competing nearest-neighbour interaction J1 and prolonged next-nearest-neighbour interaction Jp. Later Mariz et al extend Vannimenus work by addi

The history of the quadratic stochastic operators can be traced back to the work of Bernshtein (1924). For more than 80 years, this theory has been developed and many papers were published. In recent years it has again become of interest in connection with its numerous applications in many branches of mathematics, biology and physics. But most resu

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,

We study the phase diagram of the Ising model on a Cayley tree with competing prolonged next-nearest neighbour Jp and one-level next-nearest neighbour interactions J0. Vannimenus proved that the phase diagram of Ising model with competing neareast-neighbour interaction J1 and prolonged next-nearest neighbour interactions Jp contains a modulated pha