Abstract
Statistical mechanics deals with the average properties of a mechanical system. Some examples are; the water in a kettle, the atmosphere inside a room and the number of atoms in a magnet bar. These kinds of systems are made up of a large number of components, usually molecules. The observer has restricted power to consider all the components. All that can be done is to specify a few average quantities of the system such as its density, pressure or temperature. The main objective of statistical mechanics is to predict the relationship between the observable macroscopic properties of the system, given only a knowledge of the microscopic interactions between the components. The present thesis is devoted to a model whose interacting molecules are located on nearest neighbor vertices of a Cayley tree. In this thesis, ground states and Gibbs measures of λ-model on a Cayley tree of order two are investigated. This investigation is closely related to the phase transitions phenomenon for lattice models on trees, by considering the model where spin has only three values. This kind of model aims to describe all its ground states and study phase transition phenomena by using Gibbs measures
