On nonlinear dynamic systems arising in Lattice models of statistical mechanics on a Cayley tree - Ganikhodjaev, Nasir (2009)
Statistical mechanics seeks to explain the macroscopic behavior of matter on the basis of its microscopic structure. This includes the analysis of simplified mathematical models. A phase diagram of a model describes a morphology of phases, stability of phases, transitions from ne phase to another and corresponding transitions line. A Potts model just as an Ising model on a Cayley tree with competing interactions [1],[2], has recently been studied extensively because of the appearance of nontrivial magnetic orderings. The Cayley tree is not a realistic lattice; however, its amazing topology makes the exact calculation of various quantities possible. For many problems the solution on a tree is much simpler than on a regular lattice and is equivalent to the standard Bethe-Peierls theory. On the Cayley tree one can consider two type of second ( triple) neighbours: prolonged and one(two)level. For given lattice model on a Cayley tree we produce recursive relations obtained following the lines of Inawashiro et al [3]. These recursive relations 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 or abrupt, respectively characterizing second- or first- order phase transitions. We present a few typical attractors and show that for some lattice models on a Cayley tree with competing interactions their phase diagram contain modulated phase [1],[2] if and only if the Hamiltonian of corresponding lattice model contains interactions of prolonged neighbours.