Nasir Ganikhodjaev

Exact solution of an Ising model with competing interactions on a Cayley tree

The exact solution of an Ising model with competing restricted interactions on the Cayley tree, and in the absence of an external field is presented. A critical curve is defined where it is possible to get phase transitions above it, and a single Gibbs state is obtained elsewhere.

On disordered phase in the ferromagnetic Potts model on the Bethe lattice

where the sum is taken over all pairs of the nearest neighbors (jc, >>), 0. P. M. Bleher [1] proved that the disordered Phase in the ferromagnetic Ising model on the Bethe lattice is extreme for T > Γc, where Tc is the critical temperature of the spin glass model on the Bethe lattice, and it is not extreme for T < Tc. Denote θ = exp(//Γ). The main result of this paper is the following theorem

Quantum quadratic operators and processes

Covering both classical and quantum approaches, this book presents the most recent developments in the theory of quadratic stochastic operators and their Markov and related processes. The asymptotic behavior of dynamical systems generated by classical and quantum quadratic operators is investigated and various properties of quantum quadratic operators are studied, providing an insight into the construction of quantum channels

Phase diagram of the three states Potts model with next nearest neighbour interactions on the Bethe lattice

Abstract We have found an exact phase diagram of the Potts model with competing nearest neighbor and next nearest neighbor interactions on the Bethe lattice of order two. The diagram consists of five phases: ferromagnetic, paramagnetic, modulated, antiphase and paramodulated, all meeting at the multicritical point ( T = 0 , p = 1 / 3 ) . We report on a new phase which we denote as paramodulated, found at low temperatures and characterized by zero average magnetization lying inside the modulated phase. Such a phase, inherent in the Potts model has no analogues in the Ising setting.

Phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions

One of the main problems of statistical physics is to describe 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 article is to investigate phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions.

The Potts model on Zd with countable set of spin values

The Potts model with countable set Φ of spin values on Zd is considered. It is proved that with respect to Poisson distribution on Φ the set of limiting Gibbs measures is not empty.

An ising model with three competing interactions on a Cayley tree

In this paper we consider an Ising model with three competing restricted interactions on the Cayley tree J2(J3). The translation invariant and periodic Gibbs measures for these models are investigated and the problem of the phase transition in these classes is solved.

On extreme Gibbs measures of the Vannimenus model

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 the three state Potts model with competing interactions on the Bethe lattice

In the present paper the three state Potts model with competing binary interactions (with couplings J and Jp) on the second order Bethe lattice is considered. The recurrent equations for the partition functions are derived. For Jp = 0, by means of a construction of a special class of limiting Gibbs measures, it is shown how these equations are related to the surface energy of the Hamiltonian. This relation reduces the problem of describing the limit Gibbs measures to that of finding solutions of a nonlinear functional equation. Moreover, the set of ground states of the one level model is completely described. Using this fact, we find Gibbs measures (pure phases) associated with the translation-invariant ground states. The critical temperature is found exactly and the phase diagram is presented. The free energies corresponding to translation-invariant Gibbs measures are found. Certain physical quantities are calculated as well.

Mutation and Chaos in Nonlinear Models of Heredity

We shall explore a nonlinear discrete dynamical system that naturally occurs in population systems to describe a transmission of a trait from parents to their offspring. We consider a Mendelian inheritance for a single gene with three alleles and assume that to form a new generation, each gene has a possibility to mutate, that is, to change into a gene of the other kind. We investigate the derived models and observe chaotic behaviors of such models.