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We consider two models of interacting DNA molecules: First is (four parametric) bubble coalescence model in interacting DNAs (shortly: BCI-DNA). Second is (three parametric) bubble coalescence model in a condensed DNA molecules (shortly BCC-DNA). To study bubble coalescence thermodynamics of BCI-DNA and BCC-DNA models we use methods of statistical physics. Namely, we define Hamiltonian of each model and give their translation-invariant Gibbs measures (TIGMs). For the first model, we find parameters such that corresponding Hamiltonian has up to three TIGMs (three phases of system) biologically meaning existence of three states: "No bubble coalescence", "Dominated soft zone", "Bubble coalescence". For the second model, we show that for any (admissible) parameters, this model has unique TIGM. This is a state where "No bubble coalescence" phase dominates. [ABSTRACT FROM AUTHOR]

For the solid-on-solid (SOS) model with an external field and with spin values from the set of all integers on a Cayley tree, each (gradient) Gibbs measure corresponds to a boundary law (an infinite-dimensional vector function defined on vertices of the Cayley tree) satisfying a nonlinear functional equation. Recently some translation-invariant and height-periodic (non-normalizable) solutions to the equation are found. Here, our aim is to find non-height-periodic and non-normalizable boundary laws for the SOS model. By such a solution one can construct a non-probability Gibbs measure. We find explicitly several non-normalizable boundary laws. Moreover, we reduce the problem to solving of a nonlinear, second-order difference equation. We give analytic and numerical analyses of the difference equation. [ABSTRACT FROM AUTHOR]

Kittel's 1D model represents a natural DNA with two strands as a (molecular) zipper, which may be separated as the temperature is varied. We define multidimensional version of this model on a Cayley tree and study the set of Gibbs measures. We reduce description of Gibbs measures to solving of a nonlinear functional equation, with unknown functions (called boundary laws) defined on vertices of the Cayley tree. Each boundary law defines a Gibbs measure. We give a general formula of free energy depending on the boundary law. Moreover, we find some concrete boundary laws and corresponding Gibbs measures. Explicit critical temperature for occurrence of a phase transition (non-uniqueness of Gibbs measures) is obtained. [ABSTRACT FROM AUTHOR]

We consider a hard core (HC) model with a countable set ℤ of spin values on the Cayley tree. This model is defined by a countable set of parameters λ i > 0 , i ∈ ℤ ∖ { 0 }. For all possible values of parameters, we give limit points of the dynamical system generated by a function which describes the consistency condition for finite-dimensional measures. Also, we prove that every periodic Gibbs measure for the given model is either translation-invariant or periodic with period two. Moreover, we construct uncountable set of Gibbs measures for this HC model. [ABSTRACT FROM AUTHOR]

In this paper we investigate generalized Gibbs measure (GGM) for -adic Hard-Core (HC) model with a countable set of spin values on a Cayley tree of order . This model is defined by -adic parameters , . We analyze -adic functional equation which provides the consistency condition for the finite-dimensional generalized Gibbs distributions. Each solutions of the functional equation defines a GGM by -adic version of Kolmogorov's theorem. We define -adic Gibbs distributions as limit of the consistent family of finite-dimensional generalized Gibbs distributions and show that, for our -adic HC model on a Cayley tree, such a Gibbs distribution does not exist. Under some conditions on parameters , and we find the number of translation-invariant and two-periodic GGMs for the -adic HC model on the Cayley tree of order two. [ABSTRACT FROM AUTHOR]

In this paper, we give a systematic review of the theory of Gibbs measures of Potts model on Cayley trees (developed since 2013) and discuss many applications of the Potts model to real world situations: mainly biology, physics, and some examples of alloy behavior, cell sorting, financial engineering, flocking birds, flowing foams, image segmentation, medicine, sociology, etc. [ABSTRACT FROM AUTHOR]

We consider a new model which consists of a DNA together with a RNA. Here, we assume that DNA is from a mammal or bird but RNA comes from a virus. To study thermodynamic properties of this model, we use methods of statistical mechanics, namely, the theory of Gibbs measures. We use these measures to describe phases (states) of the DNA–RNA system. Using a Markov chain (corresponding to Gibbs measure) we give conditions (on temperature) of DNA–RNA renaturation. [ABSTRACT FROM AUTHOR]

The contour argument was introduced by Peierls for two dimensional Ising model. Peierls benefited from the particular symmetries of the Ising model. For non-symmetric models the argument was developed by Pirogov and Sinai. It is very general and rather difficult. Intuitively clear that the Peierls argument does work for any symmetric model. But contours defined in Pirogov–Sinai theory do not work if one wants to use Peierls argument for more general symmetric models. We give a new definition of contour which allows relatively easier proof to the main result of the Pirogov–Sinai theory for symmetric models. Namely, our contours allow us to apply the classical Peierls argument (with contour removal operation). [ABSTRACT FROM AUTHOR]

In the Letter we investigate aq-component models on a Cayley tree. The main goal of the Letter is to develop a ‘‘contour’’ argument on Cayley tree. We define contours and study some properties of these contours. Using a contour argument we show existence ofqdifferent Gibbs measures for severalq-component models. [ABSTRACT FROM AUTHOR]