T.V. Yakunina – Senior Lecturer, Department of Fundamental Training, Sayan-Shushenskiy Branch  of Siberian Federal University (Sayanogorsk)

E-mail: tatav19@mail.ru 

V.N. Udodov – Dr.Sc. (Phys.-Math.), Professor, 

Engineering and Technology Institute, N.F. Katanov Khakas State University (Abakan)

E-mail: udodov@khsu.ru

The main difficulty in solving the bond problem of percolation theory is to construct a covering lattice (or adjacency matrix) and describe it in a programming language. It is also essential that the general algorithm for constructing the covering lattice and the adjacency matrix cannot be formulated for an arbitrary percolation radius, even for the one-dimensional case. 

Aim of the work – to develop method and algorithms for solving one-dimensional bond problems for systems of finite (nanometer) size without constructing covering lattices and an adjacency matrix for an arbitrary percolation radius. 

A new algorithm for numbering the bonds of a one-dimensional lattice (without constructing a covering lattice and an adjacency matrix) and an algorithm that allows one to find the percolation threshold for lattices of an arbitrary size with an arbitrary flow radius are proposed. A method is developed for finding the free energy and critical heat capacity exponent for a one-dimensional bond problem. The analogue of free energy and the critical heat capacity exponent are calculated for one-dimensional lattice of 40 nodes with a zero external field. The dependence of the critical heat capacity exponent on the fraction of whole bonds is calculated for different flow radius in the one-dimensional percolation problem. The dependence of the analog of free energy on the fraction of whole bonds is obtained.

Algorithms are proposed for solving the bond problem for a one-dimensional lattice and for finding a percolation threshold for lattices of arbitrary size with an arbitrary flow radius without constructing a covering lattice and an adjacency matrix. These algorithms work on computers much faster than the known ones.

The presented results can be used in modeling the hopping conductivity of semiconductors at low temperatures, polytype transformations with features in the nanometer range in close-packed crystals.

Within the framework of the proposed approach, it is also possible to calculate other critical exponents.

The research was carried out with financial support from the Russian Foundation for Basic Research and the Republic of Khakassia in the framework of the scientific project No. 18-41-190003.

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