Entropic measure of the order-disorder character in lattice systems in the representation of coordination Cayley tree graphs

• Published in: Theoretical and mathematical physics 2010-07, Vol.164 (1), p.905-919
• Main Authors: Yudin, V. V., Titov, P. L., Mikhalyuk, A. N.
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical
• ISSN: 0040-5779; 1573-9333
• DOI: 10.1007/s11232-010-0072-9

We systematically expound the infodynamical method for analyzing lattice and grid systems. We establish the logic and algorithm for mapping given objects to coordination Cayley tree graphs and present their main properties. Tree graphs of grid systems are complicated objects, and the principle of cluster-type simplicial decomposition can be used to study them. Based on a simplicial decomposition, we construct the enumerating structures, from which we construct entropy-type functionals. We pose the percolation problem on Cayley tree graphs in a nonconventional sense, which may be considered for both enumerating structures and their entropies. The corresponding entropy percolational dependences and their critical indices can be considered sufficiently universal measures of order in lattice systems. The simpliciality also implies an analogy with the fractality principle. We introduce three types of fractal characteristics and give analytic expressions for fractal dimensions for the tangential and streamer representations and for the Mandelbrot shell.