The bond problem with an arbitrary percolation radius is solved!

The results of investigations of main characteristics of a one-dimensional percolation theory (percolation threshold, critical exponents of correlation radius and specific heat) are presented for the problem of bonds and sites. It is shown that for a finite-size system the stability condition is fulfilled while the scaling hypothesis is inacceptable for one-dimensional bond problem. The correlation length exponent $\nu $ in a one-dimensional problem of bonds has been found to exceed the values of $\nu $ in the problem of sites for equal-length chains, and, in general, this exponent was found to be extraordinary large compared to the 2-D and 3-D cases for ordinary phase transitions in macrosystems. The scaling hypothesis is inapplicable to random (disordered) one-dimensional nanostructures containing hundreds of structural elements. The results obtained in this work can be used in modeling hopping conduction in semiconductors at low temperatures and polytype transformations in close-packed crystals. For the first time, using the method of computer simulation, we have solved the bond problem for the model of one-dimensional percolation in finite-size systems of tens of nanometers with an arbitrary percolation radius.