Globalization of the analysis of particle placement models by cells

The title of the paper means that its goal is a general approach to the pre-asmptotic analysis of schemes with different qualities in all combinations of their distinguishability of their constituent elements (cells and particles). To do this, in each group of such schemes with general restrictions, instead of directly studying them based on the specificity of each scheme, a certain general set of algorithmic procedures for recalculating the results of their pre-asymptotic analysis in the scheme is proposed, starting with the scheme with the greatest differentiation of their outcomes, sequentially for other schemes of the group with differences as one item. The analysis of each scheme is carried out according to the traditional and in a number of new following directions: constructing a random process of formation and numbered non-repeated enumeration of the outcomes of the scheme in the order of their receipt, finding their number, solving the numbering problem for the outcomes of the scheme, which consists in establishing a one-to-one correspondence between their types and numbers, setting their probabilistic distribution and modeling the outcomes of the scheme with this probabilistic distribution.

In particular, the cases of groups of schemes without restrictions on the placement of particles and with a restriction of at most one particle in a cell are studied separately, which lead to some well-known analytical results. Under any restrictions in the considered group of circuits, their analysis is carried out by implementing algorithmic procedures for sequential transformation of the results of the analysis of one circuit of the group for another. Combinations into such pairs of schemes are made on the basis of the difference in the quality of one of their elements.