Combinatorial model

Tratch, S.S., Devdariani, R.Q. and Zefirov, N.S. (1990a). Combinatorial Models and Algorithms in Chemistry. Topological-Configurational Analogs of the Wiener Index. Zh.Org.Khim., 26, 921-932. 

Tratch, S.S., Stankevitch, I.V. and Zefirov, N.S. (1990) Combinatorial models and algorithms in chemistry. The expanded Wiener number- a novel topological index. J. Comput. Chem., 11, 899-908. 

This assumption corresponds to the Kurtz-type or combinatorial model.)... 

Equations similar to (5.36) and (5.37) can be derived using assumption (4) instead of (4) in this case we might call the equations the combinatorial model. Generally these equations are called the master equations. The structure of (5.37) is clear. Two types of elementary reactions are taken into consideration. The effect of the first class of reaction is that the state j is available from / (/ can denote different possible states). The second class describes all of the possible transitions from the state J. Therefore we can write ... 

Here the superscript K refers again to the combinatorial model. Although this equation is a general model, it is the main tool for obtaining exact solutions, and that is why it is mentioned in this section on solutions. 

As it has turned out that consistency in the mean does not hold in general, several people have presented a proof of the fact that the stochastic model of a certain simple special reaction tends to the corresponding deterministic model in the thermodynamic limit. This expression means that the number of particles and the volume of the vessel tend to infinity at the same time and in such a way that the concentration of the individual components (i.e. the ratio of the number and volume) tends to a constant and the two models will be close to each other. In addition to this the fluctuation around the deterministic value is normally distributed as has been shown in a special case by Delbriick (later head of the famous phage group) almost fifty years ago (Delbriick, 1940). To put it into present-day mathematical terms the law of the large numbers, the central limit theorem, and the invariance principle all hold. These statements have been proved for a large class of reactions for those with conservative, reversible mechanisms. Kurtz used the combinatorial model, and the same model was used by L. Arnold (Arnold, 1980) when he generalised the results for the cell model of reactions with diffusion. 

S. Tratch and N. Zefirov. Combinatorial models and algorithms in chemistry. The ladder of combinatorial objects and its application to the formalization of structural problems of organic chemistry, in N. Stepanov, editor. Principles of Symmetry and Systemology in Chemistry, pages 54-86. Moscow State University Publishers, Moscow, 1987. 

S. S. Tratch, I.V. Stankevich, and N.S. Zefirov, Combinatorial models and algorithms in chem-... 

The classical model for network formation has been proposed by Flory and Stockmayer This combinatorial model provides an adequate description for the formation of networks via step reactions and for crosslinking of linear chains. Basic assumptions are ... 

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