Kurtzs results consistency in the thermodynamic limit

The deepest and most far-reaching results on the relation of the usual deterministic and stochastic models are due to Kurtz. These results (and their generalisations by L. Arnold) can only be outlined in an informal way here. 

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. 

Kurtz s results are summarised in a very concise, but still appropriate for the chemist, form in Kurtz (1972). His book (Kurtz, 1981) contains more mathematical details, more recent results, and all the relevant references which form the necessary background. 

Write down the differential equation the solution of which is close to the number of occurrences in the reversible Lotka-Volterra reaction. 

Apply the differential equation just before (3.2) of Kurtz (1972) to this special case. 

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