Mathematical explanation of the predictive power of the X-level approach reaction noise estimator method

The X-level Approach Reaction Noise Estimator (XARNES) method has been developed previously to study reaction noise in well mixed reaction volumes. The method is a typical moment closure method and it works by closing the infinite hierarchy of equations that describe moments of the particle number distribution function. This is done by using correlation forms which describe correlation effects in a strict mathematical way. The variable X is used to specify which correlation effects (forms) are included in the description. Previously, it was argued, in a rather informal way, that the method should work well in situations where the particle number distribution function is Poisson-like. Numerical tests confirmed this. It was shown that the predictive power of the method increases, i.e. the agreement between the theory and simulations improves, if X is increased. In here, these features of the method are explained by using rigorous mathematical reasoning. Three derivative matching theoremsare proven which show that the observed numerical behavior is generic to the method.