Mean field approximation and the problem of multiple realizations in the BBGKY hierarchy on the example of a ferromagnet

The paper considers a model of a ferromagnet constructed on the basis of the mean field approximation. This model is fully described Curie-Weiss theory within a single implementation (sample).
The concept of the problem of the set of realizations (PMR) is introduced, which arises when the BBGKY hierarhy is closed — an implicit transition from the description ensemble of non interaction samples to the description of one sample.
It is shown that the closure of the BBGKY hierarhy through the mean field approximation leads to TMR. At the same time, the approximation of correlation magnetodynamics makes it possible to avoid PMR and leads to a correct description of the entire ensemble at the level of one-two-particle distribution functions.