Maximum Reaction Product Uncertainty and Factorial Schemes in Statistical Mechanics

In previous notes, we have discussed a method of obtaining equilibrium distributions by balancing reaction probabilities, i.e. taking the ln (log) of a balance equation and linking it ( up to constants) to conservation of energy e1+e2=e3+e4. We have also linked such a scenario to factorial schemes M!/ [m1! M2! …] ((1)) subject to energy and number constraints. In a recent note (1), we argued that reaction balance for the Maxwell-Boltzmann case is based on complete uncertainty of reaction products for each reaction e1+e2=E. In this note, we try to link this idea of uncertainty of reaction products to the factorial scheme ((1)). For more general cases, f(ei) is replaced by g(f(ei)) which obeys Maxwell-Boltzmann statistics and yields results for non-Maxwell-Boltzmann f(ei) such as Fermi-Dirac and Bose-Einstein cases