Maxwell-Boltzmann Distribution and Minimization of Information Versus Extremization

   Extremization is a mathematical procedure which involves taking the functional derivative of a general function of functions subject to constraints. Formally, if the general function only involves the appearance of the constituents functions without any derivatives of these functions, then the constituent functions appear as variables and one may take the functional derivative as one might take a usual variable derivative. The idea of extremization is that the general function has a certain maximum or minimum value and slight changes in the values of the constituent functions do not change it to first order. This is a formal math procedure, but nevertheless one that is used very often. One example of a quantity which may be minimized is information, i.e. the general function is a measure of information. At this point, information is simply a word, but it is linked a priori with constraints which might involve an increase in information. Thus, minimization of information should be associated with the elimination of all possible constraints, keeping only a constraint(s) which is absolutely necessary. A math process which ensures that all unnecessary constraints are removed should produce the same result as the extremization of a general function representing information (subject to the necessary constraints). 

   As examples, we consider the extremization of Shanon’s entropy - Sum over i p(ei) ln(p(ei)) subject to Sum over i ei p(ei) = Etotal and reaction balance, e1+e2= e3+e4 with p(e1)p(e2) = p(e3)p(e4).

  We have pointed this out in a previous note, but did not give a formal argument as to why this is the case. We try to do so here and show that no extremization process is needed even in the a priori derivation of Shannon’s entropy from Product over i p(ei) power (Np(ei)).