Maxwell-Boltzmann Distribution and Minimization of Information Versus Extremization Part 2

Maxwell-Boltzmann Distribution and Minimization of Information Versus Extremization Part 2 Francesco R. Ruggeri Hanwell, N.B. Dec.5 , 2025 In Part 1, we noted that Shannon’s entropy is often maximized with respect to the constraint Sum over i p(ei) ei = E total to obtain the Maxwell-Boltzmann distribution. Maximization (or extremization) is a well-known math procedure. In Part 1, we suggested that due to properties of probabilities in AND situations (i.e. p(e1)p(e2) = p(e1+e2) no normalization), one has two additive quantities in the problem, namely ei and ln(p(ei)). We then argued that to have the minimal informational solution, ln(p(ei)) = -ei/T, where -1/T is a constant. Here we wish to elaborate on this idea. We suggest that in order to maximize a function which arises a priori, e.g. Shannon’s entropy - Sum over i p(ei) ln(p(ei), one must incorporate all necessary constraints. Certainly Sum over i ei p(ei)= Etotal is an obvious constraint, but we suggest that the AND property of probability actually leads to a second constraint, namely that ln(p(ei)) = constant * ei. This is a constraint, not a result of maximization, and it must be implemented before maximization occurs. The point, however, is that ln(p(ei)) = constant* ei solves for the distribution (Maxwell-Boltzmann Cexp(-ei/T)) without any maximization being necessary. We argue that ln(p(ei)) = constant * ei follows not from minimal informational suggestions made in Part 1, but from math. Thus, a constraint in the problem actually solves the problem, we suggest. Shannon’s Entropy and Extremization Shannon’s entropy - Sum over i p(ei) ln(p(ei)) may be maximized subject to the constraint Sum over i p(ei) ei = E total to yield the Maxwell-Boltzmann distribution, p(ei) = C exp(-ei/T), as is well-known. In Part 1, we showed that Shannon’s entropy arises from calculating: Probability(N-> infinite runs) = 1/ number of states = Product over i p(ei) power (Np(ei)) ((1)) Here it is assumed that if N is very large, Np(ei) = n(ei) = number of independent ei’s. The strategy maintained in the literature is that one wishes to have as many states as possible because any restriction represents a bias and one assumes that no such bias exists. This leads to the mathematical maximization of -Sum over i p(ei) ln(p(ei)) subject to the a priori constraint Sum over i ei p(ei) = E total and yields the MB distribution. The problem with this approach, we argue, is that Sum over i p(ei) ei is not the only a priori constraint in the problem. Formally, one must include all relevant constraints and a second fundamental constraint exists. This constraint must be included before one may consider performing the maximization process, we argue. A Second Constraint Due to the Properties of Probability In Part 1, we argued that probability must adhere to the AND condition, i.e. p(ei) p(ej) = p(ei+ej) for p(ei) unnormalized ((2)) ln(p(ei)) is an additive quantity, just as ei is an additive quantity, as we pointed out in Part 1. The question becomes: Is it possible to determine the functional form of p(ei) based on ((2))? p(ei) is a function of ei. A solution to ((2)) is: p(ei) = C exp(-ei/T) ((3)) Thus, the probability property or constraint ((2)) does lead to a possible solution for p(ei) without any notion of maximization. We argue that this solution is representative of the second constraint ((3)) and must be considered before any maximization process is employed. Thus, we argue that one cannot formally maximize: - Sum over i p(ei) ln(p(ei)) + constant Sum over i ei p(ei) ((5)) without first associating ln(p(ei)) with constant * ei (e.g. -ei/T). This, however, means that the a priori Shannon’s entropy function is really: - Sum over i ei p(ei) ((6)) and so there is nothing to maximize. We thus suggest that it is formally invalid to maximize Shannon’s entropy subject to Sum over i ei p(ei), ignoring ln(p(ei)) = constant * ei which is a valid constraint which must first be utilized. Utilizing it, however, yields the solution to the problem, p(ei) = C exp(-ei/T), without any maximization required. We thus question why one maximizes Shannon’s entropy subject to the constraint Sum over i p(ei) ei = E total instead of writing p(ei) = C exp(-ei/T) from the probability constraint ((2)). Conclusion In conclusion, the maximization/extremization of a function subject to necessary constraints is a very useful math procedure which is often employed in physics. We also note, as shown in Part 1, that one may derive: 1/ number of states = exp (Sum over i p(ei) ln(p(ei)) for N→ infinite. In other words, ln (number of states) = -Sum over i p(ei) ln(p(ei)). One may then legitimately argue that one should maximize this value subject to constraints which exist. An obvious constraint is Sum over i ei p(ei) = E total and one usually proceeds with the maximization to obtain p(ei) = C exp(-ei/T). We argue here, however, that a second necessary constraint holds which follows from the AND features of probability, i.e. p(ei) p(ej) = p(ei+ej), where p(ei) is unnormalized. This constraint, however, immediately leads to the solution p(ei) = C exp(-ei/T). Furthermore, the constraint, which follows from basic AND probability behaviour, must apply, it cannot be discarded. Thus, we argue that it must be applied to the a priori Shannon’s entropy expression, - Sum over i p(ei) ln(p(ei)) before any maximization occurs. This second constraint, however, already solves the problem and so there is no maximization to be performed as argued in Part 1.