Bias and Maximization of Arrangements in the Maxwell-Boltzmann Distribution

Bias and Maximization of Arrangements in the Maxwell-Boltzmann Distribution Francesco R. Ruggeri Hanwell, N.B. Oct. 7, 2025 If one tosses a coin twice, one may obtain two heads, two tails or two cases of one head and one tail. The situation of a head and a tail allows for two permutations, while the other cases don’t allow for any (i.e. carry a weight of 1). Thus, a tail and head represents in a sense an equilibrium result and is linked to the probabilities of .5 for a head and .5 for a tail, i.e. all bias is removed. In the Maxwell-Boltzmann case, one has N particles with a total energy of E. The canonical distribution suggests that any arrangement of E among N particles carries the same weight. (Some of these violate momentum conservation, but are too few in number to be of concern.) The argument then seems to be that one wishes to have a partitioning set { n(ei) } which has the largest number of permutations. If one had a different set {.n1(ei) }, it would have fewer permutations and so one could observe a change to a more like set { n(ei) } characterized by the counting of its permutations which apparently map to experimental observations. We next consider the question of bias. We define a possible bias as n(ei)n(ej) not= n(ek)n(el) if ei+ej = ek+el. This is reminiscent of time reversal reaction balance, but here we consider only permutations. The question becomes: Is there any link between bias and permutations? We argue that a permutation should take the form of a constraint. In the simple case of only n(ei)n(ej) not= n(ek)n(el), one would require a math constraint linked to these energies only: F(ei, ej,ek,el). In the case of a bias in n(ei), one could write the maximization of permutations subject to constraints in terms of Maximize { ln ( N!/ Product over i n(ei)!) } + a1 Sum over i n(ei) + a2 Sum over i ei n(ei) + a3 F(n(ei), n(ej), n(ek), n(el)). The question then becomes: Does the above lead to more or fewer permutations than: Maximize { ln ( N!/ Product over i n(ei)!) } + a1 Sum over i n(ei) + a2 Sum over i ei n(ei) ? We argue that it must represent fewer, because a constraint constrains, i.e. removes solutions which do not conform to its rule. Thus, we argue that one cannot have n(ei)n(ej) not= n(ek)n(el) for ei+ej = ek+el. This then implies that n(ei)n(ej) = n(ek)n(el) for ei+ej = ek+el, which in turn is reaction balance and leads to the Maxwell-Boltzmann distribution directly. This result has been obtained without actually solving the maximization problem so we consider the fact that the lack of bias only occurs for N and n(ei)’s extremely large and see what this means for ln(n(ei)!). In other words, one must use an approximation linked to n(ei) extremely large in order to remove bias. Maxwell-Boltzmann Distribution and Reaction Balance If one considers time reversal balance for elastic two body collisions, one may write: p(ei)p(ej) = p(ek)p(el) for ei+ej = ek+el ((1)) This leads directly to the MB distribution: p(ei) = C exp(-ei/T) ((2)) What does such an approach have to do with the number of permutations of a set of n(ei) which have the condition that Sum over i ei n(ei) = E and Sum over i n(ei) = N with Np(ei) = n(ei)? Maxwell-Boltzmann Distributions and Permutations It is traditional to maximize: Ln ( N!/ Product over i n(ei)! ) subject to Sum over i ei n(ei) =E and Sum over i n(ei) = N ((3)) This leads to the Maxwell-Boltzmann distribution, but we ask: What is the reasoning behind this approach? Certainly, any set of { n(ei) } such that the constraints in ((3)) hold is physically viable. In fact, the microcanonical approach states that any arrangement of n(ei) values which uphold the constraints of ((3)) carry the same weight. This seems to be the same idea as tossing a coin twice. It may yield two heads, two tails or a head and tail with a weight of 2. The weight of 2 is greater than 1 and so this is like the “equilibrium” solution and reflects the information of the probabilities: p(head) = .5 and p(tail) = .5 ((4)) We argue that two heads or two tails represent bias in that they do not reflect the equal a priori probabilities for heads and tails. They suppress a priori information so to speak. In the above example, the maximum number of permutations occurs for the scenario which does not introduce bias, i.e. one head and one tail. We suggest that this simple idea applies also to the more complicated Maxwell-Boltzmann scenario and may be formalized. The Case of Bias Consider the goal of finding the set { n(ei) } with the greatest number of permutations: N!/ Product over i n(ei)! ((5)) Next, consider a very simple possible bias: n(ei)n(ej) not= n(ek)n(el) for ei+ej = ek+el ((6)) Can such a bias yield a set { n(ei) } which has a maximum number of permutations ((5))? To answer this question, one may consider the bias ((6)) as a constraint on n(ei)’s, in particular only n(ei), n(ej), n(ek) and n(el). One may write: F(n(ei), n(ej), n(ek), n(el)) as the constraint linked to the bias ((6)) Then a formal maximization process is: Maximize( N!/ Product over i n(ei)!) subject to F(n(ei), n(ej), n(ek), n(el)), Sum over i n(ei) = N and Sum over i ei n(ei) = E. ((7)) Does the n(ei) solution of ((7)) yield a higher permutation value ((5)) than: Maximize( N!/ Product over i n(ei)!) subject to Sum over i n(ei) = N and Sum over i ei n(ei) = E. ((8)) ? Now a constraint constrains, i.e. removes solutions which do not conform, so we argue that any F(n(ei), n(ej), n(ek), n(el)) would remove possibilities for n(ei)’s allowed by ((8)). Thus, we argue that ((7)) cannot lead to higher permutations ((5)), than ((7)). In other words, one wishes to remove all bias in order to obtain the highest permutation number ((5)). Removing all bias means that one must have: n(ei)n(ej) = n(ek)n(el) for ei+ej = ek+el ((9)) There other bias expressions that one could write, but ((9)) is sufficient to yield a solution for n(ei) namely: n(ei) = N C exp(-ei/T) ((10)) Lack of Bias Linked to Lack of Information in a Physical Scenario The maximization of permutations ((5)) makes no reference whatsoever to the kind of reactions which create an equilibrium scenario. It just states that one must remove bias to have a maximum number of permutations ((5)). Specific physical reactions exist. For example, one might have elastic two body collisions. One might at first consider these to be deterministic, but if a given e1 and e2 elastically scatter (with momentum conservation), then any e3, e4 which conserve momentum and energy are possible solutions. If one is to remove bias, then any such e3, e4 solution should carry the same weight as another. Generalizing this to a gas with n(ei) =N p(ei), ei particles, one would expect: n(e1)n(e2) = n(e3)n(e4) ((11)) Thus, reaction balance and maximization of permutations (subject to Sum over i n(ei) = N and Sum over i ei n(ei) = E) both make use of the same underlying idea, namely the removal of bias. In the case of reaction balance, one considers two body interactions and so the bias removal is of the form of ((11)). In the above sections, we considered permutations ((5)) and only focused on the special bias n(ei)n(ej) not= n(ek)n(el). We argued that one may consider any bias and that this one was as good as another. Moreover, this bias is directly linked to two body scattering which physically occurs in an ideal gas. By showing the removal of the bias n(ei)n(ej) not= n(ek)n(el) for ei+ej = ek+el must occur, we argued in the permutation case that one does not actually have to solve the maximization problem, but only use: n(ei)n(ej) = n(ek)n(el) for ei+ej = ek+el ((12)) This begs the question: If one actually performs the maximization, why does it yield the same result as ((12))? First of all, we note that the constraints Sum over i n(ei) = N and Sum over i ei n(ei) = E are linear in n(ei) and that taking ln of the number of permutations ((5)) also creates the linear sum: ln(N!) - Sum over i ln(n(ei)!) ((13)) The d/dn derivative must be matched with a1 Sum over i ei delta n + a2 Sum over i delta n ((14)) We also note that the above arguments only hold for N being very large and all n(ei) also being very large. Now, ((5)) is an exact expression for permutations so we argue that the removal of bias is directly linked to n(ei) being a large number. In other words, there really is bias in the system, but it only disappears in the large n(ei) limit. This seems to suggest that some kind of large n(ei) approximation must be made to ln(n(ei)!) because it is an exact expression as is ((14)). The approximation must consistent with: n(ei)n(ej) = n(ej)n(el) for ei+ej = ek+el ((15)) (i.e. bias removal) The derivative d/dn(ei) of the approximation used for ln(n(ei)!) linked with ((14)) must yield exp(-ei/T) which follows from ((15)). This implies that the approximation which must hold is: ln(n(ei)!) = n(ei) ln(n(ei)) ((16)) We note that this is arrived at with no reference to Stirling’s approximation, even though the result is equivalent. Basically, the approximation ((16)) states that: n(e)! approximately= n(ei) power n(ei) ((17)) ((17)) is a rather simplistic approximation, but it is apparently the one needed to remove bias from the system which is equivalent to ((15)). Thus, it is really the case that very large n(ei) values lead to a simplistic approximation ((17)) which is equivalent to the removal of bias and this lack of bias is represented by the Maxwell-Boltzmann distribution; p(ei) = C exp(-ei/T) ((18)) Conclusion We note that reaction balance is a statement of the removal of bias. Bias removal is directly linked to independence of n(ei) values which can only occur if N (and hence n(ei)) are extremely large. This begs the question: What happens in the case of maximizing the number of permutations of a set n(ei) which satisfy Sum over n(ei) = N and Sum over i ei n(ei) = E? The idea seems to be that the set of n(ei) which has the most number of permutations represents the equilibrium one. We argue that any bias reduces the number of permutations. Thus, a bias n(ei)n(ej) not= n(ek)n(el) for ei+ej = ek+el would reduce the number of permutations. This suggests that n(ei)n(ej)=n(ek)n(el) which is the reaction balance result leading directly to the MB solution p(ei) = Cexp(-ei/T), obtained with no maximization. The point is that maximization subject to Sum over i ei n(ei)=E and Sum over i n(ei) = N must also yield this solution. A permutation expression, however, is exact, and we argue that only a large n(ei) case leads to independence of p(ei) values. One may take ln of the permutation expression N!/ Product over i n(ei)! to make it appear linear in n(ei)!, but now one must impose an approximation on ln(n(ei)!) such that all bias is removed. This is equivalent to using the simplistic approximation: ln(n(ei)!) = n(ei) power n(ei). Thus, one is not really using the exact factorial expression, but is rather basing the entire Maxwell-Boltzmann solution on the idea of bias removal, we argue.