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Bias Removal and a Momentum Treatment of the Maxwell-Boltzmann Distribution Francesco R. Ruggeri Hanwell, N.B. Oct. 8, 2025 In (1), we argued that bias removal is the main underlying idea of both the time reversal reaction balance and maximization of ln of the number of permutations of a set { n(ei) }, (proportional to what is called entropy) subject to Sum over i n(ei) = N and Sum over i ei n(ei) = E approaches to obtaining the Maxwell-Boltzmann (MB) distribution. Furthermore, the removal of bias (independence of n(ei)’s) approach only holds for very large n(ei) values and requires the introduction of a drastic approximation for n(ei)! approx= n(ei) power n(ei). Even though this follows from Stirling’s approximation, it is still a drastic approximation of a factorial function, but is what is required to remove bias, which may be described by n(ei)n(ej) = n(ek)n(el) for ei+ej = ek+el. In (2), we argued that one may find the MB distribution solely from momentum considerations. If this is the case, one would expect that bias must again be removed, i.e. n(p1)n(p2) = n(p3)n(p4) if p1+p2 = p3+p4 (vectors) and that if any factorial expressions appear, a large n(p)! drastic approximation is required. We argue that momentum considerations (even in one dimension) are interesting because one may have n(p) and n(-p) and these should have the same value. Thus, momentum is a vector, but one cannot have its sign appear in n(p). Thus, one would expect some kind of p dot p expression, as noted in (2) or a factorial invariant under v→-v. Now, the n(ei) approach yields p(ei) = C exp(-ei/T) where ei= p dot p /2m and so one may wonder if one may have an independent even function of p for n(p). Rather, it seems that n(p) being even in p should yield the same n(ei) given the relationship between p and e=kinetic energy (nonrelativistic). Thus, a momentum treatment of the MB case should yield probabilities P(p1)P(p2) = P(p3)P(p4) for p1+p2 = p3+p4 (one dimension here for simplicity), but at the same time P(p1) should be even in p1 and should essentially capture n(ei) = N C exp(-ei/T). In the momentum case, this means one should obtain exp(- .5m v dot v/T) which is a Gaussian. Given that one anticipates a factorial expression (as ln(N!/ Product over i n(ei) !) appears in the n(ei) case), this momentum linked factorial expression must be invariant under p → -p and should reduce to a Gaussian for large factorial argument values. We argue that these conditions are met by the Galton board expression discussed in (2). This factorial expression is invariant under the interchange of k and (n-k), where v = k (dv) + (n-k) (-dv), but that v→-v under such an interchange. The large k, n-k renders the probability into a Gaussian which removes bias P1(pi)P1(pj) =P1(pk)P1(pl) for pi+pj = pk+pl (one dimension), but this function is the same as p(ei) as anticipated. Bias Removal for Reaction Balance and Maximization of Entropy for the Maxwell-Boltzmann Case In (1), we argued that bias removal is the central idea behind the MB distribution p(ei)=C exp(-ei/T). In the case of time reversal reaction balance, bias is removed by enforcing: n(ei) n(ej) = n(ek)n(el) for ei+ej = ek+el so p(ei) = Cexp(-ei/T) where n(ei) = Np(ei) ((1))
The maximization of “entropy” ( ln permutations of a set { n(ei) } such that Sum over i n(ei)=N and Sum over i ei n(ei) = E hold also seems to involve bias removal. In (1), we noted that if ((1)) does not hold in the factorial expression: Ln ( N!/ Product over i n(ei)!) ((2)) then one cannot have the maximum number of permutations. Thus, ((1)) must hold and one may actually obtain the MB distribution without performing the math of the maximization process. The point, however, is that this removal of bias only holds for very large n(ei) values which yield independence of n(ei)’s. One must introduce a drastic approximation for ln(n(ei)!) in order to force the bias removal. In particular, we showed in (1) that this means using: ln(n(ei)!) = n(ei) ln(n(ei)) or n(ei)! = n(ei) power n(ei) ((3)) ((3)) is Stirling’s approximation, but is nevertheless a very drastic approximation of a factorial function. We suggest that one must impose it to remove bias as this is the main idea behind the MB distribution and have provided arguments in (1). Now, in (2), we argue that one may obtain the MB distribution a third way, namely through only momentum considerations based on the math of a Galton board. Here we try to see if this approach removes bias. Momentum Approach to the MB Distribution and Removal of Bias If one only considers momentum (say in one dimension for simplicity), then to remove bias, it seems one must have: n(p1)n(p2) = n(p3)n(p4) if p1+p2 = p3+p4 ((4)) The issue is that p may be positive or negative and so: n(p) = n(-p) ((5)) This implies that n(p) must be even in p, for example p dot p is a possibility. Even though one wishes to only consider momentum and its conservation which is linear in v (velocity vector), n(p) must uphold ((5)). The next point is that an energy consideration which removes bias already leads to: p(ei) = Cexp(-ei/T) and ei = p dot p /2m ((6)) This begs the question: Can P(p1) equal a different function of p? It seems not as P(p) represents a unique e = pp/2m value (one dimension). In other words, P(p) must become p(ei) =C exp(-ei/T) ((7))
The question becomes: How can this be arranged? In (2), we considere a Galton board which creates a v (one dimension) through k dv units and (n-k) (-dv) units where k and n are extremely large. This leads to the binomial factor: n!/ ( k! (n-k) !) ((8)) as a weight for .5 power k * .5 power (n-k) ((9)). We consider dv and -dv as having the same probability because overall momentum is 0 in a gas. Next, we note that: Interchanging k and (n-k) leads to the same probability ((8)). ((8)) represents: v = k (dv) + (n-k) (-dv).((9)) Tthe interchange k, (n-k) yields v→-v, but the weight or probability remains unchanged. Thus, ((8)) is a candidate for P(p) as it satisfies ((5)). Bias, however, must be removed and this should require a large k, (n-k) approximation. In (2) and reference therein, it is shown that ((8)) leads to a Gaussian in v (if v is given by ((9)), i.e. exp(- C v dot v) ((10)) This is the p(ei) result for a suitable C and removes bias. In other words if one considers: Ei+ej = ek+el and p(ei)p(ej) = p(ek)p(el) ((11)) there is the implied equation P1(pi) P1(pj) = P1(pk)P1(pl) because pi+pj = pk+pk (one dimensional momentum). Now P1=p and so the energy equation removes both energy and momentum bias. The momentum factorial approach (Galton board math) yields the MB distribution which may be thought of as an energy bias removing function or a momentum bias removing function, because both momentum and energy conservation occur for an elastic collision. Thus, there is a really strong link between a P1(v) and p(ei), in fact they are the same. Conclusion In (1), we argued that the Maxwell-Boltzmann distribution really depends on bias removal, i.e. n(ei)n(ej) = n(ek)n(el) for ei+ej = ek+el, and this only occurs for n(ei) being very large. We stress that this same condition holds if one maximizes ln( N!/ Product over i n(ei)!) subject to Sum over i n(ei) = N and Sum over i ei n(ei) = E. In this case, one must introduce a drastic approximation to n(ei)!, namely n(ei) power n(ei) to ensure that bias removal occurs. This is Striling’s approximation, but is nevertheless a drastic one. In (2), we suggested that one obtain the MB distribution using only momentum considerations. This must also lead to bias removal, we argue here, i.e, P1(pi)P1(pj) = P1(pk)P1(pk) for one dimensional momentum and pi+pj = pk+pl. The issue is that P1(p) must = P1(-p) and so P1 as a
function of p must be even in p (or p dot p). The energy approach already leads to exp(-p dot p /2mT) and so given the link between p and ei= p dot p /2m, it seems that P1(p) and p(ei) must be the same function. The question then becomes: Is there a scheme involving only momentum which yields a factorial expression which reduces to exp(- p dot p /2mT)? We argue that the Galton board math introduced in (2) does just this. It considers a v (one dimension) as being composed of k (dv) units and (n-k) (-dv) ones. Here n and k are extremely large. The probability associated with a v is then: .5 power n n! / ( k! (n-k)!). This expression is invariant under the interchange of k and n-k), but v= k (dv) + (n-k) (-dv), so an interchange here leads to v→-v. Thus, P1(p)=P1(-p). The question is whether bias removal arises in the k, n extremely large value limit. In (2) and references therein it is shown that in such a limit this factorial expression becomes exp(- C v dot v) so one does obtain p(ei) = C1 exp(-ei/T) from the energy approach as well as bias removal. We note that for an elastic collision (one dimension) pi+pj = pk +pl , ei+ej = ek+el, and p(ei)p(ej)=p(ek)p(el). Thus, bias removal for ei already includes bias removal for pi as P1(p) cannot distinguish the sign of p. References 1. Ruggeri, Francesco R.Bias and Maximization of Arrangements in the Maxwell-Boltzmann Distribution (preprint, zenodo, 2025) 2. Ruggeri, Francesco R. Speculation on the Maxwell-Boltzmann exp(-ei/T) From the View of Conservation of Momentum Part 3 (preprint, zenodo, 2025)
