Bias Removal and a Momentum Treatment of the Maxwell-Boltzmann Distribution

 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.