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Bias Removal and a Momentum Treatment of the Maxwell-Boltzmann Distribution Part 3 Francesco R. Ruggeri Hanwell, N.B. Oct. 10, 2025 In previous notes, we discussed how a maximization of entropy (ln permutations of { n(ei) } sets) subject to constraints, time reversal reaction balance for 2 body elastic scattering and a random walk for v all lead to the Maxwell-Boltzmann (MB) distribution in the nonrelativistic case. The first two approaches also lead to the MB result in the relativistic scenario. Here we wish to investigate the notion of bias removal more closely. In a previous note, we argued that for 2-body elastic scattering, bias removal means that p(ei)p(ej) = p(ek)p(el) for ei+ej = ek+el and then suggested that this form of bias removal also applies to the maximization of ln factorial of permutations linked to an { n(ei) } set subject to constraints. Here, we note that bias removal should occur based on the variable in the given constraint. The key constraint is Sum over i ei p(ei) = eave, and so the constraint is on the average energy. In other words, one seeks the{ n(ei) } set which shows no energy bias other than the constraint. This seems to be the reason for maximizing permutations. It leads to this non-biased result subject to the energy constraint. Given that this represents the removal of energy bias (as much as is possible), this same removal of energy bias must apply to 2-body elastic scattering which creates the global ideal gas situation in the first place This, we argue, is why the maximization of entropy subject to constraints and 2-body elastic scattering results yields the same answer. Both of these approaches should hold relativistically and nonrelativistically. We also considered the case of a random walk in velocity space in Part 2. This leads to a Gaussian which is consistent with the nonrelativistic form of kinetic energy. Such a random walk is consistent with energy bias removal in the nonrelativistic case, but not the relativistic one, as noted in Part 2. A point we wish to make is that even though one might think that all bias is removed at the velocity level in interactions in an ideal gas, the constraint is on energy and given that v maps to a unique e (both nonrelativistically and relativistically), it is the bias of energy which must be removed and this does not necessarily mean that bias from v will be removed. In the nonrelativistic case it is, but not in the relativistic. Bias Removal In Terms of Constraints Describing a gas in terms of constraints, e.g. Sum over i ei p(ei) = eave and Sum over i p(ei)I = 1 ((1)) implies one has complete freedom (no bias) on creating a set of { p(ei) } or {n(ei)} = { Np(ei) } as long as the constraints are satisfied and the set contains no other bias linked to the constraint variable, in this case e. In other words, there may be many { p(ei) } sets which satisfy the constraints, but these may contain ei biases not imposed by the constraint. An example would be: p(ei) p(ej ) not= p(ek)p(el) for ei+ej = ek+el ((2))
The various { p(ei) } sets which satisfy the constraints ((1)), but contain hidden energy biases represent fluctuations and an equilibrium scenario is one with no biases of energy other than that forced by ((1)). This leads to a second idea, namely that given that constraint is in e, one should only consider biases linked to e. It is possible that other variables exist, such as velocity, but there is no a priori reason to assume that all bias is removed from v. One cannot state that a priori, an ideal gas means that a single particle receives velocity changes in a completely random way. Nonrelativistically, this seems to be the case, but not relativistically. Removing Energy Bias Given that the key constraint is Sum over i ei p(ei) = eave, one must remove e bias. This may be done globally for the entire gas, or locally for interactions which establish the global equilibrium. We first consider the global approach. As stated in a previous note, if one tosses a coin twice, one may obtain 2 heads, a head and tail (2 ways) and two tails. The initial constraint is that the coin is fair and so p(heads) =.5 and p(tails) = .5. As a result, the two heads and two tails results represent bias from the initial probability constraint, whereas tails-head does not. As we argued in Part 1, one may maximize permutations subject to a priori constraints because any extra bias constraint will reduce the number of permutations. This idea holds for the coin toss as well as for an ideal gas. The key idea is that the bias is considered in terms of energy because the constraint is in terms of energy. Thus one may write: Ln ( N!/ Product over i n(ei)!) and maximize it subject to ((1)) ((3)) This must represent a set {n(ei)} which has no ei bias except that imposed by ((1)). Other { n(ei) } sets which satisfy ((1)) then represent fluctuations. Now, an equilibrium is created through 2-body elastic scattering. If there is no energy bias in the entire system (except that of ((1))), then there should also be no e bias in single 2-body elastic scattering event. Assuming the probability for an ei and ej to interact is p(ei)p(ej), then e bias removal is: p(ei)p(ej) = p(ek)p(el) for ei+ej = ek+el ((4)) This must give the same result as ((3)) because both approaches are equivalent to removing all e bias except that contained in ((1)). There seems to be no other e bias in an elastic interaction than one that would occur if ((4)) did not hold. Thus, one is not forced to consider a global permutation-factorial scheme in order to remove e bias. ((4)) is sufficient, although there is nothing wrong with the factorial approach.
Bias Removal of Other Variables It is tempting to try to remove bias from other variables. For example, a given v maps to a specific e in both the relativistic and nonrelativistic cases. One might think that for a single particle one may have completely random scattering and hence random changes to the variable v. This seems to hold in the nonrelativistic case for v, as a random walk in v space leads to: v = k (dv) + (n-k) (-dv) with P(v) = n!/ ( k! (n-k)!) .5 power n ((5)) ((5)) becomes a Gaussian in v in the large n,k limit and so probability is in terms of e=.5m vv and the bias of ((4)) is automatically removed. In the relativistic case, however, one may still write ((5)), but this leads to bias removal for v dot v, but one requires bias removal for e as the constraint is in e. In the relativistic case: ((4)) must hnold, but that is no longer equivalent to: exp(-C v1v1) exp(-C(v2v2) = exp(-C v3v3) exp(-C v4v4) ((6)) because ((6)) is not linked with conservation of energy any longer. Conservation of energy is: sqrt(p1p1cc + momocccc) + sqrt(p2p2cc + momocccc) = sqrt(p3p3cc + momocccc) + sqrt(p4p4cc + momocccc) ((7)) Conclusion In conclusion, we argue that given a constraint in a statistical problem, such as Sum over i ei p(ei) = eave, one should try to remove all bias linked with the variable in the constraint, in this case e. This holds in both the relativistic and nonrelativistic cases. This may be done either globally or locally in a specific interaction. The latter is mathematically simpler and given that collisions govern the equilibrium, it should suffice to remove e bias there. In particular, for a 2-body elastic collision with p(ei)p(ej) representing the probability for ei and ej to collide, the only possible removal of ei bias which still respects energy conservation is: p(ei)p(ej)=p(ek)p(el) with ei+ej=ek+el. This then should solve the problem because one has removed all energy bias possible. In the global gas (for the entire gas), one may have { n(ei) = Np(ei)} sets which satisfy Sum over i n(ei) = N and Sum over i ei n(ei) = E. This means that all sets but one contain e bias (i..e represent fluctuations). Only one set removes bias and as argued in Part 1, this must be the set which maximizes ln( N!/ Product over i n(ei)!) subject to the constraints ((1)). Given that this removes ei bias, it must yield the same result as time reversal reaction balance. It is tempting to remove bias from variables other than the constraint one. For example, one might focus on one dimensional v instead of e and consider a random walk. In the nonrelativistic case, this yields a probability in v dot v which is proportional to kinetic energy e and so removes bias in e. As a result, in the nonrelativistic case, a random walk in v space does yield a result consistent with energy bias removal, but not in the relativistic case. We suggest that one
consider the removal of bias in the specific variable contained in the constraint as removing bias from another variable must yield a result consistent with the constraint variable bias removal, otherwise it does not yield the appropriate distribution.
