Bias Remo al and a Momen um T ea men o he Maxwell-Bol zmann Dis ibu ion Pa 3
F ancesco R. Rugge i Hanwell, N.B. Oc . 10, 2025
In p e ious no es, we discussed how a maximiza ion o en opy (ln pe mu a ions o { n(ei) }
se s) subjec o cons ain s, ime e e sal eac ion balance o 2 body elas ic sca e ing and a
andom walk o all lead o he Maxwell-Bol zmann (MB) dis ibu ion in he non ela i is ic case.
The i s wo app oaches also lead o he MB esul in he ela i is ic scena io. He e we wish o
in es iga e he no ion o bias emo al mo e closely.
In a p e ious no e, we a gued ha o 2-body elas ic sca e ing, bias emo al means ha
p(ei)p(ej) = p(ek)p(el) o ei+ej = ek+el and hen sugges ed ha his o m o bias emo al also
applies o he maximiza ion o ln ac o ial o pe mu a ions linked o an { n(ei) } se subjec o
cons ain s. He e, we no e ha bias emo al should occu based on he a iable in he gi en
cons ain . The key cons ain is Sum o e i ei p(ei) = ea e, and so he cons ain is on he
a e age ene gy. In o he wo ds, one seeks he{ n(ei) } se which shows no ene gy bias o he
han he cons ain . This seems o be he eason o maximizing pe mu a ions. I leads o his
non-biased esul subjec o he ene gy cons ain . Gi en ha his ep esen s he emo al o
ene gy bias (as much as is possible), his same emo al o ene gy bias mus apply o 2-body
elas ic sca e ing which c ea es he global ideal gas si ua ion in he i s place This, we a gue, is
why he maximiza ion o en opy subjec o cons ain s and 2-body elas ic sca e ing esul s
yields he same answe . Bo h o hese app oaches should hold ela i is ically and
non ela i is ically.
We also conside ed he case o a andom walk in eloci y space in Pa 2. This leads o a
Gaussian which is consis en wi h he non ela i is ic o m o kine ic ene gy. Such a andom walk
is consis en wi h ene gy bias emo al in he non ela i is ic case, bu no he ela i is ic one, as
no ed in Pa 2. A poin we wish o make is ha e en hough one migh hink ha all bias is
emo ed a he eloci y le el in in e ac ions in an ideal gas, he cons ain is on ene gy and
gi en ha maps o a unique e (bo h non ela i is ically and ela i is ically), i is he bias o
ene gy which mus be emo ed and his does no necessa ily mean ha bias om will be
emo ed. In he non ela i is ic case i is, bu no in he ela i is ic.
Bias Remo al In Te ms o Cons ain s
Desc ibing a gas in e ms o cons ain s, e.g.
Sum o e i ei p(ei) = ea e and Sum o e i p(ei)I = 1 ((1))
implies one has comple e eedom (no bias) on c ea ing a se o { p(ei) } o {n(ei)} = { Np(ei) } as
long as he cons ain s a e sa is ied and he se con ains no o he bias linked o he cons ain
a iable, in his case e. In o he wo ds, he e may be many { p(ei) } se s which sa is y he
cons ain s, bu hese may con ain ei biases no imposed by he cons ain . An example would
be:
p(ei) p(ej ) no = p(ek)p(el) o ei+ej = ek+el ((2))
The a ious { p(ei) } se s which sa is y he cons ain s ((1)), bu con ain hidden ene gy biases
ep esen luc ua ions and an equilib ium scena io is one wi h no biases o ene gy o he han
ha o ced by ((1)). This leads o a second idea, namely ha gi en ha cons ain is in e, one
should only conside biases linked o e. I is possible ha o he a iables exis , such as eloci y,
bu he e is no a p io i eason o assume ha all bias is emo ed om . One canno s a e ha a
p io i, an ideal gas means ha a single pa icle ecei es eloci y changes in a comple ely
andom way. Non ela i is ically, his seems o be he case, bu no ela i is ically.
Remo ing Ene gy Bias
Gi en ha he key cons ain is Sum o e i ei p(ei) = ea e, one mus emo e e bias. This may
be done globally o he en i e gas, o locally o in e ac ions which es ablish he global
equilib ium.
We i s conside he global app oach. As s a ed in a p e ious no e, i one osses a coin wice,
one may ob ain 2 heads, a head and ail (2 ways) and wo ails. The ini ial cons ain is ha he
coin is ai and so p(heads) =.5 and p( ails) = .5. As a esul , he wo heads and wo ails esul s
ep esen bias om he ini ial p obabili y cons ain , whe eas ails-head does no . As we a gued
in Pa 1, one may maximize pe mu a ions subjec o a p io i cons ain s because any ex a bias
cons ain will educe he numbe o pe mu a ions. This idea holds o he coin oss as well as
o an ideal gas. The key idea is ha he bias is conside ed in e ms o ene gy because he
cons ain is in e ms o ene gy. Thus one may w i e:
Ln ( N!/ P oduc o e i n(ei)!) and maximize i subjec o ((1)) ((3))
This mus ep esen a se {n(ei)} which has no ei bias excep ha imposed by ((1)). O he { n(ei)
} se s which sa is y ((1)) hen ep esen luc ua ions.
Now, an equilib ium is c ea ed h ough 2-body elas ic sca e ing. I he e is no ene gy bias in he
en i e sys em (excep ha o ((1))), hen he e should also be no e bias in single 2-body elas ic
sca e ing e en . Assuming he p obabili y o an ei and ej o in e ac is p(ei)p(ej), hen e bias
emo al is:
p(ei)p(ej) = p(ek)p(el) o ei+ej = ek+el ((4))
This mus gi e he same esul as ((3)) because bo h app oaches a e equi alen o emo ing all
e bias excep ha con ained in ((1)). The e seems o be no o he e bias in an elas ic in e ac ion
han one ha would occu i ((4)) did no hold. Thus, one is no o ced o conside a global
pe mu a ion- ac o ial scheme in o de o emo e e bias. ((4)) is su icien , al hough he e is
no hing w ong wi h he ac o ial app oach.
Bias Remo al o O he Va iables
I is emp ing o y o emo e bias om o he a iables. Fo example, a gi en maps o a
speci ic e in bo h he ela i is ic and non ela i is ic cases. One migh hink ha o a single
pa icle one may ha e comple ely andom sca e ing and hence andom changes o he a iable
. This seems o hold in he non ela i is ic case o , as a andom walk in space leads o:
= k (d ) + (n-k) (-d ) wi h P( ) = n!/ ( k! (n-k)!) .5 powe n ((5))
((5)) becomes a Gaussian in in he la ge n,k limi and so p obabili y is in e ms o e=.5m
and he bias o ((4)) is au oma ically emo ed.
In he ela i is ic case, howe e , one may s ill w i e ((5)), bu his leads o bias emo al o do
, bu one equi es bias emo al o e as he cons ain is in e. In he ela i is ic case: ((4)) mus
hnold, bu ha is no longe equi alen o:
exp(-C 1 1) exp(-C( 2 2) = exp(-C 3 3) exp(-C 4 4) ((6))
because ((6)) is no linked wi h conse a ion o ene gy any longe . Conse a ion o ene gy is:
sq (p1p1cc + momocccc) + sq (p2p2cc + momocccc) = sq (p3p3cc + momocccc) +
sq (p4p4cc + momocccc) ((7))
Conclusion
In conclusion, we a gue ha gi en a cons ain in a s a is ical p oblem, such as Sum o e i ei
p(ei) = ea e, one should y o emo e all bias linked wi h he a iable in he cons ain , in his
case e. This holds in bo h he ela i is ic and non ela i is ic cases. This may be done ei he
globally o locally in a speci ic in e ac ion. The la e is ma hema ically simple and gi en ha
collisions go e n he equilib ium, i should su ice o emo e e bias he e. In pa icula , o a
2-body elas ic collision wi h p(ei)p(ej) ep esen ing he p obabili y o ei and ej o collide, he only
possible emo al o ei bias which s ill espec s ene gy conse a ion is: p(ei)p(ej)=p(ek)p(el) wi h
ei+ej=ek+el. This hen should sol e he p oblem because one has emo ed all ene gy bias
possible.
In he global gas ( o he en i e gas), one may ha e { n(ei) = Np(ei)} se s which sa is y Sum
o e i n(ei) = N and Sum o e i ei n(ei) = E. This means ha all se s bu one con ain e bias (i..e
ep esen luc ua ions). Only one se emo es bias and as a gued in Pa 1, his mus be he se
which maximizes ln( N!/ P oduc o e i n(ei)!) subjec o he cons ain s ((1)). Gi en ha his
emo es ei bias, i mus yield he same esul as ime e e sal eac ion balance.
I is emp ing o emo e bias om a iables o he han he cons ain one. Fo example, one
migh ocus on one dimensional ins ead o e and conside a andom walk. In he non ela i is ic
case, his yields a p obabili y in do which is p opo ional o kine ic ene gy e and so emo es
bias in e. As a esul , in he non ela i is ic case, a andom walk in space does yield a esul
consis en wi h ene gy bias emo al, bu no in he ela i is ic case. We sugges ha one
conside he emo al o bias in he speci ic a iable con ained in he cons ain as emo ing bias
om ano he a iable mus yield a esul consis en wi h he cons ain a iable bias emo al,
o he wise i does no yield he app op ia e dis ibu ion.
