Bias Remo al and a Momen um T ea men o he Maxwell-Bol zmann Dis ibu ion Pa 2
F ancesco R. Rugge i Hanwell, N.B. Oc . 8, 2025
In Pa 1, we a gued ha i one wishes o desc ibe he dis ibu ion o ideal gas pa icles in
e ms o momen um, P1(p) = P1(-p), whe e P1 is he p obabili y dis ibu ion o momen um p.
This o ces a unc ional o m o P1 o be quad a ic in p (e.g p do p o some hing else). The poin
we made is ha one may also conside he p oblem in e ms o ene gy and elas ic collisions
ei+ej = ek+el and p(ei)p(ej) =p(ek)p(el) ((1)). The P1(p) dis ibu ion canno con adic p(ei) and
he ei one is al eady quad a ic in p, i.e. e = pp/2m and so P1(p) should equal p(ei) acco ding o
he a gumen s made in Pa 1.
Tha , howe e , does no explain why one may conside a eloci y as being cons uc ed om
k (d ) uni s and (n-k) (-d ) uni s wi h a p obabili y o : n!/ (k! (n-k)!) .5 powe n. The app oach o
((1)) deals wi h a wo body elas ic collision which also conse es momen um and his is a
physical p ocess in an ideal gas. A he same ime p(ei)p(ej) = p(ei+ej) o p unno malized
means ha one may use he bias emo al o ((1)) o ac ually c ea e a p obabili y o p(ei+ej).
We sugges hen ha he app oach o ((1)), linked o ene gy conse a ion, should allow one o
cons uc in a p obabilis ic manne which emo es bias. This app oach, howe e , should be
linked o physical eali y as well. In he case o momen um which is linea in eloci y
(non ela i is ic case), we conside ha om he poin o iew o a pa icle, one could imagine a
somewha s ange sys em in which d and -d hi s a e gi en wi h cons an p obabili y. This
means ha ene gy deli e y changes, bu he e we a e only conce ned wi h momen um, i.e. one
has a andom walk as an unbiased c ea ion o in space. Thus, he bias emo al o an AND
si ua ion (ei,ej) and (ek,el) ha ing he same join p obabili y i ei+ej=ek+el is now changed o
bias emo al o building blocks o . These seem like wo e y di e en kinds o bias emo als,
bu he goal is o emo e as much bias as possible. The cons uc ion o om k (d ) uni s and
(n-k) (-d ) uni s does no seem o iola e any cons ain s in he sys em and so a pa icle in a
Maxwell-Bol zmann gas seems o change mo ion a andom (non ela i is ic case).
In o he wo ds, he no ion o bias emo al h ough comple e andomness o momen um hi s
seems o be iden ical o he condi ion ((1)). I seems easonable ha he ac o ial p obabili y
P1(p) = n!/(k! (n-k)!) .5 powe n ( o p = 2k-n) should yield he MB dis ibu ion alue p(e) = C
exp(-ei/T). I does seem ha he e is a link be ween a Maxwell-Bol zmann gas, ime e e sal
balance o emo e bias, and he idea o bias emo al h ough andomness ep esen ed by a
Gaussian.
The issue is ha his a gumen only holds in he non ela i is ic case, whe eas ((1)) holds in
bo h he ela i is ic and non ela i is ic cases, wi h ei ollowing eiei = pcpc + momocccc in he
ela i is ic scena io. Ne e heless, he andomness (bias emo al) o a Gaussian ( andom walk
in space) seems o be ine o a non ela i is ic pic u e in which p=mo and so one can
cons uc p in essen ially he same way as .
Bias Remo al
In Pa 1, we a gued ha a Maxwell-Bol zmann dis ibu ion mus emo e bias. The ques ion is:
How does one ma hema ically desc ibe bias? We a gued ha one kind o bias ha mus be
emo ed is:
n(ei)n(ej) no = n(ek)n(el) i ei+ej = ek+el ((1))
This is done by imposing: n(ei)n(ek) = n(el)n(ek) o ei+ej = ek+el ((2))
This is a e y speci ic bias linked o physical wo body elas ic sca e ing and i leads o he
solu ion o he MB dis ibu ion:
p(ei) = C exp(-ei/T) ((3))
The ques ion we ask is: Is he e ano he seemingly di e en bias emo al which is equi alen ?
We y o a gue he e ha he e is, in he non ela i is ic si ua ion. We sugges ha he e is a hin
o his second bias con ained in ((2)). In pa icula , o unno malized p(ei):
p(ei+ej) = p(ei)p(ej) ((4))
In o he wo ds, one may ac ually conside c ea ing an ei+ej p obabili y by using a ious ei, ej
alues. One may ex end his o p(ei+ej) = p(ek)p(el)p(em) o ei+e = ek+el+em e c. As a esul ,
one may conside c ea ing ei+ej in many di e en ways and he p oduc o p obabili ies mus be
he same o bias o be emo ed. The key idea he e, we sugges , is he no ion o c ea ing ei+ej
in an unbiased manne and i s equi alence o he wo body elas ic collision ((2)).
Momen um Conside a ions
We sugges ed in he p e ious sec ion ha one may conside an unbiased c ea ion o a key
a iable (say ene gy o momen um) wi hou any bias as leading o he MB dis ibu ion. In he
abo e sec ion, we conside ed ene gy. We now ocus on momen um, which in he non ela i is ic
case is mo . Thus, his a gumen only holds in he non ela i is ic case. We sugges ha an
unbiased app oach o c ea ing is o ha e equal p obabili y o adding d o (-d ) o he exis ing
momen um, i.e a andom walk in space. This does no mean ha kine ic ene gy is being
added in cons an uni s, bu we only ocus on momen um he e. Bias emo al in he
non ela i is ic momen um case seems o be linked o a andom walk ype o building o eloci y
and in one dimension his means d and (-d ) s eps wi h equal p obabili y. In Pa 1, we a gued
ha his di e en o m o bias emo al ( andom eloci y walk) mus be equi alen o he bias
emo al o ((2)) as ul ima ely all possible bias mus be emo ed.
Thus, one may use he binomial (Gal on boa d) p obabili y:
P1(p)= P1( d (2k-n)) = n!/ ( k! (n-k)!) .5 powe n ((5))
One sees ha in e changing k and (n-k) lea es ((5)) unchanged, bu ha :
p= d k + (n-k) (-d ) becomes -p ((6))
P1(p) = P1(-p). ((5)) is he andom walk scena io and one al eady knows ha o comple e
andomness in he la ge k, n limi , a Gaussian will esul . One may, howe e , o mally show ha
ha ((5)) becomes a Gaussian as done in (1). He e we use he simples o m o S i ling’s
app oxima ion o show ha his is he case, using:
K = n/2+m/2 om (1) such ha ((5)) becomes: n!/ ( (n/2 + m/2)! (n/2-m/2)! ) ((7))
Then using b! = b powe b o b e y la ge, one has:
ln(P1) = (n/2+m/2) ln(n/2+m/2) + (n/2-m/2) ln(n/2-m/2) Fo n>>m
ln(P1) = cons an mm so p = C1 exp(C2 mm) , bu m= 2k-n which is momen um ((8))
One ob ains he andom walk Gaussian in he non ela i is ic case which emo es bias om ,
and p (because p=mo ), bu his bias emo al canno con adic he bias emo al o ime
e e sal balance ((2)). Thus, wo seemingly di e en kinds o bias emo al ac ually lead o he
same esul in he non ela i is ic case.
Rela i is ic Case
In he ela i is ic case, one has a p oblem. The app oach o bias emo al o ((2)) s ill holds, bu
eiei = ppcc + momocccc. On he o he hand, momen um p is no longe mo and so i one
wishes o cons uc p in an unbiased way, one can no longe use he k (d ) and (n-k) (-d )
app oach. The app oach o he andom walk and Gaussian as being equi alen o eac ion
balance only holds in he non ela i is ic case. Ne e heless, we a gue ha i is in e es ing ha i
holds a all.
Conclusion
In conclusion, in Pa 1, we a gued ha he Maxwell-Bol zmann dis ibu ion is d i en by
emo ing all possible bias om he ideal gas p oblem. We sugges ed ha conc e ely, his bias is
ep esen ed by n(ei)n(ej) no = n(ek)n(el) i ei+ej = ek+el. The solu ion is o o ce n(ei)n(ej)
=n(ek)n(el) o ei+ej=ek+el which yields he MB dis ibu ion. We hen showed ha a binomial
(Gal on boa d) ac o ial p obabili y o c ea ing momen um yields a dis ibu ion which is
consis en wi h he exp(-ei/T) ob ained om eac ion balance.
He e we y o explain why his is he case. We sugges ha eac ion balance is also linked o
he no ion o cons uc ing an ei+ej, i.e. p(ei+ej) = p(ei)p(ej) o an unno malized p(ei). We
sugges ha one migh y o ex end his idea o cons uc ing a a iable o he momen um case.
In he non ela i is ic scena io (and no he ela i is ic one), p=mo , so one may conside
c ea ing in an unbiased manne . We sugges ha a andom walk (i.e. equal p obabili y o d
and (-d ) s eps in eloci y space) is an unbiased cons uc ion app oach (non ela i is ically). This
leads o: P1(p) = n!/ (k! (n-k)!) .5 powe n, whe e p = (d )k + (n-k) (-d ). Thus, P1(p) = P1(-p).
This emo al o bias should lead o he same esul as he eac ion balance one because
ul ima ely one wan s all bias emo ed. This happens because he binomial ac o becomes a
Gaussian o la ge n,k as shown abo e. Thus, wo seemingly di e en bias emo als ( eac ion
balance) and andom walk cons uc ion o yield he same esul , bu only in he non ela i is ic
case.
Re e ences
1. h ps://people.ba h.ac.uk/pam28/Paul_Milewski,_P o esso _o _Ma hema ics,_Uni e si y
_o _Ba h/Pas _Teaching_ iles/s i ling.pd
