Bias Remo al and a Momen um T ea men o he Maxwell-Bol zmann Dis ibu ion
F ancesco R. Rugge i Hanwell, N.B. Oc . 8, 2025
In (1), we a gued ha bias emo al is he main unde lying idea o bo h he ime e e sal
eac ion balance and maximiza ion o ln o he numbe o pe mu a ions o a se { n(ei) },
(p opo ional o wha is called en opy) subjec o Sum o e i n(ei) = N and Sum o e i ei n(ei) =
E app oaches o ob aining he Maxwell-Bol zmann (MB) dis ibu ion. Fu he mo e, he emo al
o bias (independence o n(ei)’s) app oach only holds o e y la ge n(ei) alues and equi es he
in oduc ion o a d as ic app oxima ion o n(ei)! app ox= n(ei) powe n(ei). E en hough his
ollows om S i ling’s app oxima ion, i is s ill a d as ic app oxima ion o a ac o ial unc ion, bu
is wha is equi ed o emo e bias, which may be desc ibed by n(ei)n(ej) = n(ek)n(el) o ei+ej =
ek+el.
In (2), we a gued ha one may ind he MB dis ibu ion solely om momen um conside a ions. I
his is he case, one would expec ha bias mus again be emo ed, i.e. n(p1)n(p2) = n(p3)n(p4)
i p1+p2 = p3+p4 ( ec o s) and ha i any ac o ial exp essions appea , a la ge n(p)! d as ic
app oxima ion is equi ed. We a gue ha momen um conside a ions (e en in one dimension)
a e in e es ing because one may ha e n(p) and n(-p) and hese should ha e he same alue.
Thus, momen um is a ec o , bu one canno ha e i s sign appea in n(p). Thus, one would
expec some kind o p do p exp ession, as no ed in (2) o a ac o ial in a ian unde →- .
Now, he n(ei) app oach yields p(ei) = C exp(-ei/T) whe e ei= p do p /2m and so one may
wonde i one may ha e an independen e en unc ion o p o n(p). Ra he , i seems ha n(p)
being e en in p should yield he same n(ei) gi en he ela ionship be ween p and e=kine ic
ene gy (non ela i is ic). Thus, a momen um ea men o he MB case should yield p obabili ies
P(p1)P(p2) = P(p3)P(p4) o p1+p2 = p3+p4 (one dimension he e o simplici y), bu a he same
ime P(p1) should be e en in p1 and should essen ially cap u e n(ei) = N C exp(-ei/T). In he
momen um case, his means one should ob ain exp(- .5m do /T) which is a Gaussian.
Gi en ha one an icipa es a ac o ial exp ession (as ln(N!/ P oduc o e i n(ei) !) appea s in
he n(ei) case), his momen um linked ac o ial exp ession mus be in a ian unde p → -p and
should educe o a Gaussian o la ge ac o ial a gumen alues. We a gue ha hese condi ions
a e me by he Gal on boa d exp ession discussed in (2). This ac o ial exp ession is in a ian
unde he in e change o k and (n-k), whe e = k (d ) + (n-k) (-d ), bu ha →- unde such an
in e change. The la ge k, n-k ende s he p obabili y in o a Gaussian which emo es bias
P1(pi)P1(pj) =P1(pk)P1(pl) o pi+pj = pk+pl (one dimension), bu his unc ion is he same as
p(ei) as an icipa ed.
Bias Remo al o Reac ion Balance and Maximiza ion o En opy o he
Maxwell-Bol zmann Case
In (1), we a gued ha bias emo al is he cen al idea behind he MB dis ibu ion p(ei)=C
exp(-ei/T). In he case o ime e e sal eac ion balance, bias is emo ed by en o cing:
n(ei) n(ej) = n(ek)n(el) o ei+ej = ek+el so p(ei) = Cexp(-ei/T) whe e n(ei) = Np(ei) ((1))
The maximiza ion o “en opy” ( ln pe mu a ions o a se { n(ei) } such ha Sum o e i n(ei)=N
and Sum o e i ei n(ei) = E hold also seems o in ol e bias emo al. In (1), we no ed ha i ((1))
does no hold in he ac o ial exp ession:
Ln ( N!/ P oduc o e i n(ei)!) ((2))
hen one canno ha e he maximum numbe o pe mu a ions. Thus, ((1)) mus hold and one
may ac ually ob ain he MB dis ibu ion wi hou pe o ming he ma h o he maximiza ion
p ocess. The poin , howe e , is ha his emo al o bias only holds o e y la ge n(ei) alues
which yield independence o n(ei)’s. One mus in oduce a d as ic app oxima ion o ln(n(ei)!) in
o de o o ce he bias emo al. In pa icula , we showed in (1) ha his means using:
ln(n(ei)!) = n(ei) ln(n(ei)) o n(ei)! = n(ei) powe n(ei) ((3))
((3)) is S i ling’s app oxima ion, bu is ne e heless a e y d as ic app oxima ion o a ac o ial
unc ion. We sugges ha one mus impose i o emo e bias as his is he main idea behind he
MB dis ibu ion and ha e p o ided a gumen s in (1).
Now, in (2), we a gue ha one may ob ain he MB dis ibu ion a hi d way, namely h ough only
momen um conside a ions based on he ma h o a Gal on boa d. He e we y o see i his
app oach emo es bias.
Momen um App oach o he MB Dis ibu ion and Remo al o Bias
I one only conside s momen um (say in one dimension o simplici y), hen o emo e bias, i
seems one mus ha e:
n(p1)n(p2) = n(p3)n(p4) i p1+p2 = p3+p4 ((4))
The issue is ha p may be posi i e o nega i e and so:
n(p) = n(-p) ((5))
This implies ha n(p) mus be e en in p, o example p do p is a possibili y. E en hough one
wishes o only conside momen um and i s conse a ion which is linea in ( eloci y ec o ),
n(p) mus uphold ((5)).
The nex poin is ha an ene gy conside a ion which emo es bias al eady leads o:
p(ei) = Cexp(-ei/T) and ei = p do p /2m ((6))
This begs he ques ion: Can P(p1) equal a di e en unc ion o p? I seems no as P(p)
ep esen s a unique e = pp/2m alue (one dimension). In o he wo ds,
P(p) mus become p(ei) =C exp(-ei/T) ((7))
The ques ion becomes: How can his be a anged? In (2), we conside e a Gal on boa d which
c ea es a (one dimension) h ough k d uni s and (n-k) (-d ) uni s whe e k and n a e ex emely
la ge. This leads o he binomial ac o :
n!/ ( k! (n-k) !) ((8))
as a weigh o .5 powe k * .5 powe (n-k) ((9)).
We conside d and -d as ha ing he same p obabili y because o e all momen um is 0 in a
gas. Nex , we no e ha :
In e changing k and (n-k) leads o he same p obabili y ((8)).
((8)) ep esen s: = k (d ) + (n-k) (-d ).((9))
T he in e change k, (n-k) yields →- , bu he weigh o p obabili y emains unchanged. Thus,
((8)) is a candida e o P(p) as i sa is ies ((5)). Bias, howe e , mus be emo ed and his should
equi e a la ge k, (n-k) app oxima ion. In (2) and e e ence he ein, i is shown ha ((8)) leads o
a Gaussian in (i is gi en by ((9)), i.e.
exp(- C do ) ((10))
This is he p(ei) esul o a sui able C and emo es bias. In o he wo ds i one conside s:
Ei+ej = ek+el and p(ei)p(ej) = p(ek)p(el) ((11))
he e is he implied equa ion P1(pi) P1(pj) = P1(pk)P1(pl) because pi+pj = pk+pk (one
dimensional momen um). Now P1=p and so he ene gy equa ion emo es bo h ene gy and
momen um bias. The momen um ac o ial app oach (Gal on boa d ma h) yields he MB
dis ibu ion which may be hough o as an ene gy bias emo ing unc ion o a momen um bias
emo ing unc ion, because bo h momen um and ene gy conse a ion occu o an elas ic
collision. Thus, he e is a eally s ong link be ween a P1( ) and p(ei), in ac hey a e he same.
Conclusion
In (1), we a gued ha he Maxwell-Bol zmann dis ibu ion eally depends on bias emo al, i.e.
n(ei)n(ej) = n(ek)n(el) o ei+ej = ek+el, and his only occu s o n(ei) being e y la ge. We s ess
ha his same condi ion holds i one maximizes ln( N!/ P oduc o e i n(ei)!) subjec o Sum o e
i n(ei) = N and Sum o e i ei n(ei) = E. In his case, one mus in oduce a d as ic app oxima ion
o n(ei)!, namely n(ei) powe n(ei) o ensu e ha bias emo al occu s. This is S iling’s
app oxima ion, bu is ne e heless a d as ic one.
In (2), we sugges ed ha one ob ain he MB dis ibu ion using only momen um conside a ions.
This mus also lead o bias emo al, we a gue he e, i.e, P1(pi)P1(pj) = P1(pk)P1(pk) o one
dimensional momen um and pi+pj = pk+pl. The issue is ha P1(p) mus = P1(-p) and so P1 as a
unc ion o p mus be e en in p (o p do p). The ene gy app oach al eady leads o exp(-p do p
/2mT) and so gi en he link be ween p and ei= p do p /2m, i seems ha P1(p) and p(ei) mus
be he same unc ion.
The ques ion hen becomes: Is he e a scheme in ol ing only momen um which yields a
ac o ial exp ession which educes o exp(- p do p /2mT)? We a gue ha he Gal on boa d ma h
in oduced in (2) does jus his. I conside s a (one dimension) as being composed o k (d )
uni s and (n-k) (-d ) ones. He e n and k a e ex emely la ge. The p obabili y associa ed wi h a
is hen: .5 powe n n! / ( k! (n-k)!). This exp ession is in a ian unde he in e change o k and
n-k), bu = k (d ) + (n-k) (-d ), so an in e change he e leads o →- . Thus, P1(p)=P1(-p). The
ques ion is whe he bias emo al a ises in he k, n ex emely la ge alue limi . In (2) and
e e ences he ein i is shown ha in such a limi his ac o ial exp ession becomes exp(- C do
) so one does ob ain p(ei) = C1 exp(-ei/T) om he ene gy app oach as well as bias emo al.
We no e ha o an elas ic collision (one dimension) pi+pj = pk +pl , ei+ej = ek+el, and
p(ei)p(ej)=p(ek)p(el). Thus, bias emo al o ei al eady includes bias emo al o pi as P1(p)
canno dis inguish he sign o p.
Re e ences
1. Rugge i, F ancesco R.Bias and Maximiza ion o A angemen s in he
Maxwell-Bol zmann Dis ibu ion (p ep in , zenodo, 2025)
2. Rugge i, F ancesco R. Specula ion on he Maxwell-Bol zmann exp(-ei/T) F om he View
o Conse a ion o Momen um Pa 3 (p ep in , zenodo, 2025)
