Bias and Maximiza ion o A angemen s in he Maxwell-Bol zmann Dis ibu ion
F ancesco R. Rugge i Hanwell, N.B. Oc . 7, 2025
I one osses a coin wice, one may ob ain wo heads, wo ails o wo cases o one head and
one ail. The si ua ion o a head and a ail allows o wo pe mu a ions, while he o he cases
don’ allow o any (i.e. ca y a weigh o 1). Thus, a ail and head ep esen s in a sense an
equilib ium esul and is linked o he p obabili ies o .5 o a head and .5 o a ail, i.e. all bias is
emo ed. In he Maxwell-Bol zmann case, one has N pa icles wi h a o al ene gy o E. The
canonical dis ibu ion sugges s ha any a angemen o E among N pa icles ca ies he same
weigh . (Some o hese iola e momen um conse a ion, bu a e oo ew in numbe o be o
conce n.) The a gumen hen seems o be ha one wishes o ha e a pa i ioning se { n(ei) }
which has he la ges numbe o pe mu a ions. I one had a di e en se {.n1(ei) }, i would ha e
ewe pe mu a ions and so one could obse e a change o a mo e like se { n(ei) } cha ac e ized
by he coun ing o i s pe mu a ions which appa en ly map o expe imen al obse a ions.
We nex conside he ques ion o bias. We de ine a possible bias as n(ei)n(ej) no = n(ek)n(el)
i ei+ej = ek+el. This is eminiscen o ime e e sal eac ion balance, bu he e we conside only
pe mu a ions. The ques ion becomes: Is he e any link be ween bias and pe mu a ions? We
a gue ha a pe mu a ion should ake he o m o a cons ain . In he simple case o only
n(ei)n(ej) no = n(ek)n(el), one would equi e a ma h cons ain linked o hese ene gies only:
F(ei, ej,ek,el). In he case o a bias in n(ei), one could w i e he maximiza ion o pe mu a ions
subjec o cons ain s in e ms o Maximize { ln ( N!/ P oduc o e i n(ei)!) } + a1 Sum o e i n(ei)
+ a2 Sum o e i ei n(ei) + a3 F(n(ei), n(ej), n(ek), n(el)). The ques ion hen becomes: Does he
abo e lead o mo e o ewe pe mu a ions han: Maximize { ln ( N!/ P oduc o e i n(ei)!) } + a1
Sum o e i n(ei) + a2 Sum o e i ei n(ei) ? We a gue ha i mus ep esen ewe , because a
cons ain cons ains, i.e. emo es solu ions which do no con o m o i s ule. Thus, we a gue
ha one canno ha e n(ei)n(ej) no = n(ek)n(el) o ei+ej = ek+el. This hen implies ha n(ei)n(ej)
= n(ek)n(el) o ei+ej = ek+el, which in u n is eac ion balance and leads o he
Maxwell-Bol zmann dis ibu ion di ec ly. This esul has been ob ained wi hou ac ually sol ing
he maximiza ion p oblem so we conside he ac ha he lack o bias only occu s o N and
n(ei)’s ex emely la ge and see wha his means o ln(n(ei)!). In o he wo ds, one mus use an
app oxima ion linked o n(ei) ex emely la ge in o de o emo e bias.
Maxwell-Bol zmann Dis ibu ion and Reac ion Balance
I one conside s ime e e sal balance o elas ic wo body collisions, one may w i e:
p(ei)p(ej) = p(ek)p(el) o ei+ej = ek+el ((1))
This leads di ec ly o he MB dis ibu ion: p(ei) = C exp(-ei/T) ((2))
Wha does such an app oach ha e o do wi h he numbe o pe mu a ions o a se o n(ei) which
ha e he condi ion ha Sum o e i ei n(ei) = E and Sum o e i n(ei) = N wi h Np(ei) = n(ei)?
Maxwell-Bol zmann Dis ibu ions and Pe mu a ions
I is adi ional o maximize:
Ln ( N!/ P oduc o e i n(ei)! ) subjec o Sum o e i ei n(ei) =E and Sum o e i n(ei) = N ((3))
This leads o he Maxwell-Bol zmann dis ibu ion, bu we ask: Wha is he easoning behind his
app oach? Ce ainly, any se o { n(ei) } such ha he cons ain s in ((3)) hold is physically iable.
In ac , he mic ocanonical app oach s a es ha any a angemen o n(ei) alues which uphold
he cons ain s o ((3)) ca y he same weigh . This seems o be he same idea as ossing a coin
wice. I may yield wo heads, wo ails o a head and ail wi h a weigh o 2. The weigh o 2 is
g ea e han 1 and so his is like he “equilib ium” solu ion and e lec s he in o ma ion o he
p obabili ies:
p(head) = .5 and p( ail) = .5 ((4))
We a gue ha wo heads o wo ails ep esen bias in ha hey do no e lec he equal a p io i
p obabili ies o heads and ails. They supp ess a p io i in o ma ion so o speak. In he abo e
example, he maximum numbe o pe mu a ions occu s o he scena io which does no
in oduce bias, i.e. one head and one ail. We sugges ha his simple idea applies also o he
mo e complica ed Maxwell-Bol zmann scena io and may be o malized.
The Case o Bias
Conside he goal o inding he se { n(ei) } wi h he g ea es numbe o pe mu a ions:
N!/ P oduc o e i n(ei)! ((5))
Nex , conside a e y simple possible bias: n(ei)n(ej) no = n(ek)n(el) o ei+ej = ek+el ((6))
Can such a bias yield a se { n(ei) } which has a maximum numbe o pe mu a ions ((5))? To
answe his ques ion, one may conside he bias ((6)) as a cons ain on n(ei)’s, in pa icula only
n(ei), n(ej), n(ek) and n(el). One may w i e:
F(n(ei), n(ej), n(ek), n(el)) as he cons ain linked o he bias ((6))
Then a o mal maximiza ion p ocess is:
Maximize( N!/ P oduc o e i n(ei)!) subjec o F(n(ei), n(ej), n(ek), n(el)), Sum o e i n(ei) = N
and Sum o e i ei n(ei) = E. ((7))
Does he n(ei) solu ion o ((7)) yield a highe pe mu a ion alue ((5)) han:
Maximize( 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.
((8)) ?
Now a cons ain cons ains, i.e. emo es solu ions which do no con o m, so we a gue ha any
F(n(ei), n(ej), n(ek), n(el)) would emo e possibili ies o n(ei)’s allowed by ((8)). Thus, we a gue
ha ((7)) canno lead o highe pe mu a ions ((5)), han ((7)). In o he wo ds, one wishes o
emo e all bias in o de o ob ain he highes pe mu a ion numbe ((5)). Remo ing all bias
means ha one mus ha e:
n(ei)n(ej) = n(ek)n(el) o ei+ej = ek+el ((9))
The e o he bias exp essions ha one could w i e, bu ((9)) is su icien o yield a solu ion o
n(ei) namely:
n(ei) = N C exp(-ei/T) ((10))
Lack o Bias Linked o Lack o In o ma ion in a Physical Scena io
The maximiza ion o pe mu a ions ((5)) makes no e e ence wha soe e o he kind o eac ions
which c ea e an equilib ium scena io. I jus s a es ha one mus emo e bias o ha e a
maximum numbe o pe mu a ions ((5)). Speci ic physical eac ions exis . Fo example, one
migh ha e elas ic wo body collisions. One migh a i s conside hese o be de e minis ic, bu
i a gi en e1 and e2 elas ically sca e (wi h momen um conse a ion), hen any e3, e4 which
conse e momen um and ene gy a e possible solu ions. I one is o emo e bias, hen any such
e3, e4 solu ion should ca y he same weigh as ano he . Gene alizing his o a gas wi h n(ei) =N
p(ei), ei pa icles, one would expec :
n(e1)n(e2) = n(e3)n(e4) ((11))
Thus, eac ion balance and maximiza ion o pe mu a ions (subjec o Sum o e i n(ei) = N and
Sum o e i ei n(ei) = E) bo h make use o he same unde lying idea, namely he emo al o bias.
In he case o eac ion balance, one conside s wo body in e ac ions and so he bias emo al is
o he o m o ((11)). In he abo e sec ions, we conside ed pe mu a ions ((5)) and only ocused
on he special bias n(ei)n(ej) no = n(ek)n(el). We a gued ha one may conside any bias and
ha his one was as good as ano he . Mo eo e , his bias is di ec ly linked o wo body
sca e ing which physically occu s in an ideal gas. By showing he emo al o he bias n(ei)n(ej)
no = n(ek)n(el) o ei+ej = ek+el mus occu , we a gued in he pe mu a ion case ha one does
no ac ually ha e o sol e he maximiza ion p oblem, bu only use:
n(ei)n(ej) = n(ek)n(el) o ei+ej = ek+el ((12))
This begs he ques ion: I one ac ually pe o ms he maximiza ion, why does i yield he same
esul as ((12))?
Fi s o all, we no e ha he cons ain s Sum o e i n(ei) = N and Sum o e i ei n(ei) = E a e
linea in n(ei) and ha aking ln o he numbe o pe mu a ions ((5)) also c ea es he linea sum:
ln(N!) - Sum o e i ln(n(ei)!) ((13))
The d/dn de i a i e mus be ma ched wi h a1 Sum o e i ei del a n + a2 Sum o e i del a n
((14))
We also no e ha he abo e a gumen s only hold o N being e y la ge and all n(ei) also being
e y la ge. Now, ((5)) is an exac exp ession o pe mu a ions so we a gue ha he emo al o
bias is di ec ly linked o n(ei) being a la ge numbe . In o he wo ds, he e eally is bias in he
sys em, bu i only disappea s in he la ge n(ei) limi . This seems o sugges ha some kind o
la ge n(ei) app oxima ion mus be made o ln(n(ei)!) because i is an exac exp ession as is
((14)). The app oxima ion mus consis en wi h:
n(ei)n(ej) = n(ej)n(el) o ei+ej = ek+el ((15)) (i.e. bias emo al)
The de i a i e d/dn(ei) o he app oxima ion used o ln(n(ei)!) linked wi h ((14)) mus yield
exp(-ei/T) which ollows om ((15)). This implies ha he app oxima ion which mus hold is:
ln(n(ei)!) = n(ei) ln(n(ei)) ((16))
We no e ha his is a i ed a wi h no e e ence o S i ling’s app oxima ion, e en hough he
esul is equi alen . Basically, he app oxima ion ((16)) s a es ha :
n(e)! app oxima ely= n(ei) powe n(ei) ((17))
((17)) is a a he simplis ic app oxima ion, bu i is appa en ly he one needed o emo e bias
om he sys em which is equi alen o ((15)). Thus, i is eally he case ha e y la ge n(ei)
alues lead o a simplis ic app oxima ion ((17)) which is equi alen o he emo al o bias and
his lack o bias is ep esen ed by he Maxwell-Bol zmann dis ibu ion;
p(ei) = C exp(-ei/T) ((18))
Conclusion
We no e ha eac ion balance is a s a emen o he emo al o bias. Bias emo al is di ec ly
linked o independence o n(ei) alues which can only occu i N (and hence n(ei)) a e ex emely
la ge. This begs he ques ion: Wha happens in he case o maximizing he numbe o
pe mu a ions o a se n(ei) which sa is y Sum o e n(ei) = N and Sum o e i ei n(ei) = E? The
idea seems o be ha he se o n(ei) which has he mos numbe o pe mu a ions ep esen s
he equilib ium one. We a gue ha any bias educes he numbe o pe mu a ions. Thus, a bias
n(ei)n(ej) no = n(ek)n(el) o ei+ej = ek+el would educe he numbe o pe mu a ions. This
sugges s ha n(ei)n(ej)=n(ek)n(el) which is he eac ion balance esul leading di ec ly o he MB
solu ion p(ei) = Cexp(-ei/T), ob ained wi h no maximiza ion. The poin is ha maximiza ion
subjec o Sum o e i ei n(ei)=E and Sum o e i n(ei) = N mus also yield his solu ion. A
pe mu a ion exp ession, howe e , is exac , and we a gue ha only a la ge n(ei) case leads o
independence o p(ei) alues. One may ake ln o he pe mu a ion exp ession N!/ P oduc o e i
n(ei)! o make i appea linea in n(ei)!, bu now one mus impose an app oxima ion on ln(n(ei)!)
such ha all bias is emo ed. This is equi alen o using he simplis ic app oxima ion: ln(n(ei)!)
= n(ei) powe n(ei). Thus, one is no eally using he exac ac o ial exp ession, bu is a he
basing he en i e Maxwell-Bol zmann solu ion on he idea o bias emo al, we a gue.
