1549380323-Statistical Mechanics Theory and Molecular Simulation

86 Microcanonical ensemble

Consider a sum of the form

σ=

∑P

i=1

ai, (3.4.7)

whereai>0 for allai. Letamaxbe the largest of all theai’s. Clearly, then

amax≤

∑P

i=1

ai

Pamax≥

∑P

i=1

ai. (3.4.8)

Thus, we have the inequality

amax≤σ≤Pamax, (3.4.9)

or
lnamax≤lnσ≤lnamax+ lnP. (3.4.10)

This gives upper and lower bounds on the value of lnσ. Now suppose that lnamax>>
lnP. Then the above inequality implies that

lnσ≈lnamax. (3.4.11)

This would be the case, for example, ifamax∼eP. In this case, the value of the sum
is given to a very good approximation by the value of its maximum term (McQuarrie,
2000).
Why should this theorem apply to the sum expression for Ω(N,V,E) in eqn. (3.4.6)?
In the next section, we will see that the partition function of an ideal gas, that is, a
collection ofNfree particles, varies as Ω∼[g(E)V]N, whereg(E) is some function
of the energy. This fact motivates the more general result that the terms in the sum
vary exponentially withN. But the number of terms in the sumP also varies likeN
sinceP=E/∆ andE∼N, sinceEis extensive. Thus, the terms in the sum under
consideration obey the conditions for the application of the theorem.
Let the maximum term in the sum be defined by energiesE ̄ 1 andE ̄ 2 =E−E ̄ 1.
Then, according to the above analysis,

S(N,V,E) =kln Ω(N,V,E)

=kln ∆ +kln

[

Ω 1 (N 1 ,V 1 ,E ̄ 1 )Ω 2 (N 2 ,V 2 ,E−E ̄ 1 )

]

+klnP+klnC′. (3.4.12)

SinceP=E/∆, ln ∆+lnP= ln ∆+lnE−ln ∆ = lnE. ButE∼N, while ln Ω 1 ∼N.
SinceN >>lnN, the above expression becomes, to a good approximation,

S(N,V,E)≈kln

[

Ω 1 (N 1 ,V 1 ,E ̄ 1 )Ω 2 (N 2 ,V 2 ,E−E ̄ 1 )

]

+O(lnN) + const. (3.4.13)

Thus, apart from constants, the entropy is approximately additive: