1549380323-Statistical Mechanics Theory and Molecular Simulation

Thermal equilibrium 85

andV =V 1 +V 2 , respectively. The entropy of each system is given in terms of the
partition function for each system as

S 1 (N 1 ,V 1 ,E 1 ) =kln Ω 1 (N 1 ,V 1 ,E 1 )

S 2 (N 2 ,V 2 ,E 2 ) =kln Ω 2 (N 2 ,V 2 ,E 2 ), (3.4.2)

where the partition functions are given by

Ω 1 (N 1 ,V 1 ,E 1 ) =MN 1

∫

dx 1 δ(H 1 (x 1 )−E 1 )

Ω 2 (N 2 ,V 2 ,E 2 ) =MN 2

∫

dx 2 δ(H 2 (x 2 )−E 2 ). (3.4.3)

Of course, if we solved Hamilton’s equations forH(x) in eqn. (3.4.1),H 1 (x 1 ) and
H 2 (x 2 ) would be separately conserved becauseH(x) is separable. However, the frame-
work of the microcanonical ensemble allows us to consider the full set of microstates
for which onlyH(x) =H 1 (x 1 ) +H 2 (x 2 ) =EwithoutH 1 (x 1 ) andH 2 (x 2 ) being inde-
pendently conserved. Indeed, since the systems can exchange heat, we do not expect
H 1 (x 1 ) andH 2 (x 2 ) to be individually conserved. The total partition function is then

Ω(N,V,E) =MN

∫

dxδ(H 1 (x 1 ) +H 2 (x 2 )−E)

6 = Ω 1 (N 1 ,V 1 ,E 1 )Ω 2 (N 2 ,V 2 ,E 2 ). (3.4.4)

Because Ω 1 and Ω 2 both involveδ-functions, it can be shown that the total partition
function is given by

Ω(N,V,E) =C′

∫E

0

dE 1 Ω 1 (N 1 ,V 1 ,E 1 )Ω 2 (N 2 ,V 2 ,E−E 1 ), (3.4.5)

whereC′is an overall constant independent of the energy. In order to realize eqn.
(3.4.5) by solving Hamilton’s equations, we would need to solve the equations for all
values ofE 1 between 0 andEwithE 2 =E−E 1. We can imagine accomplishing

this by choosing a set ofPclosely spaced values forE 1 ,E
(1)
1 ,...,E

(P)
1 and solving the
equations of motion for each of theseP values. In this case, eqn. (3.4.5) would be
replaced by a Riemann sum expression:

Ω(E) =C′∆

∑P

i=1

Ω 1 (E( 1 i))Ω 2 (E−E( 1 i)), (3.4.6)

where ∆ is the small energy interval ∆ =E 1 (i+1)−E( 1 i). The integral is exact when
∆→0 andP→ ∞. When the integral is written in this way, we can make use of a
powerful theorem on sums with large numbers of terms.