1549380323-Statistical Mechanics Theory and Molecular Simulation

Harmonic systems 95

where ̄h=h/ 2 π. The integration overI′now proceeds using eqn. (A.2) and yields
unity, so that

Ω(E) =

E 0

̄hω

. (3.6.11)

Interestingly, Ω(E) is a constant independent ofE. All one-dimensional harmonic
oscillators of frequencyωhave the same number of accessible microstates! Thus, no
interesting thermodynamic properties can be derived from this partition function, and
the entropySis simply a constantkln(E 0 / ̄hω).
Consider next a collection ofN independent harmonic oscillators with different
masses and force constants, for which the Hamiltonian is

H=

∑N

i=1

[

p^2 i

2 mi

+

1

2

kir^2 i

]

. (3.6.12)

For this system, the microcanonical partition function is

Ω(N,E) =

E 0

h^3 N

∫

dNpdNrδ

(N

∑

i=1

[

p^2 i

2 mi

+

1

2

kir^2 i

]

−E

)

. (3.6.13)

Since the oscillators are all different, theN! factor is not needed. Let us first introduce
scaled variables according to

yi=

pi

√

2 mi

ui=

√

ki

2

ri, (3.6.14)

so that the partition function becomes

Ω(N,E) =

23 NE 0

h^3 N

∏N

i=1

1

ωi^3

∫

dNydNuδ

(N

∑

i=1

(

y^2 i+u^2 i

)

−E

)

, (3.6.15)

whereωi=

√

ki/miis the natural frequency for each oscillator. As in the ideal gas

example, we recognize that the condition

∑N

i=1(y

2
i+u
2
i) =Edefines a (6N−1)-
dimensional spherical surface, and we may introduce 6N-dimensional spherical coor-
dinates to yield

Ω(N,E) =

8 E 0

h^3 N

∏N

i=1

1

ω^3 i

∫

d^6 N−^1 ω ̃

∫∞

0

dR R^6 N−^1 δ

(

R^2 −E

)

. (3.6.16)

Using eqn. (3.5.14) and eqn. (A.15) allows the integration to be carried out in full with
the result:

Ω(N,E) =

23 NE 0 π^3 N

Eh^3 N

EN

Γ(3N)

∏N

i=1

1

ω^3 i

. (3.6.17)

In the thermodynamic limit, 3N− 1 ≈ 3 N, and we can neglect the prefactorE 0 /E,
leaving