1549380323-Statistical Mechanics Theory and Molecular Simulation

Ideal gas 89

∑N

i=1

y^2 i=E. (3.5.11)

Eqn. (3.5.11) is the equation of a (3N−1)-dimensional spherical surface of radius

√

E.

Thus, the natural thing to do is transform to spherical coordinates; what we need,
however, is a set of spherical coordinates for a 3N-dimensional sphere. Recall that
ordinary spherical coordinates in three dimensions consist of one radial coordinate,r,
and two angular coordinates,θandφ, with a volume element given byr^2 drd^2 ω, where
d^2 ω= sinθdθdφis the solid angle element. In 3Ndimensions, spherical coordinates
consist of one radial coordinate and 3N−1 angular coordinates,θ 1 ,...,θ 3 N− 1 with a
volume element dNy=r^3 N−^1 drd^3 N−^1 ω. The radial coordinate is given simply by

r^2 =

∑N

i=1

y^2 i. (3.5.12)

After transforming to these coordinates, the partition functionbecomes

Ω(N,V,E) =

E 0 (2m)^3 N/^2 VN

N!h^3 N

∫

d^3 N−^1 ω

∫∞

0

dr r^3 N−^1 δ

(

r^2 −E

)

. (3.5.13)

The solid angle element dnωis sinn−^1 θ 1 sinn−^2 θ 2 ···sinθn− 1 dθ 1 ···dθn, and the gen-
eral formula for the solid angle integral is
∫
dnω=
2 π(n+1)/^2
Γ

(n+1

2

), (3.5.14)

where Γ(x) is the Gamma function defined by

Γ(x) =

∫∞

0

dt tx−^1 e−t. (3.5.15)

According to eqn. (3.5.15), for any integern, we have

Γ(n) = (n−1)!

Γ

(

n+

1

2

)

=

(2n−1)!!

2 n

π^1 /^2. (3.5.16)

Finally, expanding theδ-function using eqn. (A.15), we obtain

Ω(N,V,E) =

E 0 (2m)^3 N/^2 VN

N!h^3 N

2 π^3 N/^2

Γ(3N/2)

×

∫∞

0

dr r^3 N−^1

1

2

√

E

[

δ(r−

√

E) +δ(r+

√

E)

]

. (3.5.17)

Because the integration range onris [0,∞), only the firstδ-function gives a nonzero
contribution, and, by eqn. (A.13), the result of the integration is