1549380323-Statistical Mechanics Theory and Molecular Simulation

88 Microcanonical ensemble

Ω(L,E) =

E 0 L

√

2 m

h

∫∞

−∞

dy δ

(

y^2 −E

)

. (3.5.4)

Then, using the properties of the Diracδ-function in Appendix A, in particular eqn.
(A.15), we obtain

Ω(L,E) =

E 0 L

√

2 m

h

1

2

√

E

∫∞

−∞

dy

[

δ(y−

√

E) +δ(y+

√

E

]

. (3.5.5)

Therefore, integrating over theδ-function using eqn. (A.2), we obtain

Ω(L,E) =

E 0 L

√

2 m

h

√

E

. (3.5.6)

It is easily checked that eqn. (3.5.6) is a dimensionless number.
The free particle example leads naturally into a discussion of the classical ideal
gas. An ideal gas is defined to be a system of particles that do not interact. An ideal
gas ofNparticles is, therefore, simply a collection ofNfree particles. Therefore, in
order to treat an ideal gas, we need to considerNfree particles in three dimensions
for which the Hamiltonian is

H=

∑N

i=1

p^2 i

2 m

. (3.5.7)

A primary motivation for studying the ideal gas is that all real systems approach ideal
gas behavior in the limit of low density and pressure.
In the present discussion, we shall consider an ideal gas ofNclassical particles in
a cubic container of volumeVwith a total internal energyE. The partition function
in this case is given by

Ω(N,V,E) =

E 0

N!h^3 N

∫

dNp

∫

D(V)

dNrδ

(N

∑

i=1

p^2 i

2 m

−E

)

. (3.5.8)

As in the one-dimensional case, the integrand is independent of thecoordinates, hence
the position-dependent part of the integral can be evaluated immediately as

∫

D(V)

dNr=

∫L

0

dx 1

∫L

0

dy 1

∫L

0

dz 1 ···

∫L

0

dxN

∫L

0

dyN

∫L

0

dzN=L^3 N. (3.5.9)

SinceL^3 =V,L^3 N=VN.
For the momentum part of the integral, we change integration variables toyi=
pi/

√

2 mso that the partition function becomes

Ω(N,V,E) =

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

N!h^3 N

∫

dNyδ

(N

∑

i=1

yi^2 −E

)

. (3.5.10)

Note that the condition required by theδ-function is