Liouville’s Theorem
ρ({p, q}) behaves like an incompressible fluid.
⎛
⎞
3N
3
dρ
∂ρ
∂ρ dpi
∂ρ dqi ⎠
⎝
=
+
+
= 0
dt
∂t
∂qi dt
i=1 ∂pi dt
Using Hamilton’s equations this becomes
3N
3
⎛
⎞
∂ρ
∂ρ ∂H
∂ρ ∂H ⎠
⎝
=
−
∂t
∂qi ∂pi
i=1 ∂pi ∂qi
8.044 L7B1
If the density in phase space depends only on the
energy at that point,
ρ({p, q}) = ρ(H{p, q}),
carrying out the indicated derivatives shows that
∂ρ
= 0.
∂t
This proves that ρ = ρ(H{p, q}) is a sufficient con­
dition for an equilibrium probability density in phase
space.
8.044 L7B2
1. The System
1. The System
E < energy < E+∆
∆ << E
Fixed:
E
V
N
M or H
P or E
A complete set of independent thermodynamic
variables is fixed.
Many micro-states
satisfy the conditions.
8.044 L13B1
8.044
L7B3
2. Probability Density
All accessible microscopic states are equally probable.
Classical
p({p, q}) = 1/Ω
= 0
Ω≡
�
accessible
E < H({p, q}) ≤ E + Δ
elsewhere
{dp, dq} = Ω(E, V, N )
8.044 L7B4
Quantum
p(k) = 1/Ω
= 0
Ω ≡
E < (k|H|k) ≤ E + Δ
elsewhere
�
(1) = Ω(E, V, N )
k, accessible
8.044 L7B5
Let X be a state of the system specified by a subset
{ p”,q” } of { p,q}
p(X) =
�
except {p"" ,q "" }
p({p, q}) {dp, dq}
1�
=
{dp, dq}
""
""
Ω except {p ,q }
Ω"(consistent with X)
=
Ω
volume consistent with X
=
total volume of accessible phase space
8.044 L7B6
3. Quantities Related to Ω
Φ(E, V, N ) ≡
�
H({p,q})<E
{dp, dq}
= cumulative volume in phase space
∂Φ(E, V, N )
ω(E, V, N ) ≡
∂E
= density of states as a function of energy
⇒ Ω(E, V, N ) = ω(E, V, N )Δ
8.044 L7B7
Example Ideal Monatomic Gas
qi = x, y, z
pi = mx,
˙ my,
˙ mz˙
N atoms
in a box
V = LxLy Lz
− ∞ < pi < ∞
pi2
H({p, q}) =
i=1 2m
3N
�
8.044 L7B8
Ω =
=
�
{dp, dq} =
�
{dq} {dp}
�N ��
Ly
��
Lx
dx
0
= VN
�
0
�
E<H<E+Δ
dy
�N ��
Lz
0
dz
�N �
{dp}
{dp}
Φ(E, V, N ) =
VN
�
H<E
{dp}
8.044 L7B9
pi2
E=
i=1 2m
3N
3
⇒
2mE =
3N
3
p2
i
i=1
This describes a 3N dimensional spherical surface
in the p part of phase space with a radius R =
√
2mE.
8.044 L7B10
Math:
• Volume of an α dimensional sphere of radius R is
π α/2 α
R
(α/2)!
• Sterling’s approximation for large M
ln(M !) ≈ M ln M − M
M
→ M! ≈
e
�
�M
8.044 L7B11
3N/2
π
Φ(E, N, V ) = V N
(2mE)3N/2
(3N/2)!
≈
⎧
⎪
⎨
N
V
⎪
⎫
�3N/2⎪
⎬
4πemE
�
3N
⎩
3N 1
{ }
2 E
�
ω(E, N, V ) =
�
3N Δ
{ }
2 E
�
Ω(E, N, V ) =
⎪
⎭
�
8.044 L7B12
Ω'
V N −1Ly Lz
1
p(xi) =
=
=
N
Ω
V
Lx
0 ≤ x < Lx
Ω'
V N −2Ly Lz LxLz
1
p(xi, yj ) =
=
=
= p(xi)p(yj ) ⇒ S.I.
N
Ω
V
LxLy
p(pxi ) =
�
Ω'
p({p,
q}){
dp , dq} =
����
�
��
�
Ω
p=
�
p
1/Ω
xi
Note that Ω' differs on each of the three lines, be­
ing a generic symbol for the reduced phase volume
consistent with some constraint.
8.044 L7B13
E ≡ p2
x /2m
E−E distributed over other variables
⎞(3N −1)/2
⎛
ΩI
3N − 1 Δ
4πem(E − E) ⎠
N
⎝
=
V
2
E−E
3N − 1
ΩI
3N − 1
E
4πem −1/2
=
Ω
3N � , E − E �
3
,
≈1
⎛
×
⎜ (N
⎜
⎝
,
≈1
⎞
3
N
1
− 2 +2
−1
)
⎟
3
⎟
⎠
3N
N− 2
�
A
⎛
⎜ (E
⎜
⎝
,
⎞
3N − 1
− E) 2 2 ⎟
⎟
⎠
3N
E 2
�
B
8.044 L7B14
1
A =
N −
3
but
1
1 −
3N
�
limζ→∞ 1 + xζ
so A ≈
√
�ζ
− 3N
2
1
=
N −
3
⎛
⎝1
⎞− 3N
2
⎠
1/2
+
−3N/2
= ex
N e1/2
8.044 L7B15
1
B =
√
E−E
E
1 −
E
3N
2
1
=
√
E−E
⎛
1 E/
⎜
⎝
1 −
2
<E
3N/2
⎞ 3N
>⎟ 2
⎠
where we have used < E >≡ E/3N and E = 3N <
E >.
so B ≈ √ 1
3N <E>
e−E/2<E>
8.044 L7B16
⎞
√
�√
�
3
1
−1/2
1/2
⎝
⎠
√
N e
p(px) = √
e
e−E/2<E>
3N < E >
4πm
⎛
1
= √
e−E/2<E>
4πm < E >
2
Now use E = p2
x /2m and < E >=< px > /2m.
p(px) =
1
2π < p2
x >
2 /2<p2 >
−p
x
e x
8.044 L7B17
MIT OpenCourseWare
http://ocw.mit.edu
8.044 Statistical Physics I
Spring 2013
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