Смирнов В.П. Курс статистической физики. Конспект лекций
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E = Ω + T S + µN =
X
l
f
(B−E)
l
ε
l
,
f
(B−E)
l
= n
l
f
(B−E)
l
(ε
l
)
µ
f
(B−E)
l
= n
l
T = 0
exp
ε
l
− µ
kT
1 µ ε
0
− kT n
l
1
f
(B−E)
l
≈ f
(F −D)
l
≈ exp
µ − ε
l
kT
= A exp
−
ε
l
kT
≡ f
(M−B)
l
.
l ∼ p, ε
p
=
p
2
2m
,
X
l
∼
Z
V
h
3
dp.
A
V
h
3
Z
exp
−
p
2
2mkT
dp = N, A =
h
3
n
(2πmkT )
3/2
, n =
N
V
.
A ≥ 1
A 1 n
T m
n ≈ 10
28
T = 300 h = 6.63 × 10
−34
· k = 1.38 × 10
−23
/
m ≈ 10
−26
A ≈ 10
−5
m ≈
−30
A ≈ 10
V T
∂F
∂N
T,V
(= µ) = 0,
µ = 0
ω c c /
k k = ω/c = 2πν/c = 2π/λ p = ~k = n · ~ω/c n
k ε = ~ω = pc
n
i
=
exp
~ω
i
kT
− 1
−1
,
ω
i
L
exp(ik
i
(i + L)) = exp(ik
i
i), i = x, y, z, exp(ik
i
L) = 1.
k
i
=
2π
L
n
i
, ∆k
i
= dk =
2π
L
, n
i
= 0, ±1, ...,
±
L
2
,
k
i
X
k
i
→
L
2π
3
Z
dk.
n
x
, n
y
, n
z
exp(i(k
x
x+k
y
y +k
z
z))
dk
2
L
2π
3
4πk
2
dk =
V
π
2
c
3
ω
2
dω = g(ω)dω,
ω = kc ω ω + dω
V
dN
ω
= n
ω
g(ω)dω =
V
π
2
c
3
ω
2
dω
exp
~ω
i
kT
− 1
.
V T
N =
V
π
2
c
3
Z
∞
0
ω
2
dω
exp
~ω
kT
− 1
=
V
π
2
c
3
kT
~
3
Z
∞
0
x
2
dx
exp(x) − 1
N ≈
2.4
π
2
V
kT
~c
3
≈ 20V T
3
.
T = 300 n = N/V = 5.4 · 10
14 −3
2.6 · 10
25
dN
ω
~ω ω ω + dω
dE
ω
=
V ~
π
2
c
3
ω
3
dω
exp
~ω
kT
− 1
.
ρ
ω
=
1
V
dE
ω
dω
=
~
π
2
c
3
ω
3
exp
~ω
kT
− 1
.
~ω kT
ρ
ω
= ω
2
kT
π
2
c
3
.
R
∞
0
ρ(ω)dω
E =
V ~
π
2
c
3
Z
∞
0
ω
3
dω
exp
~ω
kT
− 1
=
V ~
π
2
c
3
kT
~
4
Z
∞
0
x
3
dx
exp(x) − 1
=
π
2
15
V
(kT )
4
(c~)
3
∼ T
4
.
E/N ≈
π
4
36
kT ≈ 2.7 kT,
1.5 kT
C
V
=
4π
2
15
V k
kT
c~
3
=
π
4
9
kN.
Ω
Ω = kT
X
i
ln(1 − exp
−
~ω
i
kT
) =
V
π
2
c
3
kT
Z
∞
0
ln(1 − exp
−
~ω
kT
)ω
2
dω =
=
V ~
π
2
c
3
kT
~
4
Z
∞
0
ln(1 − exp(−x))x
2
dx = −
π
2
45
V
(kT )
4
(c~)
3
= F,
Z
∞
0
ln(1 − exp(−x))x
2
dx =
1
3
x
3
ln(1 − exp(−x))
∞
0
−
1
3
Z
∞
0
x
3
dx
exp(x) − 1
= −
π
4
45
.
S = −
∂Ω
∂T
=
4π
2
45
V k
kT
c~
3
p = −
∂Ω
∂V
=
π
2
45
(kT )
4
(c~)
3
.
E = Ω + T S
C
V
/N ≈ 11k
T = 300 K E/V = 6.13 × 10
−6
·
−3
C
V
/V = 0.817 · 10
−7
·K
−1
·
−3
p = 2.04 × 10
−6
·
−2
N/V = 0.548 × 10
15 −3
dw =
E
V
c
dΩ
4π
dΩ
T
θ
dΩ
θ ∼ cos θ
W
S
=
Z Z
cos θdw =
Ec
4πV
Z
π/2
0
cos θ sin θdθ
Z
2π
0
dϕ =
Ec
4V
= σT
4
,
σ =
π
2
k
4
60~
3
c
2
W = σST
4
S
ω = 2πc/λ; |ρ(ω)dω| = |ρ(λ)dλ|; ρ(λ) =
dω
dλ
ρ(ω);
ρ(λ) =
16π
2
~c
λ
5
1
exp
2π~c
λkT
− 1
.
λ
d ln ρ
dλ
=
1
ρ
dρ
dλ
,
dρ
dλ
= 0
d ln ρ
dλ
= 0 a =
2π~c
kT
ln ρ = −5 ln λ − ln(e
a/λ
− 1) + ln(16π
2
~c);
−
5
λ
+
1
1 − exp(−a/λ)
a
λ
2
= 0; x = a/λ; 1 − exp(−x) = x/5; x = 4.965.
λ
max
=
a
4.965
=
2π~c
4.965 · k
·
1
T
.
p
ε
p
=
p
2
2m
.
h
3
dp
V
2
V dp
h
3
.
p
p
X
p
→
2V
h
3
Z
dp.
dp = 4πp
2
dp, pdp = mdε, p =
√
2mε, dp = 4
√
2πm
3/2
ε
1/2
dε,
X
p
→
2V
h
3
Z
dp =
Z
g(ε)dε, g(ε) =
V
2π
2
2m
~
2
3/2
√
ε ≡ aV
√
ε.
Ω
Ω = −kT
X
p
ln
1 + e
µ − ε
p
kT
= −aV kT
Z
∞
0
√
ε ln
1 + e
µ − ε
kT
dε.
Ω = −
2
3
aV
Z
∞
0
ε
3/2
e
ε − µ
kT
+ 1
dε = −
2
3
Z
∞
0
εg(ε)f(ε)dε = −
2
3
E.
x =
ε
kT
µ
kT
= z z 1
Ω = −
2
3
aV (kT )
5/2
Z
∞
0
x
3/2
e
(x−z)
+ 1
dx.
I =
Z
∞
0
e
φ(x)dx
e
(x−z)
+ 1
,
e
φ(x)dx = dϕ(x)
y = x − z
I =
∞
Z
0
dϕ(x)
e
(x−z)
+ 1
=
ϕ(x)
e
(x−z)
+ 1
∞
0
+
∞
Z
0
ϕ(x)
e
(x−z)
(e
(x−z)
+ 1)
2
dx =
= −ϕ(0) +
∞
Z
−z
ϕ(z + y)
e
y
(e
y
+ 1)
2
dy ≈ −ϕ(0) +
∞
Z
−∞
ϕ(z + y)
e
y
(e
y
+ 1)
2
dy,
−z −∞
f(y) =
e
y
(e
y
+ 1)
2
f(−y) = f(y)
y → ±∞ ∼ 1
ϕ(z + y)
y |y| z
ϕ(z + y) = ϕ(z) + ϕ
0
(z)y +
1
2
ϕ
00
(z)y
2
+ ···.
y
Z
∞
−∞
exp(y)
(exp(y) + 1)
2
dy = 1,
Z
∞
−∞
y
2
exp(y)
(exp(y) + 1)
2
dy =
π
2
3
, ···,
I
I = −ϕ(0) + ϕ(z) +
π
2
6
ϕ
00
(z) + ···.
ϕ(x) =
2
5
x
5/2
, ϕ(0) = 0, ϕ(z) =
2
5
z
5/2
, ϕ
00
(z) =
3
2
z
1/2
, ··· ,
Ω z
−1
=
kT
µ
Ω = −
4
15
aV µ
5/2
"
1 +
5
8
π
2
kT
µ
2
+ ···
#
.
E(T, V, µ) =
2
5
aV µ
5/2
"
1 +
5
8
π
2
kT
µ
2
+ ···
#
.
C
V,N
=
∂E
∂T
V,N
N = N(T, V, µ)
µ = µ(T, V, N)
C
V,N
N(T, V, µ) = −
∂Ω
∂µ
T,V
=
2
3
aV µ
3/2
"
1 +
π
2
8
kT
µ
2
+ ···
#
.
µ = µ(T, V, N)
µ = µ
0
"
1 +
π
2
8
kT
µ
2
+ ···
#
−2/3
,
µ
0
≡ µ(0, V, N) =
3N
2aV
2/3
=
~
2
2m
(3π
2
n)
2/3
, (n = N/V ).
kT
µ
0
µ = µ(T, V, N)
µ(T, V, N) = µ
0
"
1 −
π
2
12
kT
µ
0
2
+ ···
#
.
µ(T, V, N) E
kT
µ
0
E =
3
5
Nµ
0
"
1 +
5π
2
12
kT
µ
0
2
+ ···
#
.
C
( )
V,N
=
π
2
2
Nk
kT
µ
0
+ ··· ∼ T.
C
( )
V,N
=
3
2
Nk.
C
( )
V,N
C
( )
V,N
=
π
2
3
kT
µ
0
1,
σ
N
1
, N
2
, N
3
N = N
1
N
2
N
3
Nσ
a
n
= n
1
a
1
+ n
2
a
2
+ n
3
a
3
,
n (n
1
, n
2
, n
3
) a
i
n
n α = 1, 2, ..., σ
Ω
a
= a
1
· a
2
× a
3