Смирнов В.П. Курс статистической физики. Конспект лекций
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T p
µ F
E T = T (E, V, N) p = p(T, V, N)
H(q, p, V, N)
g(E, V, N) E
∆W (E, V, N)
S
(E, V, N)
T, p, µ
H(q, p) =
N
X
i=1
H
i
(
i
,
i
),
H
i
(
i
,
i
)
H
i
(
i
,
i
) =
2
i
/2m + W (
i
),
W (
i
) V
L L
3
= V
W (
i
) =
∞
i
/∈ V
i
∈ V
g(E, V, N)
∆W (E, V, N) Γ(E, V, N)
H(q, p, V, N) = E
Γ(E, V, N) =
Z
H(q,p,V,N)≤E
dΓ.
√
2mE
3N
X
j=1
p
2
j
= 2mE,
L
Γ(E, V, N) =
1
h
3N
N!
Z
P
3N
j=1
p
2
j
≤2mE
dp
Z
0≤q
j
≤L
dq =
=
1
h
3N
N!
× π
3N /2
(2mE)
3N /2
Γ(3N/2 + 1)
× L
3N
= E
3N / 2
V
N
(2mπ)
3N /2
h
3N
N!Γ(3N/2 + 1)
.
n R V
n
=
π
n/2
Γ(n/2 + 1)
R
n
S(E, V, N) = kN ·
ln
E
3/2
V
N
5/2
+
3
2
ln
4πm
3h
2
+
5
2
.
x N
ln Γ(x) ≈ x ln x − x, ln N! ≈ N ln N − N.
1
T
=
3
2
kN
E
,
p
T
=
kN
V
,
µ
T
= −
S
N
+
5
2
k
E =
3
2
NkT, pV = NkT, µ = −kT ·
ln
E
3/2
V
N
5/2
+
3
2
ln
4πm
3h
2
.
T
T
V N
H(q, p, V, N) = H
1
(q
1
, p
1
, V
1
, N
1
) + H
2
(q
2
, p
2
, V
2
, N
2
),
E = E
1
+ E
2
, E
1
<< E
2
≈ E, N = N
1
+ N
2
, V = V
1
+ V
2
, ,
ρ
mc
(q, p, V, N) =
1
C
δ(H
1
(q
1
, p
1
, V
1
, N
1
) + H
2
(q
2
, p
2
, V
2
, N
2
) − E).
ρ
1
(q
1
, p
1
, V
1
, N
1
) =
Z
ρ
mc
(q, p, V, N)dΓ
2
.
6N
2
N
2
ρ
1
(q
1
, p
1
, V
1
, N
1
) =
1
C
Z
δ(H
1
(q
1
, p
1
, V
1
, N
1
) + E
2
− E)g
2
(E
2
)dE
2
=
=
1
C
g(E − H
1
(q
1
, p
1
, V
1
, N
1
)) =
1
C∆E
2
exp(
S
2
(E − H
1
(q
1
, p
1
, V
1
, N
1
))
k
).
H
1
<< E
S
2
(E − H
1
) = S
2
(E) −
∂S
2
∂E
H
1
(q
1
, p
1
, V
1
, N
1
) + ···,
∂S
2
∂E
=
1
T
.
ρ
1
ρ
1
=
1
C∆E
2
exp
S
2
(E)
k
· exp
−
H
1
(q
1
, p
1
, V
1
, N
1
)
kT
=
1
Z
exp
−
H
1
(q
1
, p
1
, V
1
, N
1
)
kT
,
1
C∆E
2
exp
S
2
(E)
k
1
Z
T V
1
N
1
ρ
c
(q, p, T, V, N) =
1
Z
exp
−
H(q, p, V, N)
kT
.
Z = Z(T, V, N)
Z(T, V, N) =
Z
exp
−
H(q, p, V, N)
kT
dΓ.
1
Z(T, V, N)
= exp
F (T, V, N)
kT
, F (T, V, N) = −kT ln(Z(T, V, N)),
F (T, V, N)
ρ
c
(q, p, T, V, N) = exp
F − H(q, p, V, N)
kT
, ln ρ
c
(H(q, p, V, N)) =
F − E
kT
,
S = −
F − E
T
, F = E − T S,
F = F (T, V, N)
dF (T, V, N) = T dS − pdV + µdN − T dS −SdT = −SdT − pdV + µdN
S = −
∂F
∂T
= k ln Z + kT
∂ ln Z
∂T
, p = −
∂F
∂V
= kT
∂ ln Z
∂V
, µ =
∂F
∂N
= −kT
∂ ln Z
∂N
.
E = F + T S
E = −kT ln Z + T · k ln Z + kT
2
∂ ln Z
∂T
= kT
2
∂ ln Z
∂T
.
p = p(T, V, N)
H = H(q, p, V, N)
Z = Z(T, V, N)
F = F (T, V, N)
S p µ
Z(T, V, N) =
Z
exp
−
H(q, p, V, N)
kT
dΓ =
=
1
h
3N
N!
Y
i
Z
exp
−
p
2
i
2mkT
dp
i
Z
exp
−
W (r
i
)
kT
dr
i
.
r
i
V
(2πmkT )
1/2
Z
∞
−∞
exp(−αx
2
)dx = (π/α)
1/2
.
Z(T, V, N) =
1
h
3N
N!
(2πmkT )
3N /2
V
N
=
(2πmkT )
3/2
h
3
eV
N
N
.
F = −NkT
"
ln
T
3/2
V
N
+ ln
2πmk
h
2
3/2
+ 1
#
S = −
F
T
+
3
2
kN, p =
NkT
V
, µ =
F
N
+ kT,
S = Nk
"
ln
T
3/2
V
N
+ ln
2πmk
h
2
3/2
+
5
2
#
,
pV = NkT,
µ = −kT
"
ln
T
3/2
V
N
+ ln
2πmk
h
2
3/2
#
.
E =
3
2
NkT, C =
3
2
Nk.
T =
2E
3Nk
S µ S µ
T V µ
T
µ
q p
E = E
1
+ E
2
, E
1
<< E
2
≈ E; N =
e
N
1
+
e
N
2
,
e
N
1
<<
e
N
2
≈ N, V = V
1
+ V
2
.
T
µ
ρ
E
(q, p, V, N) =
1
C
δ(H
1
(q
1
, p
1
, V
1
,
e
N
1
) + H
2
(q
2
, p
2
, V
2
,
e
N
2
) − E)δ
e
N
1
+
e
N
2
,N
ρ
1
=
X
e
N
2
Z
ρ
E
(q, p, V, N)dΓ
2
=
=
X
e
N
2
δ
e
N
1
+
e
N
2
,N
Z
1
C
δ(H
1
(q
1
, p
1
, V
1
,
e
N
1
) + E
2
− E)g
2
(E
2
,
e
N
2
)dE
2
=
=
1
C
g
2
(E − H
1
, N −
e
N
1
) =
1
C∆E
2
exp
S
2
(E − H
1
, N −
e
N
1
)
k
!
.
H
1
E
e
N
1
N
S
2
(E − H
1
, N −
e
N
1
) = S
2
(E, N) −
∂S
2
(E, N)
∂E
H
1
−
∂S
2
(E, N)
∂N
e
N
1
+ ··· ≈
≈ S
2
(E, N) −
1
T
H
1
+
µ
T
e
N
1
.
ρ
1
(q
1
, p
1
)
Z
−1
gc
=
1
C∆E
2
exp
S
2
(E, N)
k
,
ρ
1
(q
1
, p
1
,
e
N
1
, T, V
1
< µ) = Z
−1
gc
exp
µ
e
N
1
− H
1
(q
1
, p
1
, V
1
,
e
N
1
)
kT
!
.
ρ
gc
(q
e
N
, p
e
N
,
e
N, T, V, µ) = Z
−1
gc
exp
µ
e
N − H(q
e
N
, p
e
N
, V,
e
N)
kT
!
.
T V µ
e
N
q
e
N
p
e
N
∞
X
e
N=0
Z
ρ
gc
(q
e
N
, p
e
N
,
e
N, T, V, µ)dΓ
e
N
= 1.
Z
gc
= Z
gc
(T, V, µ
Z
gc
(T, V, µ) =
∞
X
e
N=0
exp
µ
e
N
kT
!
Z
exp
−
H(q
e
N
, p
e
N
, V,
e
N)
kT
!
dΓ
e
N
Z
gc
(T, V, µ) = exp
−
Ω(T, V, µ)
kT
, Ω(T, V, µ) = −kT ln(Z
gc
(T, V, µ)).
ρ
gc
(q
e
N
, p
e
N
,
e
N, T, V, µ) = exp
Ω + µ
e
N − H(q
e
N
, p
e
N
, V,
e
N)
kT
!
.
Ω(T, V, µ)
S = −kln(ρ
gc
) = −k
Ω + µN − E
kT
,
N =
e
N E = H
Ω(T, V, µ)
Ω = E − T S −µN = F − µN.
Ω Ω
dΩ = dF − µdN − Ndµ = −SdT − pdV − Ndµ.
S = −
∂Ω
∂T
, p = −
∂Ω
∂V
, N = −
∂Ω
∂µ
.
E = Ω + T S + µN F = Ω + µN
Ω
T, V, µ
H = H(q
e
N
, p
e
N
, V,
e
N)
Z
gc
Ω
Ω S
p N
Z
gc
(T, V, µ) =
∞
X
e
N=0
exp
µ
e
N
kT
!
1
h
3
e
N
e
N!
(2πmkT )
3
e
N/2
V
e
N
=
=
∞
X
e
N=0
1
e
N!
2πmkT
h
2
3/2
V exp
µ
kT
!
e
N
= exp
"
2πmkT
h
2
3/2
V exp
µ
kT
#
.
Ω
Ω = −kT
2πmkT
h
2
3/2
V exp
µ
kT
.
N = −
∂Ω
∂µ
= −
Ω
kT
, Ω = −NkT ;
p = −
∂Ω
∂V
= −
Ω
V
, pV = NkT ;
S = −
∂Ω
∂T
=
Ω
T
−
5
2
+
µ
kT
, S = kN
5
2
−
µ
kT
.
E = Ω + T S + µN = −NkT − NkT
−
5
2
+
µ
kT
+ µN =
3
2
NkT.
N
H =
N
X
i−1
p
2
i
2m
+
X
i<j
U
ij
, U
ij
≡ U(r
ij
), r
ij
= |r
i
− r
j
|.
U(r)
d
Z =
Z
exp
−
H
kT
dΓ =
(2πmkT )
3/2
N!h
3N
Z
exp
−
P
i<j
U
ij
kT
dq.
Z
id
=
(2πmkT )
3N /2
V
N
h
3N
N!
Z
Z = Z
id
· Q
N
,
Q
N
=
1
V
N
Z
exp
−
P
i<j
U
ij
kT
dq (dq = dr
1
dr
2
···dr
N
)
3N
Q
N
= 1
X
i<j
U
ij
= U
12
+ (U
13
+ U
23
) + (U
14
+ U
24
+ U
34
) + ···
Q
N
Q
N
=
1
V
N
Z
dr
1
Z
exp
−
U
12
kT
dr
2
Z
exp
−
U
13
+ U
23
kT
dr
3
···
···
Z
exp
−
U
1N
··· + U
N−1N
kT
dr
N
.
η
ij
= exp
−
U
ij
kT
− 1, exp
−
U
ij
kT
= 1 + η
ij
.
Q
N
r
j
J
j
(r
1
, r
2
, ···, r
j−1
) =
Z
dr
j
(1 + η
1j
)(1 + η
2j
) ···(1 + η
j−1j
) =
=
Z
dr
j
1 +
j−1
X
i=1
η
ij
+
X
il
η
ij
η
lj
+ ···
!
.
η
ij
≈ 0
J
j
≈
Z
V
(1 +
j−1
X
i=1
η
ij
)dr
j
= V + 4π(j − 1)
∞
Z
0
η(r)r
2
dr.
4π
ω(T ) = −4π
∞
Z
0
1 − exp
−
U
kT
r
2
dr.