Смирнов В.П. Курс статистической физики. Конспект лекций
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t
n
= δt · n (n = 0, 1, 2, ..., M − 1).
M
dqdp Mρdqdp
M
Mρdqdp
M
= ρdqdp = dw
dqdp
Mρ ρ
M
ρ(q, p)
A
Z
A(q, p)Mρ(q, p)dqdp
Z
Mρ(q, p)dqdp
=
Z
A(q, p)ρ(q, p)dqdp = A .
A
ρ(q, p)
ρ
1
(q
(1)
, p
(1)
) ρ
2
(q
(2)
, p
(2)
)
ρ(q, p) = ρ
1
(q
(1)
, p
(1)
) · ρ
2
(q
(2)
, p
(2)
)
dq
(1)
dp
(1)
dw
1
=
Z
(q,p)
2
ρ(q, p) dq dp = ρ
1
(q
(1)
, p
(1)
) dq
(1)
dp
(1)
Z
(q,p)
2
ρ
2
(q
(2)
, p
(2)
) dq
(2)
dp
(2)
=
= ρ
1
(q
(1)
, p
(1)
) dq
(1)
dp
(2)
,
(q, p)
2
ρ(q, p) =
Y
`
ρ
`
(q
(`)
, p
(`)
)
A(q, p)
A(q, p) = A
1
(q
(1)
, p
(1)
) · A
2
(q
(2)
, p
(2)
).
A(q, p)
A =
Z
ρ(q, p)A(q, p) dq dp =
Z
ρ
1
(q
(1)
, p
(1)
)A
1
(q
(1)
, p
(1)
) dq
(1)
dp
(2)
×
×
Z
ρ
2
(q
(2)
, p
(2)
)A
2
(q
(2)
, p
(2)
) dq
(2)
dp
(2)
= A
1
· A
2
,
A(q, p)
ρ(q, p)
q p A
∆A(q, p) = A(q, p) − A.
A(q, p)
D
A
= (∆A)
2
= A
2
− 2A · A + A
2
= A
2
− A
2
.
σ
A
=
√
D
A
δ
a
=
σ
A
A
.
n
` ` = 1, 2, ..., n
A(q, p) =
n
X
`=1
A
`
(q
`
, p
`
),
A
`
(q
`
, p
`
) `
A =
n
X
`=1
A
`
= n · a,
A
`
= a
D
A
= (A − A)
2
= (
P
`
(A
`
− a))
2
=
=
P
``
0
(A
`
− a)(A
`
0
− a) =
P
`
(A
`
− a)
2
= nb
2
.
`
0
6= `
(A
`
− a)(A
`
0
− a) = (A
`
− a)(A
`
− a) = (a − a)(a − a) = 0,
(A
`
− a)
2
= b
2
δ
A
=
√
nb
2
na
=
b
a
·
1
√
n
=
b
√
N
o
a
·
1
√
N
, δ
A
∼
1
√
N
,
N
o
= N/n
N ∼ 10
22
δ
A
∼ 10
−11
N
M
ρ(q, p)
6N
Mρ(q, p)
ρ = ρ(t, q(t), p(t)).
∂ρ
∂t
+ (ρv) = 0,
ρ(r(t), t) v(r(t), t)
t r(t)
M · ρ(q, p)
6N
6N q
j
, p
j
6N
j
,
j
∂(Mρ)
∂t
+
3N
X
j=1
∂
∂q
j
(Mρ ·
j
) +
∂
∂p
j
(Mρ ·
j
)
= 0
M
∂ρ
∂t
+
3N
X
j=1
∂ρ
∂q
j
j
+
∂ρ
∂p
j
j
+
3N
X
j=1
ρ
∂
j
∂q
j
+
∂
j
∂p
j
= 0.
∂
j
∂q
j
+
∂
j
∂p
j
=
∂
2
H
∂q
j
∂p
j
−
∂
2
H
∂p
j
∂q
j
= 0
∂ρ
∂t
+
3N
X
j=1
∂ρ
∂q
j
j
+
∂ρ
∂p
j
j
=
dρ
dt
= 0
j j
∂ρ
∂t
+ [ρ, H] = 0,
[ρ, H] =
3N
X
j=1
∂ρ
∂q
j
∂H
∂p
j
−
∂ρ
∂p
j
∂H
∂q
j
ρ H
ρ
∆q∆p ∆M
∆q∆p · (Mρ) = ∆M.
Mρ
∆q∆p ∆M
∂ρ
∂t
= 0
[ρ, H] = 0.
f
f = f(t, q(t), p(t)).
df
dt
=
∂f
∂t
+
3N
X
j=1
∂f
∂q
j
j
+
∂f
∂p
j
j
=
∂f
∂t
+ [f, H] .
f
df
dt
= 0 ,
∂f
∂t
= 0
[f, H] = 0.
ρ = ρ(q, p)
ln ρ(q, p) =
X
`
ln ρ
`
(q
(`)
, p
(`)
)
ln ρ
ρ
E = H(q, p) P M
ρ
E = H(q, p)
ρ = ρ(H(q, p)).
ρ(H(q, p))
h
3N
3N dqdp
dqdp
h
3N
.
N!
N
1/N!
N!
Z
ρ(H(q, p))dΓ = 1 , dΓ =
dqdp
h
3N
N!
,
ρ(H(q, p))
(q, p)
Γ(E
0
)
H(q, p) = E
0
dΓ(E
0
) =
∂Γ(E
0
)
∂E
0
dE
0
≡ g(E
0
)dE
0
H(q, p) = E
0
H(q, p) = E
0
+ dE
0
E
0
Z
ρ(E
0
)g(E
0
)dE
0
= 1, g(E
0
)dE
0
=
Z
E
0
≤H(q,p)≤E
0
+dE
0
dΓ
dE
0
w(E
0
) = ρ(E
0
)g(E
0
)
E
0
∼ ∆E w(E
0
)
w(E
0
)
∆E E = H(q, p)
∆E
w(E) = ρ(E)g(E)
ρ(E)g(E)∆E = 1.
∆E
∆W
∆W = g(E)∆E = ρ
−1
(E) = ρ
−1
(H(q, p)) .
S = k ln ∆W = −k ln ρ(E) = −k ln ρ(H(q, p))
k k = 1, 38 ·10
−23
·K
−1
S
∆W = exp
S
k
.
S
S = −k ln ρ(E) = −k ln
Y
`
ρ
`
(E
`
) = −k
X
`
ln ρ
`
(E
`
) =
X
`
S
`
.
E V
N
H
E
q p
H(q, p, V, N) = E
E, V, N
q p
ρ
mc
(H(q, p, V, N))
q p
H(q, p, V, N) = E
E
6N H(q, p, V, N) = E
6N − 1
ρ
mc
H(q, p, V, N)
ρ
mc
(H(q, p, V, N)) =
1
c
δ(E − H(q, p, V, N)).
1
c
δ(x)
δ(x) =
∞ x = 0
x 6= 0
Z
∞
−∞
δ(x)dx = 1.
c
1
c
Z
δ(E − H(q, p))dΓ = 1,
1
c
Z
∞
−∞
δ(E − E
0
)g(E
0
, V, N)dE
0
=
1
c
g(E, V, N) = 1, c = g(E, V, N).
h
3N
∆E
∆W (E, V, N) = g(E, V, N)∆E.
S(E, V, N) = k · ln ∆W (E, V, N) = k · ln(g(E, V, N)∆E),
g(E, V, N) =
1
∆E
exp(
S(E, V, N)
k
).
dE = T dS −pdV + µdN
dS =
1
T
dE +
p
T
dV −
µ
T
dN.
∂S
∂E
V,N
=
1
T
,
∂S
∂V
E,N
=
p
T
,
∂S
∂N
E,V
= −
µ
T
, F = E−T S