Смирнов В.П. Курс статистической физики. Конспект лекций
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κz
o
= mg
z
o
m
m =
κz
o
g
.
z
o
T
z
o
1
2
κ(∆z
o
)
2
=
1
2
kT, (∆z
o
)
2
=
kT
κ
.
∆m
∆m =
κ
g
q
(z
o
)
2
.
κ =
mg
z
o
=
10
−6
· 10
10
−2
= 10
−3
.
T = 300 K
(z
o
)
2
=
1.38 · 10
−23
· 300
10
−3
≈ 4 · 10
−18 2
, ∆m =
√
10
−3
· 4 · 10
−19
10
≈ 6 · 10
−12
,
∆m
m
=
6 · 10
−12
10
−6
= 6 · 10
−6
.
1
2
D(∆ϕ)
2
=
1
2
kT, (∆ϕ)
2
=
kT
D
.
D =
πr
4
N
2l
, N = 1.4 · 10
10
2
, r = 10
−6
l = 0.1 .
D =
3.14 · 10
−24
· 1.4 · 10
10
2 · 10
−1
≈ 2 · 10
−13
,
(∆ϕ)
2
=
300 · 1.38 · 10
−23
2 · 10
−13
≈ 2 · 10
−8
,
q
(∆ϕ)
2
≈ 1.4 · 10
−4
.
5 · 2 · 1.4 · 10
−4
= 1.4 · 10
−3
∆E
∆Q
−p∆V R
∆E = ∆Q − p∆V + R
∆Q = T ∆S
∆Q < T ∆S
R = ∆E −∆Q + p∆V > ∆E − T ∆S + p∆V = R
min
,
T p S
tot
= S
tot
(E
tot
)
E
tot
a
b −∆S
tot
∆S
tot
< 0
cb
c
b R
min
= (∆E
tot
)
c
|∆S
tot
|
tgα =
ab
bc
=
(−∆S
tot
)
∆E
tot
c
=
1
T
c
, ∆S
tot
= −
R
min
T
c
= −
1
T
c
(∆E − T
c
∆S + p
c
∆V ).
∆E ∆S ∆V
c b
T
c
p
c
a ∆W
a
= exp
S
a
k
∆W
a
b ∆W
b
= exp
S
b
k
w ∼
∆W
b
∆W
a
= e
S
b
− S
a
k
= e
∆S
tot
k
= e
−
∆E − T
c
∆S + p
c
∆V
kT
c
.
∆E
∆S ∆V
∆E = E
0
S
∆S + E
0
V
∆V +
1
2
(E
00
SS
(∆S)
2
+ 2E
00
SV
∆S∆V + E
00
V V
(∆V )
2
).
E
0
S
= T
c
E
0
V
= −p
c
∆S ∆V
∆S
tot
∆S
tot
=
1
2
(E
00
SS
(∆S)
2
+ 2E
00
SV
∆S∆V + E
00
V V
(∆V )
2
) =
=
1
2T
[∆S(T
0
S
∆S + T
0
V
∆V ) + ∆V (−p
0
S
∆S − p
0
V
∆V )] =
1
2T
(∆S∆T − ∆p∆V ),
w ∼ e
−
E
00
SS
(∆S)
2
+ 2E
00
SV
∆S∆V + E
00
V V
(∆V )
2
2kT
= e
∆p∆V − ∆S∆T
2kT
,
S V
∆S
tot
S V
∆S
tot
x ≡ (x
1
, x
2
, ..., x
n
)
t
i
= ∆x
i
x
i
w = exp
∆S
tot
k
= A exp
−
1
2
X
i,j
β
i,j
t
i
t
j
!
,
β
i,j
β
i,j
= −
∂
2
∆S
tot
∂t
i
∂t
j
t
i
,t
j
=0
A
−1
=
Z
exp
−
1
2
X
i,j
β
i,j
t
i
t
j
!
dt
1
dt
2
...dt
n
t
i
=
P
i
0
α
i,i
0
t
0
i
0
X
i,j
β
i,j
t
i
t
j
=
X
i,j
X
i
0
,j
0
β
i,j
α
i,i
0
α
j,j
0
t
0
i
0
t
0
j
0
=
X
i
0
,j
0
δ
i
0
,j
0
t
0
i
0
t
0
j
0
,
X
i,j
β
i,j
α
i,i
0
α
j,j
0
= δ
i
0
,j
0
, α
tr
βα = I.
∆
α
∆
β
α β
∆
2
α
∆
β
= 1
∆
α
= ∆
−1/2
β
J =
∂(t
1
, t
2
, ..., t
n
)
∂(t
0
1
, t
0
2
, ..., t
0
n
)
= ∆
α
A
−1
= J
n
Y
i=1
Z
∞
−∞
exp
−
t
2
i
2
dt
i
=
(2π)
n
∆
β
1/2
, A =
s
∆
β
(2π)
n
.
w =
s
∆
β
(2π)
n
exp
−
1
2
X
i,j
β
i,j
∆x
i
∆x
j
!
.
X
i
=
n
X
j
0
=1
β
i,j
0
x
j
0
.
∆X
i
=
n
X
j
0
=1
β
i,j
0
(x
j
0
− x
j
0
) =
n
X
j
0
=1
β
i,j
0
∆x
j
0
=
n
X
j
0
=1
β
i,j
0
t
0
j
= −
∂
∂t
i
∆S
tot
k
.
x
j
(j = 1, 2, ..., n)
s
∆
β
(2π)
n
Z
x
j
exp
−
1
2
X
m,l
β
m,l
(x
m
− x
m
)(x
l
− x
l
)
dx
1
dx
2
...dx
n
= x
j
.
x
i
s
∆
β
(2π)
n
Z
x
j
n
X
j
0
=1
β
i,j
0
∆x
j
0
exp
−
1
2
X
m,l
β
m,l
∆x
m
∆x
l
dx
1
dx
2
...dx
n
= δ
j,i
.
∆X
i
= 0
∆x
j
∆X
i
= δ
j,i
,
n
X
l=1
β
i,l
∆x
j
∆x
l
= δ
j,i
.
(β
−1
)
m,i
i
x
j
x
m
∆x
j
∆x
m
= (β
−1
)
j,m
, (∆x
j
)
2
= (β
−1
)
j,j
.
∆X
j
∆X
m
= β
j,m
, (∆X
j
)
2
= β
j,j
.
f = f(x
1
, x
2
, ..., x
n
) ϕ = ϕ(x
1
, x
2
, ..., x
n
)
∆f ≈
n
P
l=1
∂f
∂x
l
∆x
l
∆ϕ ≈
n
P
l=1
∂ϕ
∂x
l
∆x
l
∆f∆ϕ ≈
X
i,j
∂f
∂x
i
∂ϕ
∂x
j
∆x
i
∆x
j
=
X
i,j
∂f
∂x
i
∂ϕ
∂x
j
(β
−1
)
i,j
,
(∆f)
2
≈
X
i,j
∂f
∂x
i
∂f
∂x
j
∆x
i
∆x
j
=
X
i,j
∂f
∂x
i
∂f
∂x
j
(β
−1
)
i,j
ϕ
β
β
−1
n = 2 β ∆
β
= β
11
β
22
− β
2
12
β
−1
=
1
∆
β
β
22
−β
12
−β
12
β
11
.
T
V
∆p =
∂p
∂T
V
∆T +
∂p
∂V
T
∆V, ∆S =
∂S
∂T
V
∆T +
∂S
∂V
T
∆V.
∂S
∂T
V
=
1
T
T ∂S
∂T
V
=
1
T
∂Q
∂T
V
=
C
V
T
,
∂S
∂V
T
= −
∂
2
F
∂V ∂T
=
∂p
∂T
V
w ∼ exp
−
C
V
2kT
2
(∆T )
2
exp
∂p
∂V
T
(∆V )
2
2kT
.
x x
(∆x)
2
w(x) =
1
√
2π(∆x)
2
exp
−
(x − x)
2
2(∆x)
2
!
.
V T
w(T, V ) =
w
T
· w
V
(∆T )
2
=
kT
2
C
V
, (∆V )
2
= −kT
∂V
∂p
T
, ∆V ∆T = 0.
β
11
=
C
V
kT
2
, β
22
= −
1
kT
∂p
∂V
T
, β
12
= 0,
β
−1
11
=
kT
2
C
V
,
β
−1
22
= −kT
∂V
∂p
T
,
β
−1
12
= 0
T V
T p
∆V =
∂V
∂T
p
∆T +
∂V
∂p
T
∆p, ∆S =
∂S
∂T
p
∆T +
∂S
∂p
T
∆p.
Φ ≡ E − T S + pV = F + pV
T p
N
dΦ = −SdT + V dp − µdN.
∂S
∂p
T
= −
∂
2
Φ
∂p∂T
= −
∂V
∂T
p
,
∂S
∂T
p
=
1
T
T ∂S
∂T
p
=
1
T
∂Q
∂T
p
=
C
p
T
,
β =
1
kT
−
∂V
∂p
T
−
∂V
∂T
p
−
∂V
∂T
p
C
p
T
, ∆
β
=
1
k
2
T
2
−
C
p
T
∂V
∂p
T
−
∂V
∂T
2
p
!
.
(∆p)
2
= (β
−1
)
11
=
C
p
∆
β
kT
2
, (∆T )
2
= (β
−1
)
22
= −
∂V
∂p
T
∆
β
kT
,
(∆p)∆T = (β
−1
)
12
=
∂V
∂T
p
∆
β
kT
.
pV = NkT
∂V
∂T
p
=
Nk
p
,
∂V
∂p
T
= −
NkT
p
2
, C
V
=
3
2
Nk, C
p
=
5
2
Nk,
∆
β
=
1
k
2
T
2
C
p
T
NkT
p
2
−
N
2
k
2
p
2
=
3
2
N
2
p
2
T
2
,
(∆p)
2
=
5
2
Nk
2p
2
T
2
3N
2
kT
2
=
5
3
p
2
N
, (∆T )
2
=
NkT 2p
2
T
2
p
2
3N
2
kT
=
2
3
T
2
N
,
(∆p)∆T =
Nk
p
2p
2
T
2
3N
2
kT
=
2
3
pT
N
.
l I
−e v
n = N/V
i = −ej = −env
I = iS = −evnS = −
e
l
vN.
−
e
l
v
I = −
e
l
N
X
i=1
v
i
.
I = −
e
l
vN.
(∆I)
2
= (I − I)
2
=
e
2
l
2
(
N
X
i=1
v
i
− vN)
2
.
v
i
= v + ∆v
i
∆v
i
N =
N +∆N
(∆I)
2
=
e
2
l
2
(
N
X
i=1
v +
N
X
i=1
∆v
i
− vN)
2
=
e
2
l
2
(∆Nv +
N
X
i=1
∆v
i
)
2
.
(∆I)
2
=
e
2
l
2
(v
2
(∆N)
2
+ (∆v)
2
N) = (∆I )
2
+ (∆I )
2
.
(∆I )
2
(∆I )
2
q
(∆v)
2
1
2
m(∆v)
2
=
1
2
kT, (∆v)
2
=
kT
m
, N − nV.
E τ v
o
= 0
v
k
= −
eE
m
τ
v =
1
2
(v
o
+ v
k
) =
eE
2m
τ,
i = −env =
e
2
nτ
2m
E = σE,
n
m
=
2σ
e
2
τ
.
(∆I )
2
=
e
2
l
2
nV
kT
m
=
e
2
kT lS2σ
l
2
e
2
τ
=
2kT
l
σS
τ
=
2kT
Rτ
.
R
τ
∆f = (2τ )
−1
(∆I )
2
= 4
kT
R
∆f , (∆U )
2
= 4kT R∆f .
∆f < ∆f
(∆U )
2
= 4kT R∆f,
(P )
2
= 4kT ∆f.
T = 300 K R = 100 100
10
q
(∆U )
2
=
√
4 · 1.38 · 10
−23
· 300 · 10
4
≈ 0.13 .
(∆N)
2
= N
(∆I )
2
=
e
2
v
2
l
2
N = e
evN
l
1
l/v
=
eI
τ
.
τ = l/v
τ
2∆f
(∆I )
2
= 2eI∆f .
(∆U )
2
= 2eIR
2
∆f
P = 2eIR∆f .
∆f < ∆f
P = 2eIR∆f,
(∆U )
2
= 2eIR
2
∆f.
n ·
l = 1 S = 1
2
n = 10
15
−3
I = 10
R = ρ
l
S
= 10
0.1
0.01
= 100 .
I = e
vnS, v =
I
enS
, ∆f =
v
2l
=
I
2enV
.
∆f =
10 · 10
−3
2 · 1.6 · 10
−19
· 10
15
· 10
−3
= 30 > ∆f.
q
(∆U )
2
=
√
2 · 1.6 · 10
−19
· 10
−2
· 10
4
· 10
4
≈ 0.56 · 10
−6
= 0.56 .