Смирнов В.П. Курс статистической физики. Конспект лекций
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ρ(ξ
1
, ξ
2
, ..., ξ
N
; t) = ρ(ξ; t)
ξ
i
r
i
p
i
i ξ
η ≡ (r, p)
e
f(η; ξ) =
N
X
i=1
δ(η − ξ
i
) =
N
X
i=1
δ(r − r
i
)δ(p − p
i
).
Z
∆η
e
f(η; ξ)dη = ∆N
∆N ∆r
∆p t ∆r∆p = ∆η
ρ(...ξ
i
, ..., ξ
j
, ...) = ρ(...ξ
j
, ..., ξ
i
, ...)
f(η, t) =
Z
e
f(η; ξ)ρ(ξ; t)
dξ
N!h
3N
=
N
X
i=1
Z
δ(η − ξ
i
)ρ(ξ, t)
dξ
N!h
3N
=
= N
Z
ρ(η, ξ
2
, ξ
3
, ..., ξ
N
; t)
dξ
2
dξ
3
...dξ
N
N!h
3N
.
r p η = (r, p))
µ
Z
f(η, t)dη = N.
f(η; t)dη
dη µ
f(r, p) =
N exp
−
p
2
2mkT
(2πmkT )
3/2
·
exp
−
U(r)
kT
C
, C =
Z
exp
−
U(r)
kT
dr.
A
A(ξ) =
N
X
i=1
A(ξ
i
).
A(t) =
Z
A(ξ)ρ(ξ; t)dΓ =
N
X
i=1
Z
A(ξ
i
)ρ(ξ; t)dΓ =
=
N
X
i=1
Z
A(ξ
i
)
1
N
f(ξ
i
; t)dξ
i
=
Z
A(η)f(η; t)dη.
f(η; t)
n(r, t) =
Z
f(r, p, t)dp,
m · n(r, t) q ·
n(r, t)
j(r, t) =
1
m
Z
pf(r, p, t)dp.
6N
ρ(ξ) =
N
Y
i=1
f(r
i
, p
i
)
h
3
e
.
ln ρ =
N
X
i=1
ln[f(r
i
, p
i
)
h
3
e
],
ln ρ(H) = ln ρ(H)
S = −k ln ρ(H) = −k ln ρ(H) = −k
Z
ρ ln ρdΓ =
= −k
Z
f(r, p) ln[f(r, p)
h
3
e
]drdp.
f(r, p, t)
S(t) = −k
Z
f(r, p, t) ln[f(r, p, t)
h
3
e
]drdp.
µ ∆γ = ∆r∆p
α f(r, p, t) ≡ f
α
α N
α
= f
α
∆γ
S = −k
X
α
f
α
ln[f
α
h
3
e
]∆γ = k
X
α
N
α
ln[
∆γe
N
α
h
3
] =
= k
X
α
N
α
ln[
∆γ/h
3
N
α
/e
] = k
Y
α
G
N
α
N
α
!
= k ln ∆W,
G = ∆γ/h
3
α ∆W =
Q
α
G
N
α
N
α
!
f(r, p, t) N
α
!
N µ N
µ
µ
f(r, p, t)
µ
∂f
∂t
+
∂
∂r
(f ) +
∂
∂p
(f ) = 0.
=
∂H
∂p
, = −
∂H
∂r
,
=
p
m
, = F.
∂f
∂t
+
p
m
∂f
∂r
+ F
∂f
∂p
= 0.
F
F
ext
F
col
F = F
ext
+ F
col
F
col
∂f
∂p
= −
∂f
∂t
col
f(r, p, t)
∂f
∂t
+
p
m
∂f
∂r
+ F
ext
∂f
∂p
=
∂f
∂t
col
.
∂f
∂t
col
= −
f − f
o
τ
, f
o
=
1
(2πmkT )
3/2
exp
−
p
2
2mkT
,
f
o
∂f
∂r
= 0
F = 0
t = 0 f = f(p, 0)
∂f
∂t
= −
f − f
o
τ
.
f(p, t) = f
o
+ (f(p, 0) − f
o
) exp
−
t
τ
.
τ
τ
τ
T
τ
m
p p
1
p
0
p
0
1
m
p
2
+ p
2
1
= p
02
+ p
0
1
2
p + p
1
= p
0
+ p
0
1
.
R
c
R
R
c
=
1
2
(r + r
1
), R = r
1
+ r.
P
c
u
u
0
P
c
=
1
2
(p + p
1
) =
1
2
(p
0
+ p
0
1
),
u =
1
m
(p
1
− p), u
0
=
1
m
(p
0
1
− p
0
).
2 p · p
1
= 2 p
0
· p
0
1
.
(p
1
− p)
2
= (p
0
1
− p
0
)
2
, u = u
0
= F,
1
= F
1
, F = −F
1
m
2
= F
1
.
m
m
2
F
1
b
θ
j
dN(θ, u) dΩ =
2π sin θdθ
dN(θ, u) = jdΩσ(θ, u) = j · bdb,
σ(θ, u)
σ(θ, u) =
1
j
dN
dΩ
=
b(θ, u)
sin θ
∂b(θ, u)
∂θ
.
σ
o
(u) =
Z
σ(θ, u) sin θdθdϕ = 2π
Z
σ(θ, u) sin θdθ.
p
1
dp
1
p dr
1
dp
1
µ p
1
dp
1
u p
j = f(r, p
1
, t)dp
1
u ≡ f
1
dp
1
u.
jσ(θ, u)dΩ = f
1
dp
1
uσdΩ
p
dp
dp
1
dΩf
1
uσ · fdp.
p dp
z
1
dp = dpf
Z
dp
1
Z
dΩf
1
uσ
dp
p dp p
0
p
0
1
p p
1
p
0
dp
0
dp
0
1
dΩ
0
f
0
1
u
0
σ(θ
0
, u
0
) · f
0
dp
0
.
dpdrdp
1
dr
1
= dp
0
dr
0
dp
0
1
dr
0
1
, dpdp
1
= dp
0
dp
0
1
,
dr = dr
0
dr
1
= dr
0
1
u = u
0
θ = θ
0
u u
0
dp
1
dΩf
0
1
uσ(θ, u) · f
0
dp.
z
2
dp p dp
p
1
dΩ
z
2
dp = dp
Z
dp
1
Z
dΩσ(θ, u)uf
0
1
f
0
.
z
2
− z
1
=
∂f
∂t
col
=
Z
dp
1
Z
dΩσ(θ, u)u[f
0
1
f
0
− f
1
f],
∂f
∂t
+
p
m
∂f
∂r
+ F
ext
∂f
∂p
=
Z
dp
1
Z
dΩσ(θ, u)u[f
0
1
f
0
− f
1
f],
f ≡ f(r, p, t), f
1
≡ f(r, p
1
, t), f
0
≡ f(r, p
0
, t), f
0
1
≡ f(r, p
0
1
, t).
t
∂f
o
∂t
= 0
∂f
o
∂r
= −
f
o
kT
∂U
∂r
;
∂f
o
∂p
= −
p
mkT
f
o
.
∂f
o
∂t
+
p
m
∂f
o
∂r
+ F
ext
∂f
o
∂p
= −
p
m
f
o
kT
∂U
∂r
−
∂U
∂r
(−)
p
mkT
f
o
= 0.
f
o
[f
0
1
f
0
− f
1
f] = 0 p
2
+ p
2
1
= p
02
+ p
02
1
H
∂f
∂r
= 0 F
ext
= 0
∂f
∂t
=
Z
dp
1
Z
dΩσ(θ, u)u[f
0
1
f
0
− f
1
f].
H
H(t) =
Z
f(p, t) ln f(p, t)dp,
S
S = −kV H − kN ln
h
3
e
.
H
∂H
∂t
=
Z
∂f
∂t
ln fdp +
Z
∂f
∂t
dp =
Z
∂f
∂t
ln fdp,
Z
∂f
∂t
dp =
∂
∂t
Z
fdp =
∂
∂t
N
V
= 0.
H
1.
∂H
∂t
= −
Z
dp
Z
dp
1
Z
dΩσ(θ, u)u[f
1
f − f
0
1
f
0
] ln f.
p ↔ p
1
, p
0
↔ p
0
1
,
2.
∂H
∂t
= −
Z
dp
Z
dp
1
Z
dΩσ(θ, u)u[f
1
f − f
0
1
f
0
] ln f
1
,
p ↔ p
0
, p
1
↔ p
0
1
, u = u
0
dpdp
1
= dp
0
dp
0
1
,
3.
∂H
∂t
=
Z
dp
Z
dp
1
Z
dΩσ(θ, u)u[f
1
f − f
0
1
f
0
] ln f
0
,
p ↔ p
0
1
, p
1
↔ p
0
,
4.
∂H
∂t
=
Z
dp
Z
dp
1
Z
dΩσ(θ, u)u[f
1
f − f
0
1
f
0
] ln f
0
1
.
∂H
∂t
= −
1
4
Z
dp
Z
dp
1
Z
dΩσ(θ, u)u[f
1
f − f
0
1
f
0
][ln f
1
f − ln f
0
1
f
0
].
uσ(θ, u) ≥
0 [f
1
f − f
0
1
f
0
][ln f
1
f − ln f
0
1
f
0
] ≥ 0 f
0
1
f
0
≥ f
1
f f
0
1
f
0
≤ f
1
f
∂H
∂t
≤ 0,
∂S
∂t
≥ 0.
f
0
1
f
0
= f
1
f
T
∂f
∂t
+
p
m
∂f
∂r
+ F
ext
∂f
∂p
= −
f − f
o
τ
.
(r, p) f =
∆N
∆p∆r
∆N
∆p∆r µ
n
l
l
∆p
i
∆r
i
> h
i = x, y, z
∼ ∆r
i
∆p∆r µ ∆p
i
∆r
i
>> h i = x, y, z
∆p∆r µ
h
−3
∆p∆r
∆N = n
l
h
−3
∆p∆r
f
o
(r, p) =
∆N
∆p∆r
=
1
h
3
exp
ε(r, p) − µ
kT
± 1
−1
f
f = f
o
− τ
∂f
∂t
− τ
p
m
∂f
∂r
− τF
ext
∂f
∂p
.
τ f f
o
f − f
o
= ∆f
f f = f
o
+ ∆f ∼ τ ∆f
f = f
o
− τ
∂f
o
∂t
− τ
p
m
∂f
o
∂r
− τF
ext
∂f
o
∂p
.
f
o
n
f
f f
o
∂f
o
∂t
= 0
f = f
o
− τ
p
m
∂f
o
∂r
− τF
ext
∂f
o
∂p
.