F(X) =
: .
:
= (X) dX = 1
.
= (X) dX
N f(X),
(X)dX = N
, :
1. ;
2. ( ).
.
( ) . .
.1
1.1 ..
1.1.1 ( ).
..
.
.. 6 ; N 6 N .
..
= x z px py z = 3r · 3p
γ = 2 i N, i- .
= rk 3 pk
.. 3. N 3N.
= (*)
, (*) Ω.
1.1.2.
dt - τ ,
dW =
..
dW d, , .
(rk, pk) d3rk d3pk = 1
, . , , .
. . , .. .
,
dW1 = F1 (rk1, pk1) d3rk1 d3pk1
dW2 1 2
dW12 = dW1 · dW2
N
dW12 = dW1· dW2 . dWN
, .. .. .
1.2
(, N) . ,
N → 0 t → .
. :
|
|
X = (X - X ) = 0
, X
X = · F(p,q)dp·dq =
= · F1· F2 ·F3 Fn ·dq1·dq2·dq3 dqn·dp1·dp2·dp3·dpn
, Fi .
α = .
δ =
, δ = , - .
1.3
, , , , ( ).
= W i
= W i Wε(i)
() ℇ ( ).
:
, .
: s p .
:
1. , .
2. ℇ , .
1.4 (1.4.1) (1.4.2)
1.4.1.
Ω ( ) - , .
Ω .
N ,
Ω i = , = 3 r i · 3 p i, . ,
Ω2 = Ω(1)·Ω(2)
Ω( i )
ΩN =
, , m
Ω = Ω1 ·Ω2 ·Ω 3 ·Ωm =
, Ω= Ω(V,N,ℇ).
1.4.2.
,
S = k ,
. .
( ).
= = + + .+ =
:
S = S1 + S2 + S3 .+ Sm =
.
.
Ω = W = A
. 2
2.1
N :
S = - ,
ℇ = - .
ℇ = const, S = Smax
(), .
( ).
S < Smax, t.
, ..
|
|
.
: Q, dQ - , . .
2.2 .
2.2.1.
.
dℇ = δQ - δA (2.1)
( : Q = ℇ + A), - , . = ′ - ().
(2.1) - - . .
(N = const).
dℇ = δQ + δA′ + μdN (2.1′)
dN .
μ . . .
μ = () δQ=0, δA′=0
=0.
: , , . ( ).
, δQ = dS dA′ =- pdV,
, (2.1′)
dℇ = dS - pdV + μ dN (2.2)
.
2.2.2. (, )
ℇ S V.
,
ℇ = ℇ(S,V,N) S = S (ℇ,V,N)
(-).
dℇ = ()N,V dS + ()N,S dV + ()S,V dN
(2.2) :
= ()N,V, = - ()N,S,
, dV > 0 dℇ < 0.
2.3 .
dS.
S ; = 0.
δQ = 0 ( )
S = const
,
S = S2 S1 = (2.3′)
: = 0.
2.3.1 . - .
, , .. .
, , ( > 0).
: S 0, (2.3)
.
,
δS1 + δS2 = δS < 0
(2.3) . .
: . : .
2.3.2.
= C . , , .
:
( v= Cv ( = C : = v =
:
= = , .
(2.3′)
S =
V = const
δA′ = pdV =0, dℇ =dQ
ℇ = kNTÞ ( v= Cv kN = νRÞ = R
+ R = - . Þ = R
= γ - Þ pV γ = const
|
|
γ = , i - .
2.3.3. .
dS = dS1 + dS2 (*)
dS1 = dS2 (dS2 > 0)
: S = S2 - S1
dS (*) (2.2) :
dℇ dS - pdV + μ dN (2.4)
= , .
2.3.4. ( )
.
S. = + S0
, , ,
S. → 0, T→ 0 (2.5)
- , ( ) Ω =1 S =0.
2.3.5.
, . :
ℇ = ℇ1 + ℇ2 = const
V = V1 + V2 = const
N = N1 + N2 = const
: S = Smax
, dℇ1 = - dℇ2
,
1 = 2 (2.6)
.
,
1 = 2 μ1 = μ2
.. .
. 3
- .
3.1
( +), , Wi i - ℇ i Ni.
, ,
, S= S(ℇ,N, V).
,
Wi = ′ (*)
(*) :
Wi (ℇ i, N i) = (3.1)
.
, :
Wi (ℇ i) = B (3.2)
.
= [ -1 (**); C = [ ]-1 (***)
(**) (***) . ( ).
, N c ℇ i ℇ, ℇ + dℇ , :
Wi,N = ]-1 · (3.3)
(3.4) Nk N ℇ i .
3.2 - -
. .
() . . : 1). ;
2). : ( ), ( ).
, , Ω
|
|
= (h3N )· Ω
3.2.1 -
ℇ i.
f F = = · · ]-1}
.. N - 0 1 ( )
f F = [ 1+ ]-1 = [ +1]-1 (3.4)
(3.4) - -. , ( ).
f F . ℇ i.
(3.4):
→ 0 ℇ i < μ (- = 0 Þ
Þ f F =1
ℇ i > μ → 0 [ (+ +1]-1 = 0 Þ
Þ f F = 0 ( !)
> 0 , kT μ.
ℇ i = μ f F =
( 0,5)
μ < 0, ׀ kT ׀ μ.
ℇ i - μ kT
, :
f F = A(T)·
.
3.2.2. -
(), . .
f = · Wi,N = · [ ]-1
, ,
μ < 0 ( !). :
f = < Ni > = [ -1]-1 (3.5)
(→ 0) .. . ( ).
, = N
( !)
, f
f ·
f → ℇ → μ, .. ℇ.
f () . → 0, ׀ μ׀→ 0 , - ( ) .
, 1 (3.5) , f F.
:
f A()·
, .
.
3.3
ℇ i, , (, ).
g (ℇ) = γ , (3.6)
γ .
:
g (p) = γ , (3.7)
dℇ
dN (ℇ) = f F(ℇ)· g(ℇ) dℇ (3.8)
-.
,
F B(ℇ).
g (ℇ) g (p) .
3.3.1.
, ( )
ℇ i = =
.
= ,
V.
0 pi = p, , :
= V· , (3.9)
(3.9)
d = V· dp
, , dΩ = γ s ,
γ s - , ( ).
g (p) = = γ s V. (3.10)
,
k = V· (3.11)
ℇ (3.9)
|
|
ℇ = V· ℇ (3.12)
:
d k = V· dk
g (k) = = γ s V.
:
d ℇ = V·2 dℇ
g (ℇ) = = V· (3.13)
.
3.4
.4.1.
(3.8)
= f F(ℇ)· g(ℇ) · [ -1]-1
= f F(p)· g(p) p 2 · [ -1]-1
.
(3.8) N n ( μ =0)
ℇF = ()2 / 3 , (3.14)
n= N / V
3.4.2. -
= f B(ℇ)· g(ℇ)
. .
.4 .
4.1
, , .. . , 1 f F f B. :
f F = f B = f . = f
, 1. () .
f = (3.15)
. f = N (ℇi 1 i .
(3.15) , (3.2), C = .
4.2
() .
.
( ) , .
f F = f B (*)
,
( N (ℇi 1).
1 (**)
, 0 < ℇ < , :
1) μ / kT 1
2) μ < 0
3) ׀ μ ׀ kT
(3.8) N . :
μ = k T· ln (),
θ = (γs n)3 / 2
- , .
n nc - θ = θ. .
n nc = γs ()3 / 2
θ. .
,
λ l,
λ - l . .
4.3
(3.15)
Ni = 1
dΩ :
dN = Ni · g (ℇ)·dℇ = Ni · γs dΩ
.
μ = k T· ln )3 / 2 ] (3.16)
:
dN = · p
ℇ = ℇ + U()
dN = · p (3.17) |
U() - .
.
ℇ =( / 2 m) = - .
(3.17) :
= dW (, )= dW () dW ()(3.17, )
4.4 (..)
4.4.1 .. ,
(3.17, )
dW ()= d d d = F( ) p (3.18)
.
F( ) = (3.18, a)
F( ) = F( · F( · F( ,
:
F( =
, , ( !).
dW ()= F( )· d3 = F( )· d x y z
:
F( ) = (3.19)
F( ) = F ( x)· F ( y)· F ( z)
:
F( x) = (3.20)
. x z
1 - + F( x) .
4.4.2 ..
dΩ
dΩ = V· dp,
(3.18) (3.18, ),
:
F (p) = (3.21)
F (p) F () , p = m dp =md :
F () = (3.22)
. .
F (), :
υ =
=
()
υ = = ,
= , R - , .
:
=0 : = .
4.4.3 ..
.. - (3.17) dN dW - (3.15) - (3.16).
dΩ (3.13).
g (ℇ)
dW = F (ℇ) dℇ Þ F (ℇ) = (3.23)
(3.23) . .
, :
= kT ℇ = kT
3 = i - . , , , kT.
4.5
, .
- (3.17, ).
dW () = ·
F { U (x,y,z)} = .
, dW () =
dV = = dx·dy·dz :
dN = · dx·dy·dz
U(x,y,z) = U(z)= mgz. , no = n = :
n (x,y,z) = no (3.24)
. (p =nkT - ),
p (z) = no ,