290.Основы физической кинетики учебное пособие.pdf
TRANSCRIPT
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2006
Copyright & A K-C
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531
3673
K 89
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K 89 /.. ; . . -. : , 2006. 99 .
ISBN 5-8397-0501-2(978-5-8397-0501-2)
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, 510400
"" ( " ", ),
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531
3673
, 2006
ISBN 5-8397-0501-2(978-5-8397-0501- 2)
.. , 2006
Copyright & A K-C
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3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1. . . . . . . . . . . . 5
1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2. . . 6
1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.6. . . . . . . . . . . . . . . . . . . . 15
1.7. . . . . . . . . . . . . . . . . . . 18
1.8. . . . . . . . . . . . 19
1.9. . . . . . . . . . . . . . . . . . . . . . . 20
1.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3. . . . . . . . . . . . . . . . . . . . 32
2.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.6. . . . . . . . . . . . . . . . . . 46
3.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
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43.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1. . 64
4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4. . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.5. H- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6.
. . . . . . . . . . . . . . . . . . . . . . . . . 83
4.7. ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.8. . . . . . . . . . . . . . . . . . . . . . . 88
4.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.10. . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.11. . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.12. - . . . . . . . . . . 95
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
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1.
1.1.
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,
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, ,
,
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6
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, , , . -
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1.2.
, ,
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, , -
.
Vi, , - , -
. , Vi - , -
.
-
-
. , -
, ,
, .
, -
. ,
, -
|T | |T |/ , T - , ( ).
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7 |gradT | T/, (),
, T -
.
-
-
(, i) - :
duidt
= Tidsidt pidvi
dt
nk=1
i,kdNi,kdt
, (1)
-
.
ddt , ui, si , vi,, Ni,k
( ) , ,
.
ui = Ui/mi, si = Si/mi, vi = Vi/mi = 1/i,
Ni,k = Ni,k/mi =1
mk
k,ii
=ckmk
,
mi i- , mk k- .
(1)
si(t). V 0, u, p, , ck, k , :
s(~r, t) = s(u, p, c, ),
du
dt= T
ds
dt pd
1
dt
nk=1
kmk
d
dt(k
) (2)
S =V
s(~r, t)(~r, t)dV .
, -
,
. -
, -
, , -
.
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81.3.
, -
. -
: .
, ,
. -
, , -
. -
,
.
( ) -
(, , ),
(, , )
. Q = Q(t). , V - ,
Q =
V
G(~r, t)d~r =
V
(~r, t)G(~r, t)d~r.
G G Q, -
.
( , , ,
..) -
. ,
(1) ,
( -
). , G(~r, t) = (~r, t)G(~r, t).
Q(t)
~JG V - G,
Q
t=
~JG ~nd +V
GdV , (3)
~n - V , ~JG .
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9
: .
,
, , -
Q, .
d
dt
V0
Gd~r =
V0
G
td~r,
,
~r.
V0
[G
t+ div ~JG G]dV = 0.
V0 , - :
G
t= div ~J lG + G, (4)
Q. l , . C -
, G ,
dG
dt=G
t+G
xi
dxidt G
t+ ~v G, (5)
v0,i =dxidt . ,
~J lG = G~v DG, ,
, -
Q, D . ,
div ~J lG = G ~v +Gdiv~v div(DG)
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10
(4)
G
t= G ~v Gdiv~v + div(DG) + G. (6)
(4) (5) ,
dG
dt= G G( ~v) + div(DG). (7)
(5) , -
-
, ,
dG
dt G.
, -
vm x, y, z. -, d~r
dM . , -
J lm Jsm , ..
Jsm = Jlm ~vm 0.
JsQ Q -
~JsQ = ~JlQ G ~vm
d
dt
V0
Gd~r =
V0
Gd~r,
-
G.
V , ~vm.
V
Gd~r =
~JsQ d +
V
Qd~r.
,
G+ ~JsQ = Q,
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11
.
-
G =G
t (G ~vm).
,
dG
dt=
1
dG
dt G2d
dt. (8)
:
- , G G = G(~r), G .
.
div(G~v) + div(DG) + G = 0. (9)- G , G = 0:
G div(G(~v ~vm) + div(DG) = 0. (10)- G ,
~v G = 0; ~v = 0, :
~v G ; G = 0. G -
.
.
1.4.
. k - , k
k. (4):
kt
= k div ~J lk. (11)
~J lk(~r, t) k. k(~r, t) -. :
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12
1A1 + 2A2k1 3A3 + 4A4.
A3
forv3 = 3m3k111
22 3m3wforv3 .
1A1 + 2A2k2 3A3 + 4A4,
back3 = 3m3k233
44 3m3wback3 .
1, 2, 3, 4 , m3 - A3.
:
3 = 3m3(wforv3 wback3 ). (12)
,
A3, , R , R :
3 = m3
Rl=1
l3wl3. (13)
:
3t
= div ~J l3 +m3Rl=1
l3wl3. (14)
kt
= div ~J lk +mkRl=1
lkwlk. (15)
(5),
dkdt
= div(Dkk) kdiv(~vk) +mkRl=1
lkwlk. (16)
,
nk=1
mk
Rl=1
lkwlk = 0,
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k=1
k = 0. (17)
,
=n
k=1
k , ~Jl =
nk=1
~J lk , ~Jlk = k~vk Dkk,
t= div ~J l, d
dt= div(
Rk=1
(Dkk))Rk=1
kdiv(~vk). (18)
.
~vk(~r, t) k ~v0(~r, t), :
~J l,k = k~vk , ~J = ~v0 =
nk=1
~J,k =
nk=1
k~vk, (19)
~v0 =1
nk=1
k~vk. (20)
~v0 ,
.
~J,k = k~vk = k~v0 + k(~vk ~v0) k~v0 + k~k = k~v0 + ~Jdk , (21)
nk=1
~Jdk = 0, (22)
~k = ~vk ~v0 k- . (21) k-
( ), -
, .
(21) (15),
kt
= div(k~v0 + ~Jdk Dkk) +Kr=1
krwkr. (23)
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(17) (22)
t= div(~v0
nk=1
Dkk) (24)
:
d
dt=
t+ ~v0 = div(~v0) + div(
nk=1
Dkk). (25)
(18),
.
(16) (25)
d
dt
k
=1
{div(Dkk)kdiv(~vk~v0)+mk
Rl=1
lkwlk
kdiv(
nl=1
Dll)]}.(26)
(~r, t) = /, , (23):
d
dt=
d
dt
(
)=
2{div(~v0) div(
nk=1
Dkk)}+ 1
d
dt. (27)
t= div(~v0 D) + (),
d
dt= div(~v0) + divD) + ()
d
dt=
1
[() + div(D)]
2div(
nk=1
Dkk). (28)
, -
(~r, t), , , .
1.5.
W (t). w - m u,
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15
, ,
:
w = m + u.
m -
-
v2k/2, k(~r, t) :
m =n
k=1
k
(v2k2
+ k(~r, t)). (29)
-
,
:
w = 0 (30)
:
(w)
t= div( ~J l(w)). (31)
,
~Jkonvw = w~v0
( )
~JA
~Jq:
~J l(w) = ~Jkonvw + ~JA + ~Jq .
.
1.6.
,
k :
kd~vkdt
= ~Fk Pk,ixi
, (32)
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~Fk , k, ~Fe,k
~Fi,k, -
~Fk = ~Fe,k + ~Fi,k.
,
~Fe,k =~Fe,k/k k
~Fe,k = k (33)
kt
= 0. (34)
"" -
~Fk = ~ . , - ,
~F =1
nk=1
~Ft,k,
Pk - k- ,
Pk,i.j = pki,j + k,i,j.
pk , i,j -
, k,i,j , - .
(16),
d(k~vk)
dt= ~Fk Pk,i
xi+~vkdiv(Dkk)k~vkdiv(~vk)+~vkmk
Kl=1
lkwlk, (35)
k- . ,
d(~v0)
dt= ~F P +
nk=1
~vk{div(Dkk)kdiv(~vk)+mkKl=1
lkwlk}, (36)
d~v0dt
= ~F P +n
k=1
(~vk ~v0){div(Dkk) kdiv(~vk) +mkKl=1
lkwlk}.
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(32),
k2
dv2kdt
= ~vk (~Fk Pk,ixi
)
d(kv
2k
2 )
dt= ~vk (~Fk Pk,i
xi) +
v2k2{div(Dkk) kdiv(~vk) +mk
Kl=1
lkwlk}.
(37)
-
k - .
,
d
dt=k=1
[~vk (~Fk Pk,ixi
)+v2k2{div(Dkk)kdiv(~vk)+mk
Kl=1
lkwlk}].
(38)
=
/:
d
dt=
1
k=1
[~vk(~FkPk,ixi
)+v2k2{div(Dkk)kdiv(~vk)+mk
Kl=1
lkwlk}]
12
nk=1
(kv
2k
2) div[~v
nk=1
Dkk]. (39)
c = v20/2, -
:
dcdt
=~v0
[~F P +n
k=1
(~vk~v0){div(Dkk)kdiv(~vk)+mkKl=1
lkwlk}].
(40)
(39) (40) ,
, -
. (39) (40)
( )
.
(40)
dcdt
+ (P ~v) = ec + ic, (41)
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P ~v, ec "" ,
ec = ~v0 ~F ,ic ""
ic = P : ~v0 + ~v0[n
k=1
(~vk ~v0){div(Dkk) kdiv(~vk) +mkKl=1
lkwlk}],
- , -
~v k. , , , -
~v k, .
1.7.
k (16) k,
(33) (34), -
d
dt=
nk=1
kdiv(k~v0 + ~Jdk Dkk) +
nk=1
k
Kr=1
krwkr, (42)
d
dt=
nk=1
[kdiv(k~vk) (k )div(Dkk)]+
+n
k=1
(k )Rr=1
krwkr + div(v0),
t+div(~v0+
nk=1
k ~Jdk ) =
nk=1
(k~v0+ ~Jdk ) ~Fk+
nk=1
k
Rr=1
krwkr. (43)
, -
J = ~v0 +n
k=1
k ~Jdk ,
, -
.
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19
:
""
e = n
k=1
(k~v0 + ~Jdk ) ~Fk =
nk=1
~Fk ~vk
""
i =n
k=1
k
Rr=1
krwkr.
- , -
,
Rr=1
krwkr = 0.
1.8.
(43) (41) ,
m :
mt
+ div( ~Jm) = m, (44)
m =1
2v20 + ,
~Jm = P ~v0 + ~v0 +n
k=1
k ~Jdk , (45)
em = ~v0 ~F n
k=1
~Fk ~vk = n
k=1
~Fk (~vk ~v0), (46)
im = P : ~v0 + ~v0[n
k=1
~vk{div(Dkk) kdiv(~vk) +mkKl=1
lkwlk}]
[n
k=1 k~vk]2
2div[~v0
nk=1
Dkk] +n
k=1
k
Rr=1
krwkr. (47)
-
.
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1.9.
u
u
t= div ~J l(u) + (u).
(u) -, (29). (29)
(u) = (em + im), (48)
~J l(u) = ~J l(w) ~J lm.
~Jq ,
~J l(u) u~v:
~Jq = ~Jl(u) u~v.
~J l(u), - ,
u
t+ div ~Jq = (u),
-
:
u
t+ div ~Jsu = (u),
, -
~Jsu = ~Jq. (49)
.
du
dt+ div ~Jq = (u) + (u) ~v0.
, ""
.
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1.10.
(2)
(s)
t= div ~J ls + (s), (50)
(s) , ~J ls . ,
, (2) (28):
Tds
dt=du
dt+ p
d1
dt
nk=1
kmk
d
dt(k
) =
div~Jq
+(u) ~v0 1
nk=1
~Fk (~vk ~v0)+
+1
[P : ~v0 + ~v0[
nk=1
~vk{div(Dkk) kdiv(~vk) +mkKl=1
lkwlk}]
1[ [n
k=1 k~vk]2
2div[~v0
nk=1
Dkk] +n
k=1
k
Rr=1
krwkr]+
+p
2{t
+ ~v0 + div(~v0) div(n
k=1
Dkk)}
1
nk=1
{ kmk
[div(Dkk)kdiv(~vk~v0)+mkRl=1
lkwlk
kdiv(
nl=1
Dll)]}.(51)
(51)
(50). , , -
1
T Jq = Jq
T Jq 1
T,
kT (k~vk) = kk~vk
T (k~vk) k
T,
~J ls -
(50)
, :
~J ls =
~Jq n
k=1
kk~vk
T, (52)
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22
(s) =1
T[Rj=1
JjAj + Jq Xk +
nk=1
k~vk Xk ]. (53)
Aj , Xq , X
k -
. , .
j-
Aj n
k=1
kkj
, -
Jj j- .
Xq TT
, -
~Jq. ~Xu -
.
, -
~Fk, :
Xk ~Fk T(kT
);
~Jk. .
T(s) , - -
Jj, ~Jq, ~Jk - (53). -
T(s). , - -
,
:
Aj AjT, (54)
~Xq ~XqT
= ( 1T
), (55)
~Xk ~XkT, (56)
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(s) =Rj=1
JjAj + ~Jq ~Xq +n
k=1
~Jk ~Xk 0. (57)
, , -
, T(s).
, ,
.
(s), (54) (56), , .
. -
,
(54) (56). , -
, .
(57). (57), ,
, ,
. -
,
.
(57) -
. -
, -
.
(50)
Set
=
~JS ~nd
, -
;
Sit
=
V
(s)dV
, -
. :
S
t=Set
+Sit
(58)
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:
1. .
Set
= 0
S
t=Sit
.
,
:
S
t=Sit
0. (59) .
2. . :
. ,
:
S
t= 0,
Set
+Sit
= 0,
. , -
.
. ,
Set
< 0 |Set| > |Si
t|.
, -
,
S
t< 0.
. ,
Set
< 0 |Set| < |Si
t|.
-
,
,
S
t> 0.
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. , -
Set
> 0.
, -
, ,
S
t> 0
.
,
, -
, -
.
2.
2.1.
,
.
, .
y1, y2, y3,
, yn, y01, y02, y03, , y0n - , ,
:
S = S(y1, y2, y3, ..., yn).
-
S(y) = S(y0)+1
1!
k
(S
yk
)0
(yky0k)+1
2!
k,l
(2S
ykyl
)0
(yky0k)(yly0l )+
+ . , -
. -
, ..(S
yk
)0
= 0;
Copyright & A K-C
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26 (2S
ykyl
)0
= k,l < 0.
s =S
V
:
ds
dt=k
( syk
)0
+1
1!
k,l
(2s
ykyl
)0
yl+
+1
2!
l,m
(3s
ykylym
)0
ylym + dyk
dt,
yi = yi y0i S = S(y) S(y0)., (s
yk
)0
+1
1!
k,l
(2s
ykyl
)0
yl+1
2!
l,m
(3s
ykylym
)0
ylym+ = syk
,
= s =k
s
yk
dykdt
.
, -
S
S = 12
ni,j=1
i,jyiyj, (60)
i,j
i,j = j,i,
ds
dt=
ni=1
s
yi
dyidt .
(61)
:
Xi =s
yi(62)
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27
,
Ii =dyidt
(63)
.
ds
dt=
ni=1
XiIi. (64)
, Xi Ii , -
. ,
Ii Xi yi:
Ii = n
j=1
i,jyj
Xi =s
yi=
nj=1
i,jyj,
i,j i,j -.
1j,i ,
yj = nl=1
1j,l Xl.
,
Ii =n
j,l=1
i,j1j,l Xl =
nl=1
Li,lXl, i = 1, 2, 3, ..., n, (65)
Li,l =n
j=1
i,j1j,l .
Li,k () - , -
||L,|| ( , , -
..). L, ( 6= )
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28
,
. (65) - ,
.
-
.
Li,k - yi . - - yi,
yi(t+) = yi(t)+Ii = yi(t)+nl=1
Li,lXl = yi(t)+nl=1
Li,ls
yl.
, -
, -
. yi(t) - yi(t+ ), - . yi(t+ )yj(t) =
yi(t)yj(t)+nl=1
Li,lyj(t)syl
.
= Aesk .
A :n
Aesk d = 1.
n yj, .
yj(t)s
yl= A
n
yjs
yle
sk d =
= Ak
n1
yjesk ~el d~ Ak
n
yjyl
esk d.
(n 1)- , , - ,
Copyright & A K-C
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29
. ,
yjyl
= j,l ,
yi(t+ )yj(t) = yi(t)yj(t) kLi,j.
yj(t+ )yi(t) = yj(t)yi(t) kLj,i. ,
, i j j i,
yi(t+ )yj(t) = yj(t+ )yi(t)
Li,j = Lj,i. (66)
(66) -
, Ji,
i, Xj j, Jj Xi Li,j,
||Li,j|| . -
. ,
, -
.
.
dS =1
TdU +
p
TdV
Tdq
TdN.
T, p, , , , -
.
-
S
= S1(U1 + U,N1 + N,q) S1(U1, N1, 0)+
+St(Ut U,Nt N,q) St(Ut, Nt, 0). S1, U1, N1 , -
, St, Ut, Nt , - , U , N , q
Copyright & A K-C
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30
, -
. Nt/N ( , ,
) ,
(S
U
)=
1
T,
(S
N
)=
T,
(S
q
)=
T,
:
S
=U
T1 U
Tt 1T1
N +tTt
N 1T1
q +tTt
q
,
U dU, N dN, q dq,
dS
=
(1
T
)dU
(T
)dN
(
T
)dq. (67)
s =
(1
T
)U
(T
)N
(
T
)q. (68)
, q =eN , e . , e + = - . ,
,
~E = grad .
s =
(1
T
)U
( T
)q. (69)
-
, , jQ = U = Q ,XQ = grad(
1T ); jq = q, -
Xq = grad( T ) =~ET grad
(1T
).
2.2.
-
( , -
)
.
Copyright & A K-C
-
31
: ,
(T. Seeb ek, 1821), -
, -
, ,
;
( -
(J. Peltier, 1834)); -
,
(W.Thomson, 1856).
. :
I = q Q = U , :
~I = Li,i
[~E
T grad
(1
T
)]+ Li,qgrad(
1
T),
~Q = Lq,qgrad(1
T) + Lq,i
[~E
T grad
(1
T
)].
-
, Li,i/T = , (Lq,q Lx,x(q, i))/T 2 = , .
Li,q = Lq,i. (70)
~I = ~E + (Li,q T)grad( 1T
), (71)
~Q = T 2grad(1
T) + Li,q
~E
T. (72)
-
Li,q.
. I = 0,
I = 0 ,
= (Li,q T)T 2
T = Z T. (73)
Z , , .
Copyright & A K-C
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32
~Q = (T 2 Lq,i + Lq,iT
)grad(1
T).
. grad(
1T
)= 0 ,
~I = ~E.
Q = Lq,iE
T.
,
Q
I=Li,qT
= . (74)
.
-
,
Z = T
+
T. (75)
.
~E = 0,
. -
~I 6= 0 gradT 6= 0, ~E = 0. :
~I = (Li,q T)( 1T
),
~Q = T 2( 1T
).
~I
~Q=
(Li,q T)T 2
. (76)
2.3.
~B ,
. -
L,( ~B) = L,( ~B). ~B = 0 - L, .
Copyright & A K-C
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33
-
, .. . L,
: Ls,(~B) La,(
~B):
L,( ~B) = Ls,( ~B) + L
a,( ~B),
L,( ~B) = Ls,( ~B) La,( ~B).
Ls,( ~B) =1
2[L,( ~B) + L,( ~B)],
La,( ~B) =1
2[L,( ~B) L,( ~B)].
~Xi,
Ls(, ) ~Xi = As(, ) ~Xi +B
s(, )( ~Xi ~B) ~B, (77)
La(, ) ~Xi = Ca(, )[ ~Xi ~B], (78)
As(, ), Bs(, ) Ca(, ) ,
As(, ) = As(, ), Bs(, ) = Bs(, ), (79)
Ca(, ) = Ca(, ). (80)
, OZ (-
),
, -
L,
L, =
As(, ) Ca(, )B 0Ca(, )B As(, ) 00 0 As(, ) +Bs(, )B2
. (71), (72)
~B = 0
Copyright & A K-C
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34
~B 6= 0:
Ix = As(i, i)
ExT
+ Ca(i, i)BEyT
+ (As(i, q) As(i, i)) x
(1
T) +
+(Ca(i, q) Ca(i, i))B y
(1
T),
Iy = Ca(i, i)BEx
T+ As(i, i)
EyT (Ca(i, q) Ca(i, i))B
x(1
T) +
+(As(i, q) As(i, i)) y
(1
T),
Iz = (As(i, i) +Bs(i, i)B2)
EzT
+
+(As(i, q) +Bs(i, q)B2 (As(i, i) +Bs(i, i)B2)) z
(1
T) . (81)
Qx = (As(i, q) As(i, i))E
x
T (Ca(i, q) Ca(i, i))BE
y
T+
+(As(q, q) As(i, q)) x
(1
T) + (Ca(q, q) Ca(i, q))B
y(1
T),
Qy = (Ca(i, q) Ca(i, i))BE
x
T+ (As(q, q) As(i, q))
x
EyT
(Ca(q, q) Ca(i, q))B x
(1
T) + (As(q, q) As(i, q))
y(1
T),
Qz = (As(i, q) +Bs(i, q)B2)
EzT
+
(As(q, q) +Bs(q, q)B2 (As(i, q) +Bs(i, q)B2)) z
(1
T) . (82)
~E = (1e0+), 0 -, , e -
.
,
-
. , As(i, i)/T = x,x (As(q, q) As(i, q)) =
x,x -, Ca(i, i)/T = x,y, (C
a(q, q) Ca(i, q))/T 2 = x,y
.
As(i, q) As(i, i) x,x,
Copyright & A K-C
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35
Ca(i, q) Ca(i, i) x,y.
~E ~I, ~T , (81) (82) :
Ex = Rx,xIx +Rx,yBIy x,xT
x x,yBT
y,
Ey = Ry,xBIx +Ry,yIy y,xBT
x y,yT
y,
Ez = Rz,zIz z,zT
z, (83)
Qx = x,xIx + x,yBIy x,xTx x,yBT
y,
Qy = y,xBIx + y,yIy y,xBTx y,yT
y,
Qz = z,zIz z,zTz
. (84)
-
, (81) (82), -
:
Rx,x = Ry,y =x,x
2x,x + (x,yB)2;
Rx,y = Ry,x = x,y2x,x + (x,yB)
2;
Rz,z = T/(As(i, i) +Bs(i, i)B2);
x,x = y,y = x,xx,x + x,yx,yB2
T 2(2x,x + (x,yB)2)
;
x,y = y,x =x,xx,y alphax,yx,xT 2(2x,x + (x,yB)
2);
z,z = (As(i, i) +Bs(i, i)B2))T
(As(i, i) +Bs(i, i)B2);
x,x = y,y = x,x x,xx,x/T x,yx,yB2/T ;x,y = y,x = x,y x,xx,y/T x,yx,x/T.
-
, (83) (84) -
.
Copyright & A K-C
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36
2.4.
-
,
~B. - . -
, ,
, ,
.
.
1. ( -
Ey, Ix, - ) grad(T ) = 0 Iy = 0 (83)
Ey = Ry,xBIx. (85)
Rti, ,
Rti =EyIxB
= Ry,x.
2. ( -
Ey, Ix, -
) Iy = 0 , Qy = 0,Tx = 0 :
Ey = Ry,xBIx y,yT
y,
0 = y,xBIx y,yTy
. (86)
Rta, ,
Rti =EyIxB
= Ry,x y,xy,yy,y
.
3. ( -
, Ix
Bz) Iy = 0,Qy = 0,Tx = 0 (83):
Rta =
Ty
IxB=
y,xy,y
.
.
.
Copyright & A K-C
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37
1. grad(T ) =0, Iy = 0 (83)
Ex = Rx,xIx,Ez = Rz,zIz. (87)
Ix = R1x,xEx,
Iz = R1z,zEz. (88)
, -
( B2, ).
2.5.
, ,
. ,
, . -
, , -
,
.
.
1. -
, Ey, ,
X, ~I = 0 Tx = 0.
Ey = y,yT
y,
Qx = x,yBTy
. (89)
EyB
Qx=
y,yx,y
.
2. -
, Ey, ,
Copyright & A K-C
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38
X, ~I = 0 Qy = 0.
Ey = y,xBT
x y,yT
y,
Qx = x,xTx x,yBT
y,
0 = y,xBTx y,yT
y0. (90)
EyQxB
=y,yy,x y,xy,y
2x,x + 2x,yB
2.
3. -
,
X, ~I = 0 Qy = 0. (90) :
yTQxB
=y,x
2x,x + 2x,yB
2.
.
1. i ~I = 0 y(
1T ) = 0
Qx = x,xTx
. (91)
i = Qx/Tx
= x,x.
2. a ~I =0 Qy = 0 (90):
a = Qx/xT = x,x +2x,yxx
B2.
3. -
Qli -
Qx ~I = 0 Ty = 0.
(83) :
Ex = x,xT
x. (92)
Copyright & A K-C
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39
Qli = ExxT = x,x.
4. -
Qla Qx ~I = 0 Qy = 0. (83) (84) :
Ex = x,xT
x x,yBT
y,
0 = y,xBTx y,yT
y. (93)
Qla = ExxT = x,x +
x,yx,yx,x
B2.
3.
3.1.
, -
-
. -
, , -
, ..
, ,
. .
-
. -
, , ,
. -
, -
,
.
, -
F = U TS = U TS N , U
Copyright & A K-C
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40
, T , S , -
, N . F .
-
, , -
-
.
, , ,
.
, , -
bi (i = 1, 2, 3, . . . , s)
bit
+ div( ~Jb,i) = b,i, (94)
t , ~Jb,i, b,i , s , .
~Jbi = ~vibi Dibi, Di -
. Di , (94)
bit
= div(~vi)bi + ~vi bi +Di2bi + b,i, i = 1, 2, 3, . . . , s, (95) b,i
bi {bi}. - (95)
.
, - -
-
(95) .
, .
-
(95) -
,
. ,
{bi} {b1 , b2 , , bn} = {const}; (96)
Copyright & A K-C
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41
{~n b1 , ~n b2 , , ~n bn} = {const}, (97) ;
. ~n -
. - bi (97) - , ,
, .
(95) ,
-
.
3.2.
.
{Xi(~r, t)}, - ~r t, - (94) .
, -
- {0 6 r 6 l, = 1, 2, 3; 0 6 t < } Xi(~r, t) , .. .
1. , Xi(~r, t) -, > 0 t = t0 = (, t0), Yi(~r, t), |Xi(~r, t0)Yi(~r, t0)| < , - |Xi(~r, t) Yi(~r, t)| < t > t0. , . ,
Xi(~r, t) . 2. Xi(~r, t)
limx
|Xi(~r, t) Yi(~r, t)| = 0, (98)
, Xi(~r, t) . , ( ) ,
.
3.3.
,
, ,
Copyright & A K-C
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42
. (5.3)
, (95) ( i) . .
X(~r, t) (5.3). X(~r, t + ), , - . , -
. -
, , -
( ) .
n- , 1, 2, 3, ..., n, .
-
, .
, , -
, C -
. , C , > 0 > 0, X0, - , t0
C, t > t0 X C . C .
C t X C , C .
-
.
3.4.
-
. -
,
.
, , -
. , -
.
, -
(95). . ,
, 0() (- ) , ( = 0), , .
Copyright & A K-C
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43
, .
, -
( )
.
3.5.
bi, Jb,i (b, i) (94). - , -
, -
D , , .
- -
(94)
, . , Jb,i -
b,i bi - , . -
Di .
(95) :
dbidt
= Fi(b1, b2, b3, . . . , bs) +Di2bi, i = 1, 2, , s, (99)
Fi(b1, b2, b3, . . . , bs) ( - {bj}) - bi. (99) - . (99) -
.
, -
, .
bi -
, .
,
, -
. -
, , -
. , -
Copyright & A K-C
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44
, , -
-
.
, -
:
, , . -
,
.
. (99). (99): b1 = b1,st, b2 = b2,st,b3 = b3,st, . . . , bs = bs,st.
dbi,stdt
= 0. (100)
-
.
bi = bi,st + yi(t), (101)
yi(t) .
. (101) (99):
dyidt
= Fi(b1,st + y1, b2,st + y2, b3,st + y3, . . . , bs,st + ys). (102)
, Fi(b1,st + y1, b2,st + y2, b3,st + y3, . . . , bs,st + ys) - yi -
.
, (99)
yit
=s
j=1
ai,jyj +Di2yi, i = 1, 2, 3, . . . , s, (103)
Copyright & A K-C
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45
ai,j =
(Fibj
)st
,
bi -
.
-
, (99), ,
, ..
| yibi,st
| 1,
(..
(99)) (103).
,
(99) (103).
. (103)
, {bi,st} - (99). , {bi,st} .
(103)
yi(t) =s
j=1
Ai,jejt; i = 1, 2, . . . , s, (104)
Ai,j j , t.
j
det {Ai,j ki,j} = 0, k = 1, 2.3, . . . , s. (104) -
. j , - , .. ,
.
, -
(99) . (Thom
R. Stabilite Struturelle et Morphogenese. Benjamin, New York, 1972) -
,
Copyright & A K-C
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46
, (99)
Fi V :
Fi({bj}) = V ({bj})bi
.
, -
(99)
. -
" -
", .
3.6.
,
. -
.
(99) b1 X b2 Y
dX
dt= fx(X, Y ),
dX
dt= fx(X, Y ) . (105)
t (105),
dY
dX=
fy(X, Y )
fx(X, Y ). (106)
() -
(X, ). -
(105).
, -
. , -
-
.
(X, Y ) - dY/dX , ,
fx(X, Y ) = fy(X, Y ) = 0.
. ,
(105) -
. .
Copyright & A K-C
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47
, -
, -
(Xst, Yst). - [., , (103):
dx
dt= ax,xx+ ax,yy,
dy
dt= ay,xx+ ay,yy . (107)
ai,j =
(fij
)Xst,Yst
i = x, y; j = X, Y.
X = Xst + x,
Y = Yst + y .
(107)
x = x0ewt, y = y0e
wt. (108)
. -
(107), -
x0, y0. , -
x0, y0, .
w2 Tw + = 0, (109)
T = ax,x + ay,y,
= ax,xay,y ay,xax,y .
(109) w1 w2. (109) :
x = x01ew1t + x02e
w2t,
y = y01ew1t + y02e
w2t , (110)
Copyright & A K-C
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48
yi0 = xi0wi ax,xax,y
, i = 1, 2,
x1,0 x2,0 .
:
, Re(wi) 0, i = 1 2, (Xst, Yst).
,Re(wi) = 0, i = 1 2, - ,
, .
.
, (109) , . . T , ,
. ,
(110) , -
.
;
: T 2 4 0. > 0, wi .
(110) ,
( ).
. . 1.
6
-
I
R S
Y
X
. 1.
Copyright & A K-C
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49
, -
. ax,y =ay,x = 0, ax,x = ay,y = a 6= 0;
dx
dt= ax,
dy
dt= ay .
, -
S a < 0 S a > 0. S , -
(. 2).
6
-
@@
@
@@
@
@@I
@@R SY
X
. 2.
T 2 = 0; dx
dt= ax + ay,
dy
dt= ay, a 6= 0 .
x = x0eat + ay0e
at, y = y0eat.
t, ,
y = 0 x = y ln y+ cy, c ; . 3.
.
6
-
-
66
? ?
SY
X
. 3. -
Copyright & A K-C
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50
< 0, () w - . (110)
ew1t =y x(w2 ax,x)x0,1(w1 w2) , e
w2t =y x(w1 ax,x)x0,2(w2 w1) .
w1 > 0 w2 < 0, :(
y x(w2 ax,x)x0,1(w1 w2)
)|w2/w1|=y x(w1 ax,x)x0,2(w2 w1) .
6
-
@@
@
@@
@
@@R
@@I
RR
II
SY
X
. 4. -
,
, .
: x02 = 0 x01 = 0. w1ax,x w2ax,x. . 4 (x, y). , t , . ,
.
.
- : T 2 4 < 0 T 6= 0,
. (110) , (T 0) . .
. 5.
6
-
?
6
-
SY
X
. 5. -
Copyright & A K-C
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51
T = 0, > 0, , wi = i. - -
, , -
.
, ,
(. . 6). ,
, -
.
6
-
&%'$b??
66 SY
X
. 6.
, -
3.7.
-
,
-
. ,
, , . -
, -
, , ,
.
,
.
-
, .
-
, .
3.8.
m ,
V (x) = m(2x2 +
1
4x4).
Copyright & A K-C
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52
:
d2x
dt2= (x+ x3),
:{dX1dt = X2;dX2dt = (X1 X31).
:
1. X1 = 0, X2 = 0;
2. X1 = +,X2 = 0;
3. X1 = ,X2 = 0.
: {dX1dt = X2,
dX2dt = X1.
:
X1 = A1et + A2e
t; X2 = A1et A2et, =
.
wi , .
: {dX1dt = X2,
dX2dt = 2X1,
:
X1 = A11e
i1t + A12ei1t; X2 = i1(A11e
i1t A12ei1t), 1 =
2.
, -
.
, = 0 .
-
-
.
Copyright & A K-C
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53
n- -
n p, -. :
N 0 N+. N 0 N+ . , -
. L. 0z p , 0x 0y - . .
, -
. , -
,
-
~E, ,
Ld.- -
-
z: jnz = evsnn; jpz = evspp. vnz vpz
.
:
~jn = en(n~E +kT
en);
~jp = ep(p ~E kTep).
n,p , E - .
-
n, p, - N+ :
n
t 1e~jn = n; (111)
p
t+
1
e~jp = n +; (112)
N+
t= +. (113)
, N+ + N 0 = N .
Copyright & A K-C
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54
n, + -:
n = np+ [c1(N N+) c2nN+];+ = [c4p(NN+)c3N+]+[c1(NN+)c2nN+]+[s2p(NN+)s1N+]p., , c1, c2 -
, c3, c4 (
) -
. s1, s2 ,
( ). -
- -
~E, - s1, ,
, -
() .
( , ) -
,
~E = ~Eq + ~E
.
~Eq , -
,
,
~E , . -
, -
. ,
~Eq -
, , -
Na = p0+Nod+N+0 n0. N0d
, p0, N+0 n0
,
.
~E :
~E = e(p+N+ +N d n), (114)
N d = Nd Na. (111) (114), - ,
.
+ = 0, ~jn +~jp = ~j = const ~jn = en. (115)
Copyright & A K-C
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55
+ = 0
N+ = N/[s1p+ c2n+ c3s2p2 + c4p+ c1
+ 1] , (116)
n = np+N [c1(s1p+c3)c2np(s2p+c4)]/[s2p2+(c4+s1)p+c2n+c1+c3].(117)
:
=c1c3c2c4
c3c4
=s1s2, (118)
, n(p) - :
n = [
np][+Nc2(s2p+ c4)/(s2p2 + (c4 + s1)p+ c2n+ c1 + c3)]. (119)
,
n =j
evsn vspvsn
p. (120)
(119)
n = vspvsn
(p p1)(p p2)(p p3)(p p4)(p p5)(p p6) , (121)
p1,2 =j
2evsp
(j
2evsp)2 vsn
vsp
2e
vsnvsp, (123)
0 < p1,2 6j
evsp /,
p1 p2.
p3,4 = AA2 B, (124)
A =1
2(c4s2
+s1s2
+Nc2 c2s2
vspvsn
),
B =c2s2
j
evsn+c1s2
+c3s2
+Nc2c4s2
> 0.
p5,6 = A1 A21 B1, (125)
Copyright & A K-C
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56
A1 =1
2(c4s2
+s1s2 c2s2
vspvsn
),
B =c2s2
j
evsn+c1s2
+c3s2
> 0.
, A2B , p3 p4 - A > 0 A < 0. , .
s2, , s2 - s2 6 s2,1 s2 > s2,2, ,
s2 s2,1 6 s2 6 s2,2.
s2,1 = 2
(c3c4 +Nc2 )
2> 0, s2,2 = 2
+
(c3c4 +Nc2 )
2> 0, (126)
= c2j
evsn+ c1 +
1
2(vspc2vsn
+ c4)(c3c4
+Nc2
),
= (c2j
evsn+ c1 + c3 +N
c2c4
)(c2j
evsn+ c1 +
c2c4
vspvsn
(c3 +Nc2c4
)).
np p = p1
np|p=p1 = s2
vspvsn
(p1 p2)(p1 p3)(p1 p4)s2p21 + (c4 + s1)p1 + c2n1 + c1 + c3
< 0 . (127)
p = p2:
np|p=p2 = s2
vspvsn
(p2 p1)(p2 p3)(p2 p4)s2p22 + (c4 + s1)p2 + c2n2 + c1 + c3
> 0. (128)
n(p = 0) = +Nc1c3
c2j
evsn+ c1 + c3
> 0, (129)
n(p =j
evsp) = +N
c1(c3 + s1j
evsp)
s2(j
evsp)2 + (c4 + s1)
jevsp
+ c1 + c3> 0. (130)
(112) :
4n=1
Mn ln(p pnp0 pn ) =
vspvsn
x,
Copyright & A K-C
-
57
p0 ,
M1 =f1
(p1 p2)(p1 p3)(p1 p4) ,
M2 =f2
(p2 p1)(p2 p3)(p2 p4) ,
M3 =f3
(p3 p1)(p3 p2)(p3 p4) ,
M4 =f4
(p4 p1)(p4 p2)(p4 p3) ,
fn = (pn p5)(pn p6), n = 1, 2, 3, 4. , ( -
)
= e(1 +vspvsn
)p jvsn
+ eN/[(s1 c2 vspvsn )p+ c2
jevsn
+ c3
s2p2 + c4p+ c1+ 1].
, p1,2 p3,4 , -
p x - , ,
.
_n = n0p (p0 + c2N+0 )n (c1 + c2n0)N+0 ,
+ = ((c4 + 2s2p)(N N+0 )s2c3c4
N+0 )p c2N+0 n
((1 + s2c4p0)(c4p0 + c3) + c2n0 + c1)N
+.
p0, n0 N+0 ,
z.
,
div( ~E) =4e
(p+ N+ n).
n
t nkT
e2n = (n0 nn0
4e
)p (p0 + c2N+0 + nn0
4e
)n
(c1 + c2n0 nn04e
)N+,
Copyright & A K-C
-
58
p
t+N+
t n
t kT
e(p2p n2n) =
= (nn0 + pp0)4e
(n p N+),
N+
t= ((c4 + 2s2p)(N N+0 )
s2c3c4
N+0 )p c2N+0 n
((1 + s2c4p0)(c4p0 + c3) + c2n0 + c1)N
+.
, -
(z = 0):
n = n1 exp(t+ i~ ~r),p = p1 exp(t+ i~ ~r),
N+ = N+1 exp(t+ i~ ~r). .
-
:
[n0 nn04e
]p+ [p0 + c2N+0 + nn0
4e
+ n
kT
e2 + ]n+
+[c1 + c2n0 nn04e
]N+ = 0,
[(nn0 +pp0)4e
+kT
ep
2+]p [(nn0 +pp0)
4e
+kT
en
2+]n+
+[(nn0 + pp0)4e
+ ]N+ = 0,
[(c4 + 2s2p)(N N+0 )s2c3c4
N+0 ]p c2N+0 n
[(1 + s2c4p0)(c4p0 + c3) + c2n0 + c1 + ]N
+ = 0.
,
,
3 + A12 + A2+ A3 = 0,
-
-
, .. ,
.
Copyright & A K-C
-
59
3.9.
-
. -
. ,
. -
.
. -
.
,
, -
,
.
. -
, -
:
, ;
- .
,
. , -
, .. .
, -
.
, -
, ,
, -
.
-
, -
.
, , : ,
-
, ,
.
,
Copyright & A K-C
-
60
,
, -
.
,
-
, , -
- ,
, - , ,
. , , , -
, ,
, .
,
. , -
,
,
( ) .
, -
. ,
,
. , -
,
.
, , , - -
, , ..
, . -
. ,
-
, ,
; .
, ,
.
, -
.
. ,
, (, ,
.. ),
. , .. , -
Copyright & A K-C
-
61
x + y(z 1 + x2) + x, y = x(3z + 1 x2) + y,z = 2z( + xy) = 1, 1, = 0, 87,
, 2, 3180, 002, .. , .
3.10.
,
.
:
d = lima0
lnN
ln 1/a, (131)
N -
, ; a . d , -
.
, -
. . n
, n2 a = 1/n. (131), ,
:
d = limn
lnn2
lnn= 2.
, , -
.
, ,
-
(" ")
.
[0, 1,
(1/3, 2/3).
. -
.
, , -
. k- 2k , , 3k . k ,
Copyright & A K-C
-
62
. -
:
l =1
3+
2
32+
22
33 = 1
2
k=1
(2
3
)k=
1/3
1 2/3 = 1.
,
:
,
(i 3) . . , -
N = 2k = 1/3k -. , k = 0 N = 1, = 1. k = 1, N = 2, = 1/3; = 2 N = 4, = 1/9, k = m N = 2m, = 1/m. k . , -
(131),
d = limm
(ln 2m/ ln 3m) = ln 2/ ln 3 0, 631.
.
, ,
, .
(k = 1) -.
( , k = 2) .. k .
-
.
k = 1, N = 8 = 81, = 1/31,
k = 2, N = 8 8 = 82, = 1/32,k = 3, N = 8 8 8 = 83, = 1/33,
, , ,
k = m, N = 8m, = 1/3m ,
d = limm
(ln 8m/ ln 3m) = ln 8/ ln 3 1, 893.
Copyright & A K-C
-
63
, ,
, ,
. - "" -
.
4.
-
, -, , -
( )
, , -, - -
.
,
. -
.
f(~r, ~p, t) - . f ~r ~p, t. f(~r, ~p, t) - N
f(~r, ~p, t)d~rd~p = N,
n(~r, t)
n(~r, t) =
f(~r, ~p, t)d~p.
-
(t) =
(~r, ~p, t)f(~r, ~p, t)d~rd~p
(~r, t) =
(~r, ~p, t)f(~r, ~p, t)d~p.
-
,
"" "". , -
.
, -
r0 (108 107). r0
Copyright & A K-C
-
64
, -
.
n > r30 , "",
.
n < r30 , "", , -
. -
.
"" .
4.1. .
.
,
, -
. -
dw = W (X, Y |, t)dY X = {~r1, ~p1} dY = d~r2d~p2 Y = {~r2, ~p2} , t.
, W (X, Y |, t) :W (X, Y |, t)dY = 1, (132)
f(X, t)W (X, Y |, t)dX = f(Y, t+ ). (133) X Y
Z = {~r3, ~p3} 1 + 2, 1 X Z, 2 Z Y .
, W (X, Y |, t) - , , -
.
W (X, Y |1 + 2, t) =
W (X,Z|1, t)W (Z, Y |2, t+ 1)dZ. (134)
Copyright & A K-C
-
65
.
-
, .. t t. , X Y Y X , -
, , ..
W (X, Y |, t) = W (Y,X|, t). .
4.2.
-
, .
, -
.
.
: -
-
.
. -
-
, .
,
.
g(X), - : g(X) - |X|
g(X) 0, g(X)Xi
0, 2g(X)
XiXj 0 , . . . . (135)
(134) g(X) X:
g(Y )W (X, Y |1 + 2, t)dY =
Copyright & A K-C
-
66
=
g(Y )
W (X,Z|1, t)W (Z, Y |2, t+ 1)dZdY.
g(X) Z:
g(Y ) = g(Z) +6
i=1
g(Z)
Zi(Yi Zi) + 1
2
6i,j=1
2g(Z)
ZiZj(Yi Zi)(Yj Zj) + ...,
-
g(X): {ng(Y )
YiYjYk...
}Y=Z
=
{ng(Z)
ZiZjZk...
}.
g(Y )W (X, Y |1 + 2, t)dY =
=
g(Z)W (X,Z|1, t)W (Z, Y |2, t+ 1)dZdY+
+
6i=1
g(Z)
Zi(Yi Zi)W (X,Z|1, t)W (Z, Y |2, t+ 1)dZdY+
+
1
2
6i,j=1
2g(Z)
ZiZj(Yi Zi)(Yj Zj)W (X,Z|1, t)W (Z, Y |2, t+ 1)
dZdY. (132), (
Z Y ) g(Y )(W (X, Y |1 + 2, t)W (X, Y |1, t))dY =
= 2
6i=1
g(Z)
Ziai(Z, 2, t+ 1)W (Z, Y |1, t)dZ+
+21
2
6i,j=1
2g(Z)
ZiZjbi,j(Z, 2, t+ 1)W (Z, Y |1, t)dZ,
ai(Z, 2, t+ 1) =1
2
(Yi Zi)W (Z, Y |2, t+ 1)dY
Copyright & A K-C
-
67
bi,j(Z, 2, t+ 1) =1
2
(Yi Zi)(Yj Zj)W (Z, Y |2, t+ 1)dY,
(132) W (Z, Y |1, t)dY = 1.
ai(Z, 1, t) 1 Z -
t. bi,j(Z, 1, t) i- k-
1. 1 0 2 ,
{W (X,Z|, t)
g(Z)
6
i=1
ai(X, t)W (X,Z|, t)g(Z)Zi
6
i,j=1
bi,j(X, t)W (X,Z|, t) 2g(Z)
ZiZj...}dZ = 0,
ai(Z, t) = lim20
ai(Z, 2, t),
bi,j(Z, t) = lim20
bi,j(X, 2, t),
, 6i=1
ai(Z, t)W (X,Z|, t)g(Z)Zi
dZ =
=6
i=1
ai(Z, t)W (X,Z|, t)g(Z)dZ i|Zi=+Zi=
6
i=1
g(Z)
Zi[ai(Z, t)W (X,Z|, t)]dZ.
-
-
- g(X) .
Copyright & A K-C
-
68
6i,j=1
bi,j(Z, t)W (X,Z|, t)
2g(Z)
ZiZjdZ =
6i,j=1
g(Z)
Zibi,j(Z, t)W (X,Z|, t)dZ j|Zj=+Zj=
6
i,j=1
g(Z)
Zj[bi,j(Z, t)W (X,Z|, t)]dZ i|Zi=+Zi=+
+6
i,j=1
g(Z)
2
ZjZi[bi,j(Z, t)W (X,Z|, t)]dZ =
=6
i,j=1
g(Z)
2
ZjZi[bi,j(Z, t)W (X,Z|, t)]dZ.
g(Z){W (X,Z|, t)
6i=1
Zi[ai(Z, t)W (X,Z|, t)]
12
6i,k=1
2
ZkZi[bi,k(Z, t)W (X,Z|, t)]}dZ = 0.
g(X) ,
W (X,Z|, t)
=6
i=1
Zi[ai(Z, t)W (X,Z|, t)]
12
6i,k=1
2
ZkZi[bi,k(Z, t)W (X,Z|, t)]. (136)
W (X,Z|, t). f(X, t) X (133), f(Z, t+ ):
f(Z, t+ )
=
6i=1
Zi[ai(Z, t)f(Z, t+ )]
Copyright & A K-C
-
69
12
6i,k=1
2
ZkZi[bi,k(Z, t)f(Z, t+ )]. (137)
t , (136) (137)
t = 0.
-
:
f(Z, t)
t+
6i=1
jiZi
= 0, (138)
ji = ai(Z, t)f(Z, t) 12
6k=1
Zk[bi,k(Z, t)f(Z, t)].
. x, y, z - , .
, ,
, -
:
f(~r, t)
t+ div~j = 0, (139)
~j = ~a(~r, t)f(~r, t) 12div[b(~r, t)f(~r, t)],
~a(~r, t) , b(~r, t)
.
-
.
.
.
-
-
:
W (~r1, ~r2|, t) = W (|~r1 ~r2|).
Copyright & A K-C
-
70
~a(~r, t) = lim0
1
W (|~r ~r2|)(~r ~r2)d~r2 = 0
bi,j(~r, t) = lim0
1
W (|~r ~r2|)(ri r2i)(rj r2j)d~r2 = bi,j,
b = lim0
1
W (|~r|)x2d~r = const.
:
f
t=
b
22f,
f
t= D2f. (140)
b
D:
b = 2D.
t = 0 ~r0, - (140) :
f(~r, ) = (4D)1/2e|~r~r0|
2
4D .
(x)2 =
(x x0)2f(~r, )d~r = 6D,
..
.
. .
, -
~a 6= 0. b , -
,
~j:
~j = ~af b2f.
Copyright & A K-C
-
71
~j = 0 :
f(~r) = AeU(~r)kT .
,
~j = (~a+b
2kTU(~r))eU(~r)kT = 0.
~a = b2kT
U(~r) = DkTU(~r) = q ~F ,
q = DkT ,~F = U(~r) .
-
, -
.
4.3.
,
.
-
, -
. ,
X Y t :
W (X, Y |, t) = A(X, t)(X Y ) + P (X, Y, t). P (X, Y, t) X Y t , A(X, t)(X Y ) "" X Y , -, -
X. (132) ,
A(X, t) = 1
P (X, Y, t)dY.
,
W (X, Y |, t) = {1
P (Z, Y, t)dZ}(X Y ) + P (X, Y, t). (141)
Copyright & A K-C
-
72
, f(Y, t + ) Y t+ f(X, t) X t (133). (132) :
f(Y, t+ ) f(Y, t) ={W (X, Y |, t)f(X, t)W (Y,X|, t)f(Y, t)}dX.
(141) 0, :
f(X, t)
t=
{P (X, Y, t)f(Y, t) P (Y,X, t)f(X, t)}dY . (142)
- -
( ):
i(t)
t=j=1
{Pi,j(t)j(t) Pj,i(t)i(t)}. (143)
, Pi,j(t) Pj,i(t) - j(t), - (142) (143) , -
, -
.
, -
. 1916
. ,
.
,
1 2 (1 < 2), N1 N2. 1 < 2, 1 2 , 1 2 - . ,
(, T ) = (2 1)/h:
B1,2N1(, T ).
:
Copyright & A K-C
-
73
1. (), A2,1N2,
2. ( ),
(, T ), .. - B2,1N2(, T ).
,
B2,1N2(, T ) + A2,1N2 = B1,2N1(, T )
(, T ) =A2,1B2,1
/[N1N2 B2,1B1,2
].
N1N2
=g1g2e(21)/kT ,
g1 g2 .
:
g1B1,2 = g2B2,1,
, 2 1 = h,
(, T ) =g2A2,1g1B2,1
/[ehkT 1].
(, T ) = 3f(
T).
g2A2,1g1B2,1
= A3.
(, T ) = A3/[ehkT 1].
h kT :
(, T ) =8
c3kT2.
:
(, T ) =8h
c33/[e
hkT 1].
Copyright & A K-C
-
74
4.4.
-
.
. -
~r ~p . -
, ,
, -
. ,
, -
. ,
" -
".
- -
, (-
) :
df
dt=f
t+
3i=1
[f
xixi +
f
pipi] = 0. (144)
, , -
.
"" -
" " -
(
). ,
(144)
.
, f(~p, ~r, t) :
f
t+
3i=1
[f
xixi +
f
pipi] = q q, (145)
q
q
-
, . -
.
Copyright & A K-C
-
75
q
q
=
B(, ~q)[f(~r, ~v, t)f(~r, ~v
1, t)f(~r,~v, t)f(~r, ~v1, t)]ddd3v1,
~v -
,
~q = ~v1 ~v, ~v, ~v1, -
~v , ~v
1 .
B(, ~q) - .
(, -
), .
W (~p, ~p1)d~p1 - , ~p1 ~p1 +d~p1, ~p. W (~p, ~p1) - ~r t. , ~p,
, :
q
=
W (~p, ~p1)f(~p1, ~r, t)(1 f(~p, ~r, t))d~p1,
~p1 , .
, ~p, , ~p1 ~p1 + d~p1, ,
q
=
W (~p1, ~p)f(~p, ~r, t)(1 f(~p1, ~r, t))d~p1,
~p1 -, .
f
t=
6i=1
[f
xi xi + f
pi pi]+
+
{W (~p, ~p1)f(~p1, ~r, t)(1f(~p, ~r, t))W (~p1, ~p)f(~p, ~r, t)(1f(~p1, ~r, t))}d~p1,
f
t= (~v rf) (~F pf)+
Copyright & A K-C
-
76
+
{W (~p, ~p1)f(~p1, ~r, t)(1 f(~p, ~r, t))
W (~p1, ~p)f(~p, ~r, t)(1 f(~p1, ~r, t))}d~p1. (146) . -
W (~p|~p1) - f(~p, ~r, t), (146) .
~F W (~p|~p1). q -
~E ~H ,
~F = q ~E +q
c[~v ~H],
c .
W (~p|~p1) - .
(f
t
)
= (~v rf) (~F pf)
f
, -
(f
t
)
=
{W (~p, ~p1)f(~p1, ~r, t)(1 f(~p, ~r, t))
W (~p1, ~p)f(~p, ~r, t)(1 f(~p1, ~r, t))}} d~p1 f .
-
, ~p
~p1, ~p - ~p1:
W (~p, ~p1)f0(~p1, ~r)(1 f0(~p, ~r)) = W (~p1, ~p)f0(~p, ~r)(1 f0(~p1, ~r)).
:
f0(~p, ~r) =[e(~p,~r)
kT + 1]1
,
Copyright & A K-C
-
77
W (~p, ~p1)
W (~p1, ~p)= exp
((~p1, ~r) (~p, ~r)
kT
).
, (~p1, ~r) > (~p, ~r) W (~p, ~p1) > W (~p1, ~p), .. ~p ~p1 , - .
, .
, ..
(~p1, ~r) = (~p, ~r)
W (~p|~p1) = W0(p, )(p p1), (p p1) , - p = p1, ~p ~p1.
, (
| f f0 | f0) W ,
.
f
t= (~v rf) (~F pf)+
+2
0
{W0(p, )(f(~p1, ~r, t) f(~p, ~r, t))p2sin()d. (147)
(147) , -
(D.Enskog, 1917),
f = f0 + f1, (148)
f0 f0((p, ~r)) , f1(~p) -, . -
f F0 , f0 , f1(~p) ~p. ,f1(~p) ,
~p-.
Copyright & A K-C
-
78
f (147),
(~v rf) + (~F pf) = f1(~p)/(p), (149) :
1
(p)= 2
0
W0(p, )[1 f1(~p1)/f1(~p)]p2d. (150)
(p) ( ),
. ,
~F = 0 rT = 0. t = 0 , (149) ,
f f0 :
f
t+f(~p, t) f0(p)
(p)= 0.
:
(f f0) = (f f0)t=0exp(t/). , , -
e .
,
(146) -
(148) -
. ,
-
f1. - , -
, f1(~p1)/f1(~p) , ,
.
-
.
. , ,
-
,
Copyright & A K-C
-
79
. -
. , -
, -
. -
(
(155) , ).
.
, " ".
-
, "" , ..
- -
,
.
, -
.
. -
,
. , -
, -
, ,
. ,
.
4.5. H-
,
. , -
.
~p1 ~p2, -
~p
1 ~p
2.
~p1 + ~p2 = ~p
1 + ~p
2,
p21 + p21 = (p
1)2 + (p
1)2.
Copyright & A K-C
-
80
( )
~P =1
2(~p1 + ~p2),
~u = ~p1 ~p2 (151)
~P
, ~u
,
(151)
~P = ~P
,
|~u| = |~u|. (152) (
f(~p1)
t
)
= [(~p1, ~p2|~p1, ~p
2)f(~p
1)f(~p
2) (~p
1, ~p
2|~p1, ~p2)f(~p1)f(~p2)]d3p2d3p
1d3p
2.
(~p1, ~p2|~p1, ~p
2) = (~p
1, ~p
2|~p1, ~p2). (153) (151), , , -
~p1 , ~p2 -
, ~p
1 ~p
2 ~p1, ~p2 ~.(
f(~p1)
t
)
=
d
d3p2(~p, ~p2|)[f(~p1)f(~p
2) f(~p1)f(~p2)]. (154)
-
, (~p, ~p2|) () -
(~p, ~p2|) = (~p, ~p2|)|~p1 ~p2|.
f
t+
3i=1
[f
xixi +
f
pipi] =
=
d
d3p2()|~p1 ~p2|[f(~p1)f(~p
2) f(~p1)f(~p2)]. (155)
Copyright & A K-C
-
81
-
f(~p, ~r, t). , H- -
, -
-
(1872 .).
, .
, -
-
S -,
S k0
(6N)
(p, q)ln((p, q)d3Npd3Nq
h3N, (156)
(p, q) , k0 .
, .
-
V (f/~r) = 0, F = 0. f ~p t.
f(~p1)
t=
d
d3p2()|~p1 ~p2|[f(~p1)f(~p
2) f(~p1)f(~p2)] (157)
f(~p, t)d3p =N
V= n = const. (158)
, H(t)
H(t)
f(~p, t)ln(f(~p, t)d3p. (159)
,
H
t=
t(fln(f)) d3p =
f
tln(f)d3p,
(158)f(~p, t)
td3p =
n
t= 0.
Copyright & A K-C
-
82
(158)
H
t=
ln(f(~p1, t))
d()|~p1~p2|[f(~p1)f(~p
2)f(~p1)f(~p2)]d3p1d3p2.(160)
~p1
~p2, () . -
.
H
t=
=1
2
ln(f(~p1)f(~p2))
d()|~p1~p2|[f(~p1)f(~p
2)f(~p1)f(~p2)]d3p1d3p2. {~p1, ~p2} {~p1, ~p
2}, - . ,
H
t=
=1
2
ln(f(~p
1)f(~p
2))
d
()|~p1~p
2|[f(~p1)f(~p2)f(~p
1)f(~p
2)]d3p
1d3p
2.
, d3p
1d3p
2 = d3p1d
3p2,|~p1 ~p
2| = |~p1 ~p2|
() = (),
H
t= 1
4
(ln(f(~p1)f(~p2)) ln(f(~p1, t)f(~p
2))
()|~p1 ~p2|[f(~p1)f(~p2) f(~p1)f(~p
2)]dd3p
1d3p
2. (161)
[ln(f(~p1, t), ~p2, t)ln(f(~p1, t), ~p
2, t)][f(~p1)f(~p2)f(~p
1)f(~p
2)] (ln(x)ln(y))(x y), x y -.
H
t 0, (162)
f(~p1)f(~p2) = f(~p
1)f(~p
2). (163)
H S . -
-
Copyright & A K-C
-
83
(p, q, t) f
(~pi, ~ri, t):
(p, q, t) =Ni=1
f
(~pi, ~ri, t), (164)
f
(~p, t) =h3
nf(~p, t).
(164) -
f
(~p, t) f
(~p, t)d3pd3r
h3= 1,
S = k0( 1NH + ln(
h3
n)). (165)
,
H S .
H- .
. H , (163), ..
.
" ".
. H , .. -
, - -
.
4.6.
-
. , -
~p :
= g +p2
2m.
g , m
.
~v =
~p=
~p
m.
Copyright & A K-C
-
84
(149)
(148), f1 ~F , - , ,
f1 = f1(~p, ~r) :
f1(~p, ~r) =f0
(~(, ~r) ~v) ,
~(, ~r) , f0() - -
:
f0(, ~r) =1
exp(k0T ) + 1,
, T , k0
. , -
~r, ~F
~F = (e)[ ~E + 1c[~v ~H]],
-
f0. , ,
.
,
rf rf0 = f0
[+ ( )TT
].
rf1 .
-
pf = pf0 +pf1,
pf0 = f0
~v,
pf1 = 2f02
~v(
~() ~v)
+f0
(~
m+ (~v
)~v).
,
e[ ~E + 1c[~v ~H]] pf0 = ef0
[ ~E +
1
c[~v ~H]] ~v = ef0
~E ~v,
Copyright & A K-C
-
85
[~p ~H] ~p = [~p ~p] ~H = 0. ,
, -
.
ec[~v ~H] pf1 = e
c [
2f02
(~v [(~v ~H])(
~() ~v)
+f0
(~ [~v ~H]
m+
+(~v
)(~v [~v ~H]))] = ec
f0
([ ~H ~] ~v)m
.
,
f0
([+()TT
]~v)ef0
~E~vec
f0
([ ~H ~] ~v)m
= 1
f0
(~ ~v) ,
~ [~v ~H] = [ ~H ~] ~v. ~v ,
~ = {+ ( )TT
+ e ~E +e
c
[ ~H ~]m
}. (166)
~
~ = ~a+ [~b ~], (167)
~a = [+ ( )TT
+ e ~E],
~b =e ~H
cm.
~b (167) ~b [~b ~] = 0,
~b ~ = ~b ~a. (168) ~ (167) ~a + [~b ~],
~ = ~a+ [~b ~a] + [~b [~b ~a]].
[~b [~b ~a]] = ~b(~b ~) ~b2
Copyright & A K-C
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86
(168),
~ = ~a+ [~b ~a] +~b(~b ~a) ~b2.
~ =~a+ [~b ~a] +~b(~b ~a)
1 + b2.
, b2
,
-
,
.
, .. -
:
~ = ~a+ [~b ~a] +~b(~b ~a).
~ = {[( )TT
+ e ~Et] +e
cm[ ~H [( )T
T+ e ~Et]]+
+(e
cm)2( ~H [( )T
T+ e ~Et]) ~H},
~Et = ~E +1e
, -
( ).
4.7.
,
, ,
~j = eV
~p,sz
~vf1 = 2eh3
f0
(~ ~v)~vd3p. (169)
, -
~, ~ = {0, 0, },
(~ ~v) = v cos ,
~v =~iv cos sin +~jv sin sin + ~kv cos .
Copyright & A K-C
-
87
,
20
sind = 0,
20
cosd = 0,
0
cos2 sin d =2
3.
~j = 8e3h3
0
f0
~v2p2dp = 8e23/2m
3h3
0
f0
~3/2d. (170)
< n >3/2 8e2
3h3(e
cm)n1
0
f0
n()3/2d,
>3/2 8e3h3
(e
cm)n1
0
f0
n()( )3/2d,
~j =>3/2TT
+ < >3/2 ~Et+ >3/2 [ ~H T
T]+
+ < 2 >3/2 [ ~H ~Et]+ >3/2 ( ~H TT
) ~H+ < 3 >3/2 ( ~H ~Et) ~H. ,
~w =1
V
~p,sz
~vf1 =2
h3
f0
(~ ~v) ~vd3p (171)
~w =8
3h3
0
f0
~v2p2dp,
~w =>5/2TT
+ < >5/2 ~Et+ >5/2 [ ~H T
T]+
+ < 2 >5/2 [ ~H ~Et]+ >5/2 ( ~H TT
) ~H+ < 3 >5/2 ( ~H ~Et) ~H.
Copyright & A K-C
-
88
, , ,>3/2< >5/2,>3/2< 2 >5/2,
, -
.
4.8. .
V (r) = e2/r. . -
-
. ,
, -
: -
, -
.
, ,
() Ek
E
:
E
Ek 1.
E
Ek 1
, (
32kT )
, -
, -
, .
E
Ek 2
3e2/kTr0,
r0 n1/3 , n .
, -
. -
n 1012 3 kT 5 . EEk 2 104, .
Copyright & A K-C
-
89
10 , , n 1017
3.
.
-
,
.
4.9.
,
: ( -
) -
, -
( ""),
.
f(t,r ,p ) f(t,r ,p ) = f0(p) + f1(t,r ,p ),
f0(p) = exp(p2/2m
kT ) , f1 - ,
1
h3
f0(p)d
p = n,n .
exp(
kT) = n
h3
(2mkT )3/2. (172)
,
+0 = 0 =e
h3
f0(p)d
p ,
=e
h3
f1(t,
r ,p )dp ,
~j =e
h3
~p
mf1(t,
r ,p )dp .
,
f1t
+ ~v rf1 e( ~E + 1c[~v ~H]) pf0 = f1, (173)
Copyright & A K-C
-
90
div ~E = 4eh3
f1(t,
r ,p )dp (174)
div ~H = 0, (175)
rot ~H =1
c
~E
t 4e
ch3
~p
mf1(t,
r ,p )dp (176)
rot ~E = 1c
~H
t. (177)
-,
f1 = f~,exp(i(t ~ ~r)), ~E = ~E~,exp(i(t ~ ~r)),~H = ~H~,exp(i(t ~ ~r))
: -
OX, OZ, :
~ = (, 0, 0), ~H~, = (0, 0, H~,), ~E~, = ((E~,)x, (E~,)y, 0).
if~, + ivxf~, + ekT
[vx(E~,)x + vy(E~,)y]f0 = f~,; (178)
i(E~,)x = 4eh3
f~,d~p; (179)
ic(E~k,)x
4e
h3c
vxf~k,d~p = 0; (180)
iH~, = ic(E~,)y 4e
h3c
vyf~,d~p; (181)
i(E~k,)y = i
cH~k,. (182)
f~,
f~, = i ekT
vx(E~,)x + vy(E~,)y vx + i f0.
Copyright & A K-C
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91
, , -
i(E~,)x = i4e2
h3kT
[vx(E~,)x + vy(E~,)y]f0
vx + i d~p; (183)
i
c(E~,)x = i
4e2
h3ckT
vx
[vx(E~,)x + vy(E~,)y]f0 vx + i d~p; (184)
i[
c
2c
](E~,)y = i
4e2
h3ckT
vy
[vx(E~k,)x + vy(E~k,)y]f0
vx + i d~p. (185), - vy
vy(E~,)yf0 vx + id~p = 0,vxvy(E~,)yf0 vx + i d~p = 0,
(172) vx(E~,)xf0 vx + id~p =
n2mkT
vx(E~,)xexp( p2x
2mkT )
vx + i dpx,
v2y(E~,)yf0
vx + id~p =mkT2m2
n
(E~,)yexp( p2x
2mkT )
vx + i dpx.
(184) (183),
v2x(E~,)xexp( p2x
2mkT )
vx + i dpx =
vx(E~,)xexp( p
2x
2mkT)dpx+
+ + i
vx(E~,)xexp( p2x
2mkT )
vx + i dpx =
vx(E~,)xexp( p2x
2mkT )
vx + i dpx.
, -
: :
(E~,)x =4e2n
2mkTkT
vx(E~,)xexp(
p2x2mkT )
vx + i dpx; (186)
[
c
2c
](E~,)y =
4e2n
cm
2mkT
(E~,)yexp(
p2x2m )
vx + i dpx. (187)
Copyright & A K-C
-
92
4.10.
. -
(186) (E~,)x,
1 =4e2nm
2mkTkT
vxexp(mv2x2kT )
vx + idvx. (188)
,
vx =+i
,
.
lim0
1
x i =Px i(x),
P -, (x) - . ,
1 = J(, ) + iI(, ), (189)
J(, ) =4e2nm
2mkTkTP
vxexp(mv2x2kT )
vx dvx; , (190)
I(, ) =42e2nm
3
2mkTkTexp(
m222kT
). (191)
J(, ) , - (.., .. -
, .: , 1954; Fried B. D., Conte S.D. The
plasma Disp ersion Funtion, Aademi Press, New York, 1961.),
.
1
1 vx = 1 +
vx
1
1 vx =
= 1 +vx + (
vx)2 + (
vx)3 + (
vx)4 + (
vx)5 + (
vx)6(
1
1 vx ).
, .
, ,
Copyright & A K-C
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93
|vx | < 1. -
vx = v3x = v5x = 0, v
2x =
kT
m, v4x = 3(
kT
m)2, v6x = 15(
kT
m)2.
(189)
1 20
2(1+6
202
(LD)2 +60
404
(LD)4 +
m
kT5
5(v7x
1 vx ))+ iI(, ) = 0,
(192)
I(, ) =
4
0
1
(LD)3exp
[1
4
2
20
1
(LD)2
], (193)
0 =
4e2n/m , LD =kT/(4ne2)
.
202 (LD)
2 1 ( ) ,
..
1 20
2(1 + 6
202
(LD)2) + iI(, ) = 0. (194)
I(, ) 6=0, . = i, ,
1
2=
1
( i)2 1
2(1 + 2i
),
I(, ) I(,).
1 20
2(1 + 6
202
(LD)2) 2i
20
2(1 + 12
202
(LD)2)
+ iI(,) = 0.
,
= 0(1 + 3202
(LD)2 + ),
=
2
220I/(1 + 12
202
(LD)2 + ).
,
0(1 + 3(LD)2 + ), (195)
Copyright & A K-C
-
94
=02I(, )/(1 + 12(LD)
2)
8
0(LD)3
exp
[ 1
4(LD)2
]. (196)
(195), (196) () , -
. -
, .
, -
, = ~ ~v. , , ,
.
, -
,
.
~0, ~0,
Al 15,0 Be 18,9
Mg 10,5 Si 16,9
Ge 16,0 Na 5,7
202 (LD)
2 1 J(, )
J(, ) =4e2nm
2
2mkTkTP
vxexp(mv2x2kT )
vx dvx =
4e2n
2kT(1+
+m
2mkTP
exp(mv2x
2kT )
vx dvx
. (197) vx = u +
,
:
m
2mkTP
exp(mv2x
2kT )
vx dvx =
m
2mkTP
exp(m(u+
)2
2kT)du
u=
=me
m2
2kT2
2mkTP
emu2
2kT (1 ukT
+1
2(u
kT)2 1
6(u
kT)3 + . . . )
du
u
Copyright & A K-C
-
95
2e
m2
2kT2
2kT.
J(, ) 4e2n
2kT
(1
2em2
2kT2
2kT
). (198)
, 0 .4.11.
(187) (E~,)y,
2 = (c)2 +4e2n
m
2mkT
exp(p
2x
2m )
vx + idpx. (199)
, -
,
2 = c22 + 20(1 +202
(LD)2 + 3
404
(LD)4 + )
i20
0LDexp
[ 1
4(LD)2
].
.
, (-
= mv2
2
), -
, vx =
c. -
,
2 = c22 + 20(1 + (LD)2 + ), = 0. (200)
4.12. -
( , -
: n-GaAs, InSb, InAs, CdS .), -
, (-
, : Si, Ge, PbTe, PbSe n-).
Copyright & A K-C
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96
- -
, me mn .
, , . . qe = e,qi = +e ne = ni = n. -, , me mi e i - ,
fet
+ ~v fe~r
e ~E e~p
= efe,fit
+ ~v fi~r
+ e ~E i~p
= ifi,
div ~E = 4eh3
(fe(t,
r ,p ) fi(t,r ,p ))dp ,
div ~H = 0,
rot ~H =1
c
~E
t 4e
ch3
~p(fe(t,
r ,p )me
fi(t,r ,p )mi
)dp ,
rot ~E = 1c
~H
t,
e = (2mekTe)3/2exp( p
2
2mekTe),
i = (2mikTi)3/2exp( p
2
2mikTi),
Te, Ti . -
1954 .. ,
Ti Te. -
ife,~, + i pxme
fe,~, +e
mekT[px(E~,)x + py(E~,)y]e = fe,~,;
ifi,~, + i pxmi
fi,~, emikT
[px(E~,)x + py(E~,)y]i = fi,~,;
i(E~,)x = 4eh3
(fe,~, fi,~,)d~p;
0 = ic(E~k,)x
4e
h3c
px(
fe,~k,me
fi,~k,mi
)d~p;
Copyright & A K-C
-
97
iH~, = ic(E~,)y 4e
h3c
py(
fe,~k,me
fi,~k,mi
)d~p;
i(E~k,)y = i
cH~k,.
fe,~, fi,~, -
,
:
i(E~,)x = i4e2
h3k
[px(E~,)x + py(E~,)y][
e/Teme px + ime+
+i/Ti
mi px + imi ]d~p; (201)
c(E~,)x =
4e2
h3ck
px[px(E~,)x + py(E~,)y][
e/Tememe px + ime+
+i/Timi
mi px + imi ]d~p; (202)
[
c
2c
](E~,)y =
4e2
h3ck
py[px(E~,)x + py(E~,)y][
e/Tememe px + ime+
+i/Timi
mi px + imi ]d~p; (203)
, -
1 =20
px[
exp( p2x2meTe )kTe
2mekTe( px me + i)+
memiexp( p2x2miTi )
kTi
2mikTi( px mi + i)]dpx
(204)
c
2c
=20c
[
exp( p2x2meTe )2mkTe( px me + i)
+
memiexp( p2x2miTi )
2mikTi( px mi + i)]dpx.
(205)
mi me , , ,
. -
Copyright & A K-C
-
98
0. , -,
20
px[
memiexp( p2x2miTi )
kTi
2mikTi( px mi + i)]dpx
20,i2
(1+620,i2
(LD,i)2)iI(, );
(206)
I(, ) =
4
0,i
1
(LD,i)3exp
[1
4
2
20,i
1
(LD,i)2
],
0,i =
4e2n/mi , LD,i =kTi/(4ne2)
.
= 0
J(, ) 4e2n
2kTe
1 2e me22kTe22kTe
. 0,i 0 Te Ti, -
, -
. -
1+4e2n
2kTe
1 2e me22kTe22kTe
20,i2
(1+620,i2
(LD,i)2)+i
4
0,i
1
(LD,i)3 0.
= i -, ():
kTeme
(1 1
2
0mikTe
2 + . . .
), (207)
=
me8mi
.
.
-
- -
- ,
Copyright & A K-C
-
99
. (207)
kTeme
, -
.
, - -
: -
( me/mi
).
1. . ,
/. , . . .: - , 1973.
280 .
2. , .. -
/.. . .: - , 1978. 128 .
3. . .
/. . .: - , 1974.
304 .
4. .. /..
. .: - , 1971. 415 .
5. .. -
/.. , .. . .:
- , 1976. 480 .
6. , .. .
/... .: - . 1987.
559 .
7. , .. /.., ...
.: - , 1979. 527 .
8. . /. ,
. . .: - , 1979. 512 .
9. , .. .
/.. , .. . M.: - , 1977. 552 .
10. .. .
/.. . .: , 2000. 312 .
Copyright & A K-C
-
100
, ..
..
23.11.2006. 60x84/16. .
. . . 5,11. .-. . 4,8.
100 . .
-
-
. ...
150000 , . , 14.
"" 06151 26.10.2001.
. , . , 94, . 37, . (4852) 73-35-03
Copyright & A K-C