Елесин В.Ф., Кашурников В.А. Физика фазовых переходов
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ɜɚɹ ɮɭɧɤɰɢɹ ɬɨɠɞɟɫɬɜɟɧɧɨ ɪɚɜɧɚ ɧɭɥɸ, ɱɬɨ ɨɬɪɚɠɚɟɬ ɩɪɢɧɰɢɩ
ɉɚɭɥɢ ɜ ɮɟɪɦɢɨɧɧɨɣ ɫɬɚɬɢɫɬɢɤɟ, ɡɚɩɪɟɳɚɸɳɢɣ ɞɜɭɦ ɮɟɪɦɢɨ-
ɧɚɦ ɧɚɯɨɞɢɬɫɹ ɜ ɨɞɧɨɦ ɫɨɫɬɨɹɧɢɢ. ɇɚɨɛɨɪɨɬ, ɩɪɢ ɡɧɚɤɟ “+”
ɞɥɹ ɛɨɡɨɧɨɜ ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɯɨɠɞɟɧɢɹ ɜ ɨɞɧɨɦ ɫɨɫɬɨɹɧɢɢ ɧɚɢ-
ɛɨɥɶɲɚɹ.
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɫɢɫɬɟɦɭ ɷɥɟɤɬɪɨɧɨɜ, ɥɨɤɚɥɢɡɨɜɚɧ-
ɧɵɯ ɜ ɩɨɥɟ ɤɪɢɫɬɚɥɥɢɱɟɫɤɨɣ ɪɟɲɟɬɤɢ. ɍɱɬɟɦ ɬɚɤɠɟ ɫɩɢɧɨɜɭɸ
ɤɨɦɩɨɧɟɧɬɭ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ, ɬɚɤ ɱɬɨ
ɩɨɥɧɚɹ ɜɨɥɧɨɜɚɹ
ɮɭɧɤɰɢɹ (ɤɨɬɨɪɚɹ ɞɨɥɠɧɚ ɛɵɬɶ ɚɧɬɢɫɢɦɦɟɬɪɢɱɧɚ) ɢɦɟɟɬ ɜɢɞ:
)
(r
1
,r
2
,
V
1
,
V
2
) =
<
(r
1
,r
2
)
F
(
V
1
,
V
2
),
<
(r
1
,r
2
) = 1/ 2 {
<
a
(r
1
)
<
b
(r
2
) r
<
a
(r
2
)
<
b
(r
1
)}.
ɉɪɢ ɷɬɨɦ ɚɧɬɢɫɢɦɦɟɬɪɢɱɧɨɣ ɫɢɬɭɚɰɢɢ “-” ɞɨɥɠɧɚ ɫɨɨɬɜɟɬɫɬ-
ɜɨɜɚɬɶ ɫɢɦɦɟɬɪɢɱɧɚɹ ɫɩɢɧɨɜɚɹ ɤɨɦɩɨɧɟɧɬɚ (ɫɭɦɦɚɪɧɵɣ ɫɩɢɧ
S=1):
F
(
V
1
,
V
2
) =
F
(1/2,
V
1
)
F
(1/2,
V
2
),
F
(
V
1
,
V
2
) =
F
(-1/2,
V
1
)
F
(-1/2,
V
2
),
F
(
V
1
,
V
2
) =
F
(1/2,
V
1
)
F
(-1/2,
V
2
) +
F
(-1/2,
V
1
)
F
(1/2,
V
2
),
ɚ ɫɢɦɦɟɬɪɢɱɧɨɣ ɫɢɬɭɚɰɢɢ “+” ɞɨɥɠɧɚ ɫɨɨɬɜɟɬɫɬɜɨɜɚɬɶ ɚɧɬɢ-
ɫɢɦɦɟɬɪɢɱɧɚɹ ɫɩɢɧɨɜɚɹ ɤɨɦɩɨɧɟɧɬɚ (ɫɭɦɦɚɪɧɵɣ ɫɩɢɧ S=0):
F
(
V
1
,
V
2
)=
F
(1/2,
V
1
)
F
(-1/2,
V
2
)-
F
(-1/2,
V
1
)
F
(1/2,
V
2
).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɢɦɦɟɬɪɢɹ ɤɨɨɪɞɢɧɚɬɧɨɣ ɜɨɥɧɨɜɨɣ ɮɭɧɤɰɢɢ
ɡɚɜɢɫɢɬ ɨɬ ɨɪɢɟɧɬɚɰɢɢ ɫɩɢɧɨɜ:
<
nn
(r
1
,r
2
) = 1/
2
{
<
a
(r
1
)
<
b
(r
2
) -
<
a
(r
2
)
<
b
(r
1
)}, (1.16)
<
np
(r
1
,r
2
) = 1/ 2 {
<
a
(r
1
)
<
b
(r
2
) +
<
a
(r
2
)
<
b
(r
1
)}, (1.17)
20
ɍɱɬɟɦ ɬɟɩɟɪɶ ɜ ɩɟɪɜɨɦ ɩɨɪɹɞɤɟ ɩɨ ɬɟɨɪɢɢ ɜɨɡɦɭɳɟɧɢɣ ɤɭɥɨ-
ɧɨɜɫɤɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɷɥɟɤɬɪɨɧɨɜ
U
e
rr
oo
2
12
. ɉɨɩɪɚɜɤɚ ɤ
ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɡɚ ɫɱɟɬ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɢɦɟɟɬ ɜɢɞ:
E
e
rr
rr
oo
³
\\
(, ) *(, )12 12
2
12
1
dd
33
2
. (1.18)
ɉɨɞɫɬɚɜɥɹɹ ɜ (1.18) ɜɨɥɧɨɜɵɟ ɮɭɧɤɰɢɢ (1.16)-(1.17), ɢɦɟɟɦ
E
nn
=E
0
J
12
,E
np
=E
0
J
12
, (1.19)
ɝɞɟ E
0
- ɷɧɟɪɝɢɹ, ɧɟ ɡɚɜɢɫɹɳɚɹ ɨɬ ɨɪɢɟɧɬɚɰɢɢ ɫɩɢɧɨɜ ɱɚɫɬɢɰ:
^`
E
e
rr
r r r r drdr
ab a b
oo
³
1
2
2
12
1
2
2
2
2
2
1
2
3
1
3
2
\\ \\
() () () ()
, (1.20)
ɚ J
12
- ɨɛɦɟɧɧɵɣ ɢɧɬɟɝɪɚɥ:
^`
E
e
rr
r r r r drdr
ab a b
oo
³
2
12
12 21
3
1
3
2
\\\\
() () () ()
**
. (1.21)
ȼɫɥɭɱɚɟJ
12
>0 ɫɩɢɧɚɦ ɜɵɝɨɞɧɨ ɜɵɫɬɪɨɢɬɫɹ ɩɚɪɚɥɥɟɥɶɧɨ, ɟɫɥɢ
J
12
< 0, ɬɨ ɚɧɬɢɩɚɪɚɥɥɟɥɶɧɨ. ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɭɩɨɪɹɞɨɱɟɧɢɟ
ɩɪɢɜɨɞɢɬ ɤ ɮɟɪɪɨɦɚɝɧɟɬɢɡɦɭ, ɜɨ ɜɬɨɪɨɦ - ɤ ɚɧɬɢɮɟɪɪɨɦɚɝɧɟ-
ɬɢɡɦɭ.
Ɉɰɟɧɢɦ ɜɟɥɢɱɢɧɭ J
12
. Ʉɚɤ ɜɢɞɧɨ, ɦɚɫɲɬɚɛ J
12
ɨɩɪɟɞɟ-
ɥɹɟɬɫɹ ɤɭɥɨɧɨɜɫɤɢɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ, ɬ.ɟ. J
12
~e
2
/a~10ɷȼ, ɝɞɟ a
- ɦɟɠɚɬɨɦɧɨɟ ɪɚɫɫɬɨɹɧɢɟ. ȿɳɟ ɧɚ ɩɨɪɹɞɨɤ ɜɟɥɢɱɢɧɭ J
12
ɦɨɠɟɬ
ɭɦɟɧɶɲɢɬɶ ɦɧɨɠɢɬɟɥɶ exp(-2a/a
Ȼ
), ɝɞɟ ɚ
Ȼ
- ɛɨɪɨɜɫɤɢɣ ɪɚɞɢɭɫ,
ɢɡ-ɡɚ ɩɟɪɟɤɪɵɬɢɹ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɵɯ “ɯɜɨɫɬɨɜ” ɜɨɥɧɨɜɵɯ
ɮɭɧɤɰɢɣ (ɫɦ. (1.21)). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, J
12
~ 1ɷȼ~10
4
K, ɱɬɨ ɫɨ-
ɝɥɚɫɭɟɬɫɹ ɫ ɨɰɟɧɤɨɣ ɜɟɥɢɱɢɧɵ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ (
O
~ 10
4
).
21
Ɍɟɩɟɪɶ ɪɚɫɫɦɨɬɪɢɦ ɪɟɚɥɶɧɵɟ ɦɨɞɟɥɢ, ɨɩɢɫɵɜɚɸɳɢɟ
ɫɢɫɬɟɦɭ ɫɩɢɧɨɜ ɫ ɨɛɦɟɧɧɵɦ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ. Ɍɚɤ, ɟɫɥɢ ɜɡɚɢ-
ɦɨɞɟɣɫɬɜɢɟ ɞɜɭɯ ɜɵɞɟɥɟɧɧɵɯ ɫɩɢɧɨɜ ɫɨɝɥɚɫɧɨ (1.19) ɢɦɟɟɬ
ɜɢɞ:
22
S
2
EEJS
SS
12
0121
oo
oo
,
, (1.22)
ɬɨ ɜ ɪɟɚɥɶɧɨɦ ɬɜɟɪɞɨɦ ɬɟɥɟ ɫ ɪɟɲɟɬɤɨɣ ɫɩɢɧɨɜ ɜɜɨɞɢɦ ɬɚɤ ɧɚ-
ɡɵɜɚɟɦɵɣ ɝɚɦɢɥɶɬɨɧɢɚɧ Ƚɟɣɡɟɧɛɟɪɝɚ:
HJSSH
ij i j
ij
i
i
oo o o
¦¦
12/.S
(1.23)
Ɂɞɟɫɶ ɦɵ ɭɱɥɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɫ ɜɧɟɲɧɢɦ ɩɨɥɟɦ, ɚ ɨɬɫɱɟɬ
ɷɧɟɪɝɢɢ ɩɪɨɜɨɞɢɬɫɹ ɨɬ ɬɨɣ ɟɟ ɱɚɫɬɢ, ɤɨɬɨɪɚɹ ɧɟ ɡɚɜɢɫɢɬ ɨɬ
ɨɪɢɟɧɬɚɰɢɢ ɫɩɢɧɨɜ ɢ ɧɟ ɜɥɢɹɟɬ ɧɚ ɦɚɝɧɢɬɧɨɟ ɭɩɨɪɹɞɨɱɟɧɢɟ
ɫɢɫɬɟɦɵ. ȿɫɥɢ J
ij
>0, ɬɨ ɝɚɦɢɥɶɬɨɧɢɚɧ (1.23) ɨɩɢɫɵɜɚɟɬ ɮɟɪɪɨ-
ɦɚɝɧɢɬɧɭɸ ɦɨɞɟɥɶ Ƚɟɣɡɟɧɛɟɪɝɚ, ɟɫɥɢ J
ij
< 0, ɬɨ ɚɧɬɢɮɟɪɪɨɦɚɝ-
ɧɢɬɧɭɸ.
ȼ ɦɨɞɟɥɢ (1.23) ɨɪɢɟɧɬɚɰɢɹ ɫɩɢɧɨɜ ɞɨɩɭɫɤɚɟɬɫɹ ɥɸɛɨɣ.
ȿɫɥɢ ɠɟ ɭɱɟɫɬɶ ɬɨɬ ɮɚɤɬ, ɱɬɨ ɨɛɦɟɧɧɵɣ ɢɧɬɟɝɪɚɥ J
ij
ɦɨɠɟɬ
ɛɵɬɶ ɪɚɡɥɢɱɟɧ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɤɪɢɫɬɚɥɥɨɝɪɚɮɢɱɟɫɤɢɯ ɧɚɩɪɚɜ-
ɥɟɧɢɣ ɜ ɪɟɚɥɶɧɨɦ ɚɧɢɡɨɬɪɨɩɧɨɦ ɤɪɢɫɬɚɥɥɟ, ɬ.ɟ. ɜɨɡɦɨɠɟɧ ɜɢɞ
ɝɚɦɢɥɶɬɨɧɢɚɧɚ:
HJSSJSSJSSH
ij
x
i
x
j
x
ij
y
i
y
j
y
ij
z
i
z
j
z
ij
i
i
¦¦
oo
12/( ) S ,
ɝɞɟ J
x
, J
y
ɢ J
z
ɪɚɡɥɢɱɧɵ, ɬɨ ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ ɦɚɝɧɢɬɧɵɯ ɦɨɦɟɧ-
ɬɨɜ ɦɨɝɭɬ ɛɵɬɶ ɨɝɪɚɧɢɱɟɧɵ ɤɨɧɤɪɟɬɧɵɦ ɜɢɞɨɦ ɷɬɢɯ ɜɟɥɢɱɢɧ.
Ɍɚɤ, ɜɨɡɦɨɠɧɚ ɫɢɬɭɚɰɢɹ, ɤɨɝɞɚ ɦɚɤɫɢɦɚɥɶɧɨɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ
ɪɟɚɥɢɡɭɟɬɫɹ ɜɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ ɩɪɢɥɨɠɟɧɧɨɝɨ ɩɨɥɹ H. Ɍɨɝɞɚ
ɦɨɞɟɥɶ ɭɩɪɨɳɚɟɬɫɹ, ɨɝɪɚɧɢɱɢɜɚɟɬɫɹ ɱɢɫɥɨ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵ ɢ
ɨɫɬɚɟɬɫɹ ɥɢɲɶ ɞɜɚ ɜɨɡɦɨɠɧɵɯ ɫɨɫɬɨɹɧɢɹ - ɩɨ ɩɨɥɸ ɢ ɩɪɨɬɢɜ (
P
i
=r1) - ɤɚɤ ɜ ɩ.1.2 ɢ 1.3, ɢ ɦɵ ɩɨɥɭɱɚɟɦ ɦɨɞɟɥɶ ɂɡɢɧɝɚ:
HVH
ij i j
ij
i
i
¦¦
12/
PP P
. (1.23)
Ɂɞɟɫɶ ɩɟɪɟɨɛɨɡɧɚɱɟɧɨ J
ij
oV
ij
V(r
i
-r
j
), S
i
o
P
i
.
Ɉɫɧɨɜɧɨɟ ɫɨɫɬɨɹɧɢɟ ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɫɨɜɩɚɞɚɟɬ ɫ ɨɫɧɨɜ-
ɧɵɦ ɫɨɫɬɨɹɧɢɟɦ ɦɨɞɟɥɢ Ƚɟɣɡɟɧɛɟɪɝɚ ɩɪɢ ɧɭɥɟɜɨɣ ɬɟɦɩɟɪɚɬɭ-
ɪɟ, ɤɨɝɞɚ ɫɩɢɧɵ “ɡɚɦɨɪɨɠɟɧɵ” ɢ ɨɪɢɟɧɬɢɪɨɜɚɧɵ ɥɢɛɨ ɩɨ ɩɨ-
ɥɸ, ɥɢɛɨ ɩɪɨɬɢɜ, ɢ
npnppnnnppnn
npnppnnnppnn
ȼɛɥɢɡɢ ɨɛɥɚɫɬɢ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ ɧɚɢɛɨɥɟɟ ɚɞɟɤɜɚɬɧɨ
ɨɩɢɫɚɧɢɟ ɫ ɩɨɦɨɳɶɸ ɦɨɞɟɥɢ Ƚɟɣɡɟɧɛɟɪɝɚ, ɪɟɚɥɶɧɨ ɭɱɢɬɵɜɚɸ-
ɳɟɣ ɜɫɟ ɫɬɟɩɟɧɢ ɫɜɨɛɨɞɵ ɫɩɢɧɨɜɨɣ ɫɢɫɬɟɦɵ. Ȼɨɥɟɟ ɬɨɝɨ, ɢ ɩɪɢ
ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ɬɚɤɢɟ ɜɨɛɭɠɞɟɧɢɹ, ɤɚɤ ɫɩɢɧɨɜɵɟ ɜɨɥɧɵ,
ɧɟɜɨɡɦɨɠɧɨ ɨɩɢɫɚɬɶ (ɤɚɤ ɛɭɞɟɬ ɩɨɤɚɡɚɧɨ ɧɢɠɟ) ɛɟɡ ɦɨɞɟɥɢ
Ƚɟɣɡɟɧɛɟɪɝɚ. Ɉɞɧɚɤɨ ɦɨɞɟɥɶ ɂɡɢɧɝɚ ɧɚɢɛɨɥɟɟ ɩɪɨɫɬɚ, ɧɚɝɥɹɞɧɚ
ɢ ɞɨɫɬɚɬɨɱɧɨ ɭɞɨɛɧɚ ɩɪɢ ɢɡɥɨɠɟɧɢɢ ɨɬɞɟɥɶɧɵɯ ɩɪɨɛɥɟɦ ɬɟɨ-
ɪɢɢ ɦɚɝɧɟɬɢɡɦɚ.
Ɂɚɞɚɱɢ
1.4.
1. Ⱦɚɧɚ ɬɪɟɯɦɟɪɧɚɹ ɤɥɚɫɫɢɱɟɫɤɚɹ ɦɨɞɟɥɶ Ƚɟɣɡɟɧɛɟɪ-
ɝɚ ɛɟɡ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ:
HSHS
i
i
i
oo o
¦
P
0
,| | S
.
Ɉɩɪɟɞɟɥɢɬɶ ɫɬɚɬɢɫɬɢɱɟɫɤɭɸ ɫɭɦɦɭ Q.
Ɋɟɲɟɧɢɟ
. ɋɬɟɩɟɧɢ ɫɜɨɛɨɞɵ ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɹɸɬɫɹ ɭɝ-
ɥɨɦ
4
i
ɦɟɠɞɭ ɩɨɥɟɦ H ɢ ɫɩɢɧɨɦ S
i
, ɩɨɷɬɨɦɭ:
>@
QSH HS
ii
i
N
³
dsh: exp( cos ) ( { } )
EP T S EP EP
00
4
HS
0
.
1.4.
2. ɇɚɣɬɢ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɶ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɟɣ
ɦɨɞɟɥɢ Ƚɟɣɡɟɧɛɟɪɝɚ, ɢɫɯɨɞɹ ɢɡ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɣ ɮɨɪɦɭɥɵ
MF
ww
H ɢ ɫɪɚɜɧɢɬɶ ɫ ɪɟɡɭɥɶɬɚɬɨɦ, ɩɨɥɭɱɚɟɦɵɦ ɢɡ ɨɩ-
ɪɟɞɟɥɟɧɢɹ:.MN S SH
Zi
³
oo
[ / ] cos exp( )4
00
SP T EP
d:
23
Ɉɬɜɟɬ:
MFH
H
QNF HSFx x x
LL
ww
w
wE
PEP
ln ( ), ( ) cth( ) ,
00
1
F
L
(x) - ɮɭɧɤɰɢɹ Ʌɚɧɠɟɜɟɧɚ. ȼɬɨɪɨɣ ɪɟɡɭɥɶɬɚɬ ɢɡ ɨɩɪɟɞɟɥɟɧɢɹ
ɫɨɜɩɚɞɚɟɬ ɫ ɷɬɢɦ ɜɵɪɚɠɟɧɢɟɦ.
1.4.
3. ɂɫɫɥɟɞɨɜɚɬɶ ɩɨɜɟɞɟɧɢɟ ɧɚɦɚɝɧɢɱɟɧɧɨɫɬɢ ɜ ɫɥɭɱɚ-
ɹɯ T
o
0 ɢ T
of
.
Ɋɟɲɟɧɢɟ
. Ɏɭɧɤɰɢɹ Ʌɚɧɠɟɜɟɧɚ F
L
(x) (ɫɦ. ɩɪɟɞɵɞɭɳɭɸ
ɡɚɞɚɱɭ) ɢɦɟɟɬ ɚɫɢɦɩɬɨɬɵ:
Fx x x x
Fx x x x x
L
L
( ) / exp( ), ;
() / / / , .
| of
| o
11 2 2
3 45 2 945 0
35
Ɉɬɫɸɞɚ ɧɚɯɨɞɢɦ:
.MNST MNSHTT
PP
00
22
03,; /,of
1.4.
4. ɇɚɣɬɢ ɦɚɝɧɢɬɧɭɸ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ ɫɢɫɬɟɦɵ.
Ɉɬɜɟɬ
.
FP
o
() | /13
00
22
NMH S T
H
dd .
1.4.
5. Ɋɚɫɫɱɢɬɚɬɶ ɬɟɩɥɨɟɦɤɨɫɬɶ ɫɢɫɬɟɦɵ ɋ(T) ɤɥɚɫɫɢɱɟ-
ɫɤɢɯ ɫɩɢɧɨɜ ɜ ɦɨɞɟɥɢ Ƚɟɣɡɟɧɛɟɪɝɚ ɛɟɡ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ. ɂɫɫɥɟ-
ɞɨɜɚɬɶ ɩɪɟɞɟɥɶɧɵɟ ɫɥɭɱɚɢ T
o
0 ɢ T
o
f.
Ɉɬɜɟɬ
:
CTFTTTQT
HS
T
FHS
Fx F x
L
LL
ww w w
P
EP
222 2
0
2
2
0
(ln )
()
'( ),
'( ) ,dd
SHST
CHST
o
|of
14 2 0
3
0
2
0
0
2
( ) exp( ), ,
()/, .
EP EP
EP
CH
1.4.
6. ɇɚɣɬɢ ɷɧɬɪɨɩɢɸ ɫɢɫɬɟɦɵ S(T)=-
w
F/
w
T. Ɋɚɫɫɦɨɬ-
ɪɟɬɶ ɩɪɟɞɟɥɶɧɵɟ ɫɥɭɱɚɢ.
_____________
24
2. ɆɈȾȿɅɖ ɂɁɂɇȽȺ
ȼ ɷɬɨɣ ɝɥɚɜɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɦɨɞɟɥɶ ɂɡɢɧɝɚ ɜ ɩɪɢɛɥɢ-
ɠɟɧɢɢ ɦɨɥɟɤɭɥɹɪɧɨɝɨ ɩɨɥɹ ȼɟɣɫɫɚ ɢ ɩɨɥɭɱɢɦ ɨɫɧɨɜɧɵɟ ɪɟ-
ɡɭɥɶɬɚɬɵ ɩɨ ɬɟɩɥɨɟɦɤɨɫɬɢ ɢ ɦɚɝɧɢɬɧɨɣ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɢ. ɇɚ-
ɪɹɞɭ ɫ ɷɬɢɦ ɛɭɞɭɬ ɢɡɭɱɟɧɵ ɬɨɱɧɨɪɟɲɚɟɦɵɟ ɨɞɧɨɦɟɪɧɚɹ ɢ ɞɜɭ-
ɦɟɪɧɚɹ ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɢ ɢɫɫɥɟɞɨɜɚɧɚ ɩɪɨɛɥɟɦɚ ɮɚɡɨɜɨɝɨ ɩɟɪɟ-
ɯɨɞɚ ɜ ɧɢɯ.
2.1. Ɇɨɞɟɥɶ ɂɡɢɧɝɚ. ɉɪɢɛɥɢɠɟɧɢɟ ɫɪɟɞɧɟɝɨ ɩɨɥɹ
Ɋɚɫɫɦɨɬɪɢɦ ɦɨɞɟɥɶ ɂɡɢɧɝɚ (1.23):
HVH
ij i j
ij
i
i
¦¦
12
PP P
.
ɢ ɢɫɩɨɥɶɡɭɟɦ ɩɪɢɛɥɢɠɟɧɢɟ ɫɪɟɞɧɟɝɨ ɩɨɥɹ. ɉɨɤɚɠɟɦ, ɤɚɤ ɜ
ɞɚɧɧɨɣ ɦɨɞɟɥɢ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɭɪɚɜɧɟɧɢɟ ȼɟɣɫɫɚ. Ɍɚɤ ɠɟ, ɤɚɤ
ɢɜɩ.1.3, ɭɱɬɟɦ, ɱɬɨ ɧɚ ɜɵɞɟɥɟɧɧɵɣ ɦɚɝɧɢɬɧɵɣ ɦɨɦɟɧɬ ɛɭɞɟɬ
ɞɟɣɫɬɜɨɜɚɬɶ ɜɧɟɲɧɟɟ ɩɨɥɟ H ɢ ɩɨɥɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɨɬ ɜɫɟɯ
ɨɫɬɚɥɶɧɵɯ ɦɨɦɟɧɬɨɜ:
HV
iijj
j
¦
P
H
. (2.1)
ɉɨɫɥɟ ɭɫɪɟɞɧɟɧɢɹ ɫɭɦɦɚɪɧɨɟ ɩɨɥɟ ɛɭɞɟɬ ɬɨɝɞɚ ɪɚɜɧɨ
HV
iijj
j
¦
P
H
H
r
. (2.2)
ȼ ɫɢɥɭ ɬɪɚɧɫɥɹɰɢɨɧɧɨɣ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ (<
P
j
> ɧɟ ɡɚɜɢɫɢɬ ɨɬ j)
ɜ ɫɪɟɞɧɟɦ ɧɚ ɜɵɞɟɥɟɧɧɵɣ ɫɩɢɧ ɞɟɣɫɬɜɭɟɬ ɪɟɡɭɥɶɬɢɪɭɸɳɟɟ
ɷɮɮɟɤɬɢɜɧɨɟ ɩɨɥɟ:
HHRV
eff i ij
j
!
¦
. (2.3)
ɝɞɟ R =<
P
j
>. Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɟɥɢɱɢɧɚ
VVVr
ij
j
ij
j
() (| |)0
¦¦
25
26
o
ɹɜɥɹɟɬɫɹ ɧɭɥɟɜɨɣ ɮɭɪɶɟ-ɤɨɦɩɨɧɟɧɬɨɣ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
V
(
q
):
Vq V r r iqr r
ij
j
ij
() (| |)exp{ ( )}
ooooo
¦
. (2.4)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɦɟɟɦ:
HHRV
eff i
()0 H . (2.5)
ȼ ɫɥɭɱɚɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɥɢɲɶ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟɞɟɣ
ɜɜɨɞɹɬ ɨɛɨɡɧɚɱɟɧɢɟ V(0) = ZJ, ɝɞɟ Z - ɱɢɫɥɨ ɛɥɢɠɚɣɲɢɯ ɫɨɫɟ-
ɞɟɣ; J - ɨɛɦɟɧɧɵɣ ɢɧɬɟɝɪɚɥ. ɂɫɩɨɥɶɡɭɹ ɩɪɢɛɥɢɠɟɧɢɟ ɫɪɟɞɧɟɝɨ
ɩɨɥɹ, ɡɚɦɟɧɢɦ ɢɫɬɢɧɧɨɟ ɩɨɥɟ ɷɮɮɟɤɬɢɜɧɵɦ (2.5). Ɍɨɝɞɚ ɫɪɟɞ-
ɧɢɣ ɦɚɝɧɢɬɧɵɣ ɦɨɦɟɧɬ
P
P
EP
EP
EP
EP
EP EP
i
iii
ii
ii ii
ii ii
Sp H
Sp H
HH
HH
!
!
! !
! !
exp[ ]
exp[ ]
exp[ ] exp[ ]
exp[ ] exp[ ]
,
(
E
=1/T), ɨɬɤɭɞɚ ɧɟɦɟɞɥɟɧɧɨ ɩɨɥɭɱɚɟɦ ɫɚɦɨɫɨɝɥɚɫɨɜɚɧɧɨɟ
ɭɪɚɜɧɟɧɢɟ ȼɟɣɫɫɚ:
>@
RHV th
E
()0 R
j
. (2.6)
ȿɫɥɢ ɫɪɚɜɧɢɬɶ ɟɝɨ ɫ ɭɪɚɜɧɟɧɢɟɦ (1.11), ɬɨ ɥɟɝɤɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ
ɪɨɥɶ ɬɟɦɩɟɪɚɬɭɪɵ Ʉɸɪɢ
4
ɢɝɪɚɟɬ ɜɟɥɢɱɢɧɚ V(0). ɗɬɨ ɭɪɚɜɧɟ-
ɧɢɟ ɦɵ ɭɠɟ ɢɫɫɥɟɞɨɜɚɥɢ ɜ ɩ.1.3. Ɉɬɦɟɬɢɦ ɬɨɥɶɤɨ, ɱɬɨ ɩɪɢ V(0)
> 0 ɢ ɩɪɢ
E
V(0) >1 (T <
4
) ɭɪɚɜɧɟɧɢɟ (2.6) ɨɩɢɫɵɜɚɟɬ ɢɦɟɧɧɨ
ɮɟɪɪɨɦɚɝɧɢɬɧɨɟ ɫɨɫɬɨɹɧɢɟ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢɢ ɫ (1.19).
ȿɳɟ ɪɚɡ ɢɫɫɥɟɞɭɟɦ ɜɨɩɪɨɫ, ɜ ɤɚɤɨɦ ɩɪɟɞɟɥɶɧɨɦ ɫɥɭɱɚɟ
ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɩɪɢɛɥɢɠɟɧɢɟ ɦɨɥɟɤɭɥɹɪɧɨɝɨ ɩɨɥɹ ȼɟɣɫɫɚ ɬɨɱ-
ɧɵɦ. Ɉɬɜɟɬ ɨɱɟɜɢɞɟɧ: ɱɢɫɥɨ ɱɚɫɬɢɰ, ɜɧɨɫɹɳɢɯ ɜɤɥɚɞ ɜ ɩɨɥɟ
, ɞɟɣɫɬɜɭɸɳɟɟ ɧɚ ɨɬɞɟɥɶɧɵɣ ɫɩɢɧ, ɞɨɥɠɧɨ
ɛɵɬɶ ɜɟɥɢɤɨ. Ɍɨɝɞɚ ɧɚɩɪɚɜɥɟɧɢɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɫɩɢɧɚ ɧɟ
ɦɨɠɟɬ ɫɢɥɶɧɨ ɜɥɢɹɬɶ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɫɩɢɧɚ, ɞɚɸɳɟɝɨ ɜɤɥɚɞ ɜ
ɩɨɥɟ, ɞɟɣɫɬɜɭɸɳɟɟ ɧɚ ɩɟɪɜɵɣ ɫɩɢɧ, ɬɚɤ ɤɚɤ ɷɬɨɬ ɜɬɨɪɨɣ ɫɩɢɧ
ɭɞɟɪɠɢɜɚɟɬɫɹ ɧɚ ɦɟɫɬɟ (Z-1) ɨɫɬɚɥɶɧɵɦɢ ɫɩɢɧɚɦɢ, ɝɞɟ Z - ɱɢɫ-
ɥɨ ɫɩɢɧɨɜ ɜ ɫɮɟɪɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ (ɜ ɩɪɢɛɥɢɠɟɧɢɢ ɛɥɢɠɚɣ-
ɲɢɯ ɫɨɫɟɞɟɣ ɫɨɜɩɚɞɚɸɳɟɟ ɫ ɢɯ ɱɢɫɥɨɦ). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɚɪɚ-
ɦɟɬɪ ɦɚɥɨɫɬɢ ɜ ɩɪɢɛɥɢɠɟɧɢɢ ɫɪɟɞɧɟɝɨ ɩɨɥɹ: 1/Z << 1.
HV
iij
j
¦
P
Ɂɚɞɚɱɢ
2.1.
1. Ⱦɚɧ ɝɚɦɢɥɶɬɨɧɢɚɧ ɂɡɢɧɝɚ ɫ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟɦ ɜ
ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
HJSSSHS
ij i j
ij
i
i
i
r
¦¦
12 1
0
2
0
/,
PP
,
J
ij
>0. ɇɟɨɛɯɨɞɢɦɨ ɡɚɩɢɫɚɬɶ ɟɝɨ ɜ ɩɪɢɛɥɢɠɟɧɢɢ ɫɪɟɞɧɟɝɨ ɩɨ-
ɥɹ, ɩɪɟɧɟɛɪɟɝɚɹ ɤɜɚɞɪɚɬɢɱɧɵɦɢ ɮɥɭɤɬɭɚɰɢɹɦɢ ɦɚɝɧɢɬɧɵɯ ɦɨ-
ɦɟɧɬɨɜ ([<S> - S]
2
o
0 ). ȼɜɟɫɬɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɛɥɢɠɚɣɲɢɯ
ɫɨɫɟɞɟɣ Z.
Ɋɟɲɟɧɢɟ
. ɍɱɢɬɵɜɚɹ ɪɚɡɥɨɠɟɧɢɟ, ɫɨɯɪɚɧɹɸɳɟɟ ɬɪɟɛɭɟ-
ɦɭɸ ɬɨɱɧɨɫɬɶ ɫɪɟɞɧɟɝɨ ɩɨɥɹ, ɜ ɜɢɞɟ:
SS S S S S S S
kk kk kk k k'''
| !
'
!!!,
ɩɨɥɭɱɚɟɦ:
HNHRSH
i
i
¦
12
00 0 0
/(
PP
H ),
JR
P
ɝɞɟ - ɫɪɟɞɧɟɟ ɩɨɥɟ;
R=<S
HJRZ
ij
j
00 0
¦
P
i
> - ɩɚɪɚ-
ɦɟɬɪ ɩɨɪɹɞɤɚ, ɫɪɟɞɧɢɣ ɦɚɝɧɢɬɧɵɣ ɦɨɦɟɧɬ.
2.1.
2. Ɋɚɫɫɱɢɬɚɬɶ ɫɬɚɬɢɫɬɢɱɟɫɤɭɸ ɫɭɦɦɭ Q ɜ ɩɪɢɛɥɢɠɟ-
ɧɢɢ ɫɪɟɞɧɟɝɨ ɩɨɥɹ, ɢɫɯɨɞɹ ɢɡ ɜɢɞɚ ɝɚɦɢɥɶɬɨɧɢɚɧɚ, ɩɨɥɭɱɟɧɧɨ-
ɝɨ ɜ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɟ.
Ɉɬɜɟɬ
:
>@
QHRSH
NHR HH
i
S
i
N
i
r
¦
exp( / ) exp[ ( )]
exp( / ) { ( )}
EP EP
EP EP
00 0 0
1
00 0 0
2
22ch
H
27
2.2.
ɋɜɨɛɨɞɧɚɹ ɷɧɟɪɝɢɹ ɮɟɪɪɨɦɚɝɧɟɬɢɤɚ
ɜ ɦɨɞɟɥɢ ɂɡɢɧɝɚ
.
Ɇɟɬɨɞ Ȼɪɷɝɝɚ
-
ȼɢɥɶɹɦɫɚ
Ⱦɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɩɟɪɟɯɨɞɨɜ ɮɟɪɪɨɦɚɝɧɟɬɢɤ-
ɩɚɪɚɦɚɝɧɟɬɢɤ ɜ ɦɨɞɟɥɢ ɂɡɢɧɝɚ ɩɨɥɟɡɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɫɜɨɛɨɞ-
ɧɭɸ ɷɧɟɪɝɢɸ ɫɢɫɬɟɦɵ ɢ ɢɡɭɱɢɬɶ ɫɢɬɭɚɰɢɸ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ
ɷɧɟɪɝɟɬɢɱɟɫɤɨɣ ɜɵɝɨɞɧɨɫɬɢ ɮɚɡɨɜɵɯ ɫɨɫɬɨɹɧɢɣ. ɉɨ ɨɩɪɟɞɟɥɟ-
ɧɢɸ ɫɜɨɛɨɞɧɚɹ ɷɧɟɪɝɢɹ F = E - TS, ɝɞɟ E - ɜɧɭɬɪɟɧɧɹɹ ɷɧɟɪɝɢɹ
ɫɢɫɬɟɦɵ; S - ɷɧɬɪɨɩɢɹ. ɗɧɟɪɝɢɸ E ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɬɟɪ-
ɦɨɞɢɧɚɦɢɱɟɫɤɨɟ ɫɪɟɞɧɟɟ ɨɬ ɝɚɦɢɥɶɬɨɧɢɚɧɚ:
EH V H
ij i j
ij
i
i
¦¦
12/
PP P
. (2.7)
ȼ ɩɪɢɛɥɢɠɟɧɢɢ ɫɪɟɞɧɟɝɨ ɩɨɥɹ ɩɟɪɟɦɟɧɧɵɟ
P
i
ɧɟɡɚɜɢɫɢɦɵ,
ɩɨɷɬɨɦɭ<
P
i
P
j
>=<
P
i
><
P
j
>. ɍɱɢɬɵɜɚɹ, ɱɬɨ <
P
i
>
{
<
P
>=R,
ɩɨɥɭɱɚɟɦ:
E = - R
2
4
N / 2 - R H N, (2.8)
ɝɞɟ ɭɱɬɟɧɨ, ɱɬɨ
VN Vr
l
lj
() ( )
,
0
¦
, V(0) =
4
, ɚ ɬɚɤɠɟ ɢɫɩɨɥɶ-
ɡɨɜɚɧɚ ɬɪɚɧɫɥɹɰɢɨɧɧɚɹ ɢɧɜɚɪɢɚɧɬɧɨɫɬɶ. Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɷɧ-
ɬɪɨɩɢɢ ɭɱɬɟɦ, ɱɬɨ ɩɚɪɚɦɟɬɪ ɩɨɪɹɞɤɚ R ɹɜɥɹɟɬɫɹ ɬɟɪɦɨɞɢɧɚɦɢ-
ɱɟɫɤɨɣ ɜɟɥɢɱɢɧɨɣ, ɩɨɷɬɨɦɭ ɷɧɬɪɨɩɢɸ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ, ɩɨɞ-
ɫɱɢɬɵɜɚɹ ɱɢɫɥɨ ɤɨɧɮɢɝɭɪɚɰɢɣ ɩɪɢ ɡɚɞɚɧɧɨɦ ɡɧɚɱɟɧɢɢ
. ɗɬɨ ɧɟɫɥɨɠɧɨ ɫɞɟɥɚɬɶ (ɱɬɨ ɛɵɥɨ ɩɨɤɚɡɚɧɨ
Ȼɪɷɝɝɨɦ ɢ ȼɢɥɶɹɦɫɨɦ), ɬɚɤ ɤɚɤ ɜɫɟ ɬɚɤɢɟ ɤɨɧɮɢɝɭɪɚɰɢɢ ɢɦɟɸɬ
ɜ ɩɪɢɛɥɢɠɟɧɢɢ ȼɟɣɫɫɚ ɨɞɧɭ ɢ ɬɭ ɠɟ ɷɧɟɪɝɢɸ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ R
= (Nn - Np)/N, ɝɞɟ N = Nn + Np, Nn - ɱɢɫɥɨ ɫɩɢɧɨɜ, ɧɚɩɪɚɜ-
ɥɟɧɧɵɯ ɩɨ ɩɨɥɸ, Np
- ɩɪɨɬɢɜ ɩɨɥɹ. ɗɧɬɪɨɩɢɸ ɦɨɠɧɨ ɜɵɪɚ-
ɡɢɬɶ ɱɟɪɟɡ ɥɨɝɚɪɢɮɦ ɫɬɚɬɢɫɬɢɱɟɫɤɨɝɨ ɜɟɫɚ: S = ln W, ɩɪɢ ɷɬɨɦ
ɫɬɚɬɢɫɬɢɱɟɫɤɢɣ ɜɟɫ W ɩɪɢ ɡɚɞɚɧɧɨɦ R ɪɚɜɟɧ:
RN
l
l
¦
1/
P
28
W
N
NN
np
!
!!
. (2.9)
ɂɫɩɨɥɶɡɭɹ ɩɪɢ ɛɨɥɶɲɢɯ ɡɧɚɱɟɧɢɹɯ N ɚɫɢɦɩɬɨɬɢɱɟɫɤɭɸ ɮɨɪ-
ɦɭɥɭ ɋɬɢɪɥɢɧɝɚ: ln N! = N ln N - N ɢ ɭɱɢɬɵɜɚɹ, ɱɬɨ
Nn
1
= (1+R) / 2, Np
/ N = (1-R) / 2,
ɧɚɯɨɞɢɦ ɨɤɨɧɱɚɬɟɥɶɧɨ:
S = - N/2 { (1+R) ln[(1+R)/2] +(1-R) ln [(1-R)/2]}. (2.10)
Ʉɚɤ ɢ ɫɥɟɞɨɜɚɥɨ ɨɠɢɞɚɬɶ, ɩɪɢ ɩɨɥɧɨɦ ɭɩɨɪɹɞɨɱɟɧɢɢ, ɤɨɝɞɚ ɜɫɟ
ɦɚɝɧɢɬɧɵɟ ɦɨɦɟɧɬɵ ɧɚɩɪɚɜɥɟɧɵ ɩɨ ɩɨɥɸ (R=1), ɷɧɬɪɨɩɢɹ
S=0. ɇɚɩɪɨɬɢɜ, ɩɪɢ ɩɨɥɧɨɦ ɪɚɡɭɩɨɪɹɞɨɱɟɧɢɢ (R=0) ɷɧɬɪɨɩɢɹ
ɪɚɫɯɨɞɢɬɫɹ (S = N ln 2
of
) .
ɋɜɨɛɨɞɧɚɹ ɷɧɟɪɝɢɹ ɮɟɪɪɨɦɚɝɧɟɬɢɤɚ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɪɟ-
ɡɭɥɶɬɚɬɟ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
F/N = -R
2
4
/2 - RH +
+ T/2{(1+R) ln[(1+R)/2] + (1-R) ln[(1-R)/2]}. (2.11)
ɂɫɫɥɟɞɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɜ ɩɪɟɞɟɥɟ ɦɚɥɨɝɨ ɜɧɟɲɧɟɝɨ
ɩɨɥɹ H
o
0.
ȼ ɬɨɱɤɟ ɪɚɜɧɨɜɟɫɢɹ ɫɜɨɛɨɞɧɚɹ ɷɧɟɪɝɢɹ ɞɨɥɠɧɚ ɛɵɬɶ ɦɢɧɢ-
ɦɚɥɶɧɚ, ɬ.ɟ. ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɭɫɥɨɜɢɟ
w
F /
w
R = 0:
ww T
FR N R H
R
R
§
©
¨
·
¹
¸
®
¯
½
¾
¿
0
1
2
1
1
()ln
, (2.12)
ɨɬɤɭɞɚ ɧɟɦɟɞɥɟɧɧɨ ɫɥɟɞɭɟɬ ɭɪɚɜɧɟɧɢɟ ȼɟɣɫɫɚ (2.6) ɞɥɹ ɩɚɪɚ-
ɦɟɬɪɚ ɩɨɪɹɞɤɚ R. Ʉɪɨɦɟ ɬɨɝɨ, ɞɥɹ ɚɧɚɥɢɡɚ ɧɚɦ ɩɨɧɚɞɨɛɢɬɫɹ
ɜɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ:
1
1
22
2
N
FR
T
R
ww T
/
. (2.13)
ɉɪɢ T >
4
, R=0 ɢɢɡ (2.13)
1
22
N
FR T
ww T
/ > 0,
ɬɚɤ ɱɬɨ ɫɜɨɛɨɞɧɚɹ ɷɧɟɪɝɢɹ ɢɦɟɟɬ ɜɫɟɝɨ ɨɞɢɧ ɦɢɧɢɦɭɦ (ɩɪɢ
R=0) (ɪɢɫ.5). ȼɫɥɭɱɚɟT <
4
ɜɨɡɧɢɤɚɟɬ ɟɳɟ ɨɞɢɧ ɷɤɫɬɪɟɦɭɦ
29