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. . .

4- ( - )



- . .

2008


531.19

. . , . - .: , 2008 , - 38 .

, 4- . , , [4].



« », , 1964 . , - ( ) , , , , . 2- 1- 2- . . [2]-[3], 1991 [1], : , , . [1-4].

1


:::::::::::::::::::::::::::::::::::::::::::

1.

::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

§1.
1. W V (r), W , , h1 h2 , .
(. [2], . 155)

2. , , D, P E , .
(. [2], . 157)

3. , B , M H , .
(. [2], . 158-159)

4. , , 1 .
(. [2], . 152-153)

2


§2.
5. , x, y , z f (x, y , z ) = 0, : x y z x y z = -1,
x

z

y

.
( . [2], . 145, [1], . 160)

6. , p = p(, v ) cV cP , N = const I Q .
(. [1] . 183, [2] . 162)

7. , , 10-10 .
(. [2], . 154-155)

8. cV cP I ( c) pv = , cV = const, cP = cV +1 = const. , c 0,cV ,cP , .
(. [2], . 162-163 [1], . 184-185)

9. c , (v, p) v , 3


, , c .
(. [2], . 163)

10. - , , ( ). , .
( . [2], . 199-201 [1], . 231233)

11. - , ( ).
(. [2], . 200 [1], . 233)

12. , 0o C +4o C ( ), +4o C cV cP .
(. [2], . 163-164 [1], . 186-187)

13. Q, E W , ( , ) ( ).
(. [2], . 163 177-178 [1], . 185 198-201)

4


14. , , 1 2 . , .. , , .
(. [2], . 181 [1], . 204)

15. , , , , .
( . [2], .182-183 [1], . 206-208)

§3. II
16. , cP - cV -- ( a = 0 b = 0 - ) cV .
( . [2], 165-166 [1], . 188-189)

17. , cV -- ( ) .
( . [2], . 160)

18. , cV = const cp = const p(v - b) = .
( . [2], . 161)

5


19. EM M , .
( . [2], . 170-171)

20. , , CM M M = H .
2H CM = - 2 = 0, M H (, M ) , H = M/ H = const( 0 )M - . : ( )

21. CH CM H , C () .
( . [2], . 165)

22. - , , CM = a3 ,a = const.
: . 5 II () H =-
S

S

H

S H

=-


CH

M

,
H

- M CH . 12. CH - CM a3 0 bH = 4. H S a

23. - M = bH/( - 0 ),CM = a3 : H 1 > 0 , . 6


2 .
( . [2], . 171-173 [1], . 197-198)

24. , 1 2 , , 2 1 , .
( . [2], . 192-193 [1], . 221-222)

25. 1 - 2 = d I (), ( E / V ) p = p(, V ) ( N ).
( . [2], . 189-190 [1], . 217-218)

26. I -.
( . [2], . 81 [1], . 219)

27. : ) , ) , ) , ) .
( . [2], . 193-196 [1], . 223227)

7


§4.
28. (, V , a, ) (, p, a, ) .

29. B = E- F = S ?

30. ,

p S

V S

-

p S

V

S

= 1.

: , . , (u, v ) = (x, y ) u x v x u y v y

y y

x x

v = y , , , ( y / x)y = 0, (u, v ) = (x, y ) u x .
y

. , , p V - S S (p, V ) (, · = (, V ) (, p V S V) p = S) =
S

/
V

(p, V ) = (, S ) S V

dF = -Sd - pdV ,

8


F S V =


p

.
V

31. --.
( . [2], . 165-166 [1], . 188)

32. , R h = 1 , , m.
( . [2], . 166-167 [1], . 190-191)

33. , pv , v = V/N , pv = k, p - v , p - - v . , cV a (a > 0), (, v ) f (, v ), p(, v ) cp - cV .
( . [2], . 168-169 [1], . 193-194)

34. E , = () . a D/4 , P E.
( . [2], . 170 [1], . 195-196)

35. , 9


cH cM -, H .
( . [2], . 171-172 [1], . 196-197)

36. (, V , N ), S (E ,V ,N ) = N ln e5/2 V N m 2
3/2 2

2E 3N

3/2

(, V , ) = -V m 2 2
3/2

e/ .

37. , ( - ) f = - ln 2ch H .

38. , ( - ) f = - ln 1+ e
-/

.

39. CV = bv 3 + ... , b - . , Cp - CV , Cp , . 10


( . [2], . 173-174)

§5.
40. , (, V , N ) ( , ), , Cp > CV .
[2], . 100-103 ([1], . 120-124) U (r) = 0. . [2], . 96-97 ([1], . 117-118)

41. (, V , ) ( , , ).
: , V , , , - , |,V
,

= (F (, V , N ) - N )
2

,V ,

=

1 F (, V , N ) F (, V , N ) - N + (N )2 + ... > 0 N 2 N 2 N 0, = F (, V , N ) N - N = N (, V , )

2 F (, V , N ) = N 2 N > 0,
V

11


, (N )2 = N > 0.
V

42. , , .
(, v (r)) + U (r) = const n(r) = 1/v (r), , . [2], . 103 [1], . 125.

§6. 1-
43. - a) v /p; , . ,v = ) , -1o C , v - v = -0, 0913 /.
( ) . [2], . 197-198 [1], . 228-229) ). dp = p, d = -1o , = 273o K, q = 80 / , 1 = 4, 1868 = 41, 868 · 3 , p = q = 135 , (v - v )

34 c â 0,2 P 135 · 7 = 945 1 , = .

44. , , --, q p() . 12


( . [2], . 201-202 [1], . 236)

§7. 2- .
45. -- z = - 1 = - (p/ v ) Cp - CV = p/p = 1 = v/v = 1 z 0.
( . [2], . 208-211)

46. H H () = H
0

1-

0

2



0 ; H () = 0 > 0 ,

H H = 0.
( . [2], . 221-223 [1], . 246249)

47. 2- 1, H 0 , | - 0 |/0 F (, H, ) = F0 ()+ a( - 0 ) 2 + b 3 - H, - , F / = 0, a, b - . , , .
( . [2], . 224-226)

13


48. f (, H, ) = - ln 2ch H + 0 + 0 2 , 2

f / = 0. ( ) 0 .
( . [2], . 226-229)

49. , = 0 , , CH - , M (M / H ) = - , = (0 - )/0 < 0 | | 1, > 0, > 0, > 0,

, .
( . [2], . 229-230 [1], . 256)

50. , H = M · (, M (, M ,
1/ 1/

),

) - ( , M
1/

) = (, M

1/

),

M | | H = 0 < 0, - | | ( = ), H M = 0 = 1 + / C | |- = 2 - 2 + .
( . [2], . 233-234)

14


:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

2. .
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

§1. .
51.
+ + x
2

Ik () =
-

xk e-

dx, Jk () =
0

xk e-

x

2

dx

k = 2n k = 2n +1 n = 0, 1, 2,... > 0.
( . [1], . 349-350 [3], . 78)

52. v = (vx ,vy ,vz ) w(v )dv = w(vx ,vy ,vz )dvx dvy dvz - C exp - mv 2
2

dv

v, p = mv , vx ,v = |v | = mv 2 /2, 2 vx , vx , v, v 2 , 1/v , 2 . v v 2 (v ) , 2 () .
( . [1], . 401-403 [3], . 114)

53. vx , v , , E N .
( [1], . 403-404 [3], . 115)

54. , 1 2 15


. , v0 .
( . . [1], . 407-408 [3], . 118)

55. , , n .
( . [1], . 405-406 [3], . 116)

56. ( ) , .
( . [1], . 408-409 [3], . 119)

57. wN (E )dE E N , , .
( . [1], . 415-418 [3], . 123-125)

58. , Ep = p2 c2 + m2 c4 , (p2 ) .
: = (p/mc)2 2 · exp - mc +1 , (p2 ) = 2m 2 + mc = 2m
2 2 c

mc

2 2

+1 =



mc2 , (Ep p2 /2m), = 2 , (E pc). mc p=

16


59. , (Ep = pc) Ep = (Ep )2 /Ep = 1/ 3 0, 577. =

60. w(p) = C exp {-Ep / } Ep = p2 c2 + m2 c4 ,p2 = p2 + p2 + p2 x y z px .
: (p2 + p2 )/m2 c2 = w(p) y z py pz , dpy dpz m2 c2 d , w(px ) = C 2 c2
2

1+

mc

2

mc p2 x +1 exp - 2 c2 m

2

p2 x +1 . m2 c2

§2.
61. , , wn S=-
n

wn ln wn = -ln wn .

( . [1], . 384-385 [3], . 101-102)

62. , S = -
n

wn ln wn ;

wn - En wn , = ( II ) wn , 17
n


, S/ = 1/.
( . [1], . 385-387 [3], . 102-103)

63. : ) (, V , N ) N , V , - ; ) Z (, V , N ); ) (, V , ).
( . [1], . 418-420 [3], . 125-126)

64. , Ep pv =
( Z (, V , N ))

65. /mc2 1 cV .
: Ep / = p2 p2 mc2 1 + - · + ... 2m 2m 2 mc2 , /mc2 , p2 c2 + m2 c4 / =
2

z () = = e-
mc2

e-
0

Ep /

4p2 dp = p2 2m
2



e
0

p2 2m

4p2 dp 1+

·

1 + ... 2 mc2

,

, () = 2 ln z ()/ cV = 3 15 + + .... 2 4 mc2

18


(. [1], . 411-414 [3], . 121-122)

66. , , /2, - , /2. «» .
( . [1], . 424-427 [3], . 129131)

§3. -
67. , pV = 2 , 3 p2 Ep = 2m , pV = 1 , Ep = pc. 3
( . [1], . 524-525 [3], . 212)

68. - . , , , , .
( . [1], . 525-528 [3], . 213-214)

69. -, 1 2 , ( p0 p2 /2m - F 0 ).
( . [1], . 530-532 [3], . 216)

70. , U (z ) = mg z = 0. 19


( . [1], . 534-535 [3], . 220)

71. Ep pF (p - pF ) : =m , Ep = + p . p p2 p = (p-mF ) , exp{-p /} 2 p p = pF , CVN .
( . [1], . 536 [3], . 220-221)

72. , ( = 0) , , , 0 < < (. ). ( ) - ( ) , , , m m .
. Ep , , p2 Ep = + 2m , m - , . «» ( ) Ep , , , 2 m , Ep = - 2p . exp{ 2 } 1 m 3 K , 10 1 exp{ } 1, exp{ - } ( 71) -: , Ep > > 0,

20


np = 1 e
Ep -

+1

= e

1
p2 2m

+

-

+1

e =

-

e-

p2 2m

,

«» , Ep < 0, 1 - np = 1 - e 1
Ep -

+1

=

1 e
-
Ep

+



+1

e- e- =

p2 2m

.

- , N (, V ) =
p

np =
p

(1 - np ),

, , N (, V ) = 2V e (2 )3
-

e-

p2 2m

dp =

2V e- (2 )3



e-

p2 2m

dp.

(2m)3/2 (2m )3/2 , ( ) . , m 3 + ln = =, 2 2 m 2 2V N (, V ) = (2 mm )3/2 e- 2 . 3 (2 ) = - 0 , 0 - =
p

Ep n p +
p
p2 2m

Ep (1 - np ) = dp + e-


p2 - p 2 e 2m d p 2m m m = = 2V e (2 )3
-

(

p2 +)e- 2m

=

2V (2m)3/2 (3 +)e- (2 )3

2

N (, V ), =

21


15 2V 1 = +3 + CV = (2m)3/2 3 (2 ) 2 2 1 = 2
2

2

e-

2

=

N (, V )

, , 0. ( . [3], . 221-222)

73. N = lim M/H H 0 , ( ). .
( .[1], . 537-539 [3], . 224-225)

74. = 0 F M (H ) , ± , H , .
( .[1], . 539-541 [3], . 225-227)

75. (, V , N , H ) . .
( .[1], . 541-546 [3], . 227-230)

76. -, .
( .[1], . 554-555 [3], . 235)

77. - , , = 0. 22


.[1], . 558-559 [3], . 237-238, : = -pV = - = - gV (2 )3
p

ln (1 + e- 1 e
Ep -

Ep -

)=

0

1 Ep 4 3 p dp, +1 p 3

, = 0

F

I (F ) =
0

4 d
2

1 = F 8 +1

2 2 1+ F (2F - 3)+

3 + ln (F + 8

2 1+ F ),

= 0 2 3 p2 F 3 5 2m + ..., pF 2 mc pv = 2 I (F ) = 1 3 F 3 4 pF c + ..., pF

mc, mc.

§4. -
78. , p = 0 < F = -pV = - 2 . 3
( .[1], . 566-567 [3], . 251-252)

79. - (p - v ) - , CVN .
( .[1], . 567 [3], . 252)

80. - mg z 23


1 2 . n(r) .
( .[1], . 571-573 [3], . 251-255)

81. - E (p) E (p) = pc («»), c - , p = p0 , E (p) (p - p0 ), E (p) E (p0 ) = , -p 2 E (p) = + (p2m0 ) («»), N N , cVN - .
( .[1], . 574-575 [3], . 256-258)

82. -, .
( .[1], . 565-566 [3], . 250-251)

§5.
83. , E0 E1 = E0 + . , .
( .[1], . 599-603 [3], . 276278)

84. H2 , l. 24


( .[1], . 579-582 [3], . 262-265)

85. m, U (r) = k (r - r0 )2 , 2r - . , kr2 , 2 exp (-kr0 /), kr2 , 0 .
( .[1], . 584-587 [3], . 265-267)

86. , , U (x) = x2 + x3 + x4 , > 0, x - .
( .[1], . 587-589 [3], . 268-269)

87. H . H , , .
( .[1], . 593-595 [3], . 272-273)

§6.
88. , n( ).
( .[1], . 608-609 [3], . 282)

89. . 25


( .[1], . 507-508 [3], . 199-200)

90. .
( .[1], . 508 [3], . 200)

91. , , .
( .[1], . 512-513 [3], . 203-204)

92. , .
( .[1], . 615-616 [3], . 289-290)

§7.
93. , .
( .[1], . 717 [3], . 371)

94. F2 (R).
( .[1], . 720-721 [3], . 373-374)

95. h(R) = F2 (R) - 1 - c(R) p - v .
( .[1], . 729-730, . 699 [3], . 378-379 . 357)

26


96. c(R), - h(R), ck =

- «» F2 (R) - 1 = c0 - ak 2 + ...,

, h(R) .
( .[1], . 699-700 [3], . 357-358)

97. ij = (|ri -rj |) F2 (R) .
( .[1], . 735-736 [3], . 382-383)

98. F2 (r1 , r2 ) = F12 , , , F3 (r1 , r2 , r3 ) = F123 , F123 = F12 · F23 · F13 .
( .[1], . 742-743 [3], . 387-389)

99. Q N 1/v 1 () .
( . , [1], . 747-759 [3], . 390-402). Q Q(, V , N ) = 1 VN exp -
(V ) as

1

(|ri - rj |) dr1 ...dr
1i
N

=

as

= [q (, v )]N ,

27


q , 1 /2, q (, v ) = 1 + 11 · 1 ()+ ... ; v2 ln q = 11 · 1 ()+ ... . v2

p= ln Z V =
V

11 1 - · 1 ()+ ... . v v2

, Q(, V , N ) , «» q (, v ): Qas = 1+ 11 · 1 ()+ ... v2
N

=1+N ·

11 · 1 ()+ ... . v2

, - , 1 N 1, ( , , N1 2v N - . , , Qas N/v , . Q N/v , ( N , v = const) 1 1 (). 2 ( - ) , . . 1 fij = f (|ri - rj |) = exp - (|ri - rj |) - 1 Q. Q= 1 VN 1 VN dr1 ...dr
(V ) N i
(1 + fij ) =

=

dr1 ...drN (1 + f12 )(1 + f13 )...(1 + f(
(V )

N -1)N

).

28


, N (N 2 fij , Q= 1 VN dr1 ...drN (1 +
(V )

-1)

N (N -1) 2

- - 1

fij + ...).
1i
V N , V N -2 dr3 ...drN , 1 i < j N , Q=1+ N (N - 1) 1 ·2 2 V dr1 dr2 f (|r1 - r2 |)+ ... .

V -2-N , n 1. r1 - r2 = R, r2 = r2 , dr2 = V , ( , N/v ) Q=1+N · 11 · v2 f (R)dR + ... .
(V )

1 (), ,


1 () =

f (R)dR =
0

e-

(R)

- 1 4R2 dR.

pv = 1 - 1 () 2v -, 1/v . , (R) = -U (R) R < d = 2r0 , R > d, .

, R > d U (R) f (R) = -1 U (R)/

R < d R > d

29


pv 123 1 =1+ d - v3



U (R)4R2 dR + ... .
d

-- v a 1 pv a = - =1+ b- v-b v v + ... ,

-- 2 b = d3 , 3


a=
d

U (R)4R2 dR.

, 1 () c -- , .. - v v , .

100.
N -1 N

H = -I
i=1

i i+1 - h
i=1

i ,

a) ) )

(1 ,2 , ..., N ) , i = I = 0 h = 0 .

, ±1, , ,

( .[1], . 771-774 [3], . 410-412)

30



[1] .., , . . , 1991 . 800 . [2] .., , 1, , . . , ., 2002., 283 . [3] .., , 2, , . . , ., 2002., 429 . [4] .., .., , I. . , 1981 ., 88 .

31


: 1. I . . 2. II . II . 3. . 4. --. 5. -. 6. . . 7. II . . 8. II . . 9. III . . 10. . . 11. . . 12. . . 13. , ( ­). 14. . p(, V , N ) 32


(, V , )? 15. . 16. . 17. 0 (v ). 18. . ­ . 19. . 20. . 21. . 22. . 23. I . ­. 24. II . . 1. , . 2. , . . 3. , . . 4. . . . 5. , . 33


, . 6. ( ­ ). 7. , . 8. . -. 9. . -. 10. . . 11. -. . . 12. -. 13. -. -. 14. - . 15. . . 16. . . 17. , , . ­ . 18. . 19. . . 34


20. . 21. . 22. . 23. . . . 24. . . 25. .

35


, , , ( ) 1. . Q = dE + p dV + A da - dN. 2. . dS = Q

3. . = + - +
-

4. . = - s + pv 5. . dS > Q

6. .
0

lim S (, V , a, N ) = 0

7. : F , H , G, «» . F = E - S, H = E + pV , G = F + pV = N , = F - N = -pV

8. . cp - cv = 1, = cv +
0

(cv = const, 0 = const)

36


9. --. p+ a (v - b) = , v2 cv = const;

10. . 1 (, p) = 2 (, p) 11. . cv > 0 ( ), p V <0


12. . (, v )+ U = const. 13. . wn (E , E , V , a, N ) = (En -E ) , (E ,V ,a,N ) (E ,V ,a,N ) =
n

(En -E ),

(En -E ) =

1, |En -E | E , 0, |En -E | > E

S (E ,V ,a,N ) = ln (E ,V ,a,N ) 14. . wn (, V , a, N ) = 1 e Z (, V , a, N )
-
En

,

Z (, V , a, N ) =
n

e-

En

,

F (, V , a, N ) = - ln Z (, V , a, N ) 15. . w
Nn

(, V , a, ) =

1 e (, V , a, )

-

ENn -N

,

(, V , a, ) =
N n

e-

ENn -N

,

(, V , a, ) = - ln (, V , a, ) 37


16. . , Np 0, 1 17. . w(p)dp = 1 (2m)
3/2

e

-

|p |2 2m

dp,

w(v )dv =

m 2

3/2

e

-

m|v |2 2

dv

18. - . pF = 6 2 n
1 3

,

F

2 p2 = F= 2m 2m

6 2 n

2 3

19. . u( ) = 2 2 c3 e


-1

20. ­ ­. , e ±1 «+» ­, «-» -- ­ np =
Ep -

1

21. ? ? ? , , 22

38