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.. .

4- ( - )



- . .

2009


531.19

. . , . - .: , 2009 , - 19 .

, « » 4- . , ( , , , ), , [3].


§1.
1. N V . V1 V
( . [1], 46 [2], . 1, 2.)

2. , n = 1/v = N/V ( «») R. N ,v = const r N1 N2 , N1 N2 - .
: , r2 r V1 = 2 R - 2 2R + 2 , 1. 3 (, [2], . 1, 4)

3. , 1, N1 V1 ( ) N1 (N1 )2 . N, N 1,v = const, N1 - ; 1) N1 N1 , 1 N1 N, v = const, V1 /V = const - (2) |N1 - N1 | ).
( . [1], 45-46 [2], . 1, §1)

4. 200 . «» , 1/100 . ? 1,2,3 .. ?
: (. 3). N1 = 200/100 = 2, w
N
1

(N ) =

2N1 -2 1 4 e ; w0 = e-2 ; w1 = w2 = 2w0 ,w3 = w0 .. N1 ! 9 3

(. [3], . 1, 1.)

5. , t 1


, , (I )2 I = ej t (I )2 = eI /t
( . [1], 49 [2], . 1, 6.)

6. N 1/2, = e /2mc, H , z . M = (N+ - N- ) = (2N+ - N ) , N+ N- = N - N+ - , p = exp {H/} /2ch H/.
( . [1], 48 [2], . 1, 5.)

§2.
7. . 1 cm3 - ( ), .
( . [1], 54-56 [2], . 1, 12.)

8. , . , .
( . [1], 55-57 [2], . 1, 13.)

9. . -.
( . [1], 65 [2], . 1, 18.)

10. - , .
: µ| = - µ v v 2 n = - p v v 3 n,





2


(µ)2 | =

p p 1 v 6 (n)2 = v 2 - . v N v , (n)2 = (N )2 /V 2 7 3 N F , 2= 2 F (N ) N F , 2 F F , (µ)2 | = 32 N F . N -

11. , , .
: s| = -
s v

(N )2 /V 2 7, v p 2 4 (N )2 p . (s)2 | = v = 2 v V v p N p =
2 CV 3v
N



v 2 n = -

p

v

v 2 n, , (n)2 =

2 3v

,

p

, (N )2 10
F 1 N

v

=





F , (s)2 | = F .

6

F

3

1 N



F

(s)2 | =



12. - ( ), - .
( . [1], .57, [2], .1, 14)

13. V , .
( . [1], .65, [2], .1, 19)

14. , . . 3


( . [1], .74, [2], 1, 27)

15. . ,

5 , 0,1 .?

( . [1], .74, [2], 1, 26)

16. , . .
( . [1], .75-76, [2] .1, 28)

§3. .
17. , , .
( . [1], .78, [2] .1, 30)

18. ()2 , (V )2 , (S )2 , (V )N , N N N (S )N ,(pV )N .
( . [1], .79, [2] .1, 31)

19. CVN , , . .
( . [1], .81, [2] .1, 36)

20. , ( ) , (N )N , (N µ)N , (S )N .
( . [1], .79, [2] .1, 32)

4


21. , , . .
( . [1], .80, [2] .1, 34)

22. , , , .
( . [1], .83, [2] .1, 38)

23. , , N1 N2 . .
( . [1], .84, [2] .1, 39)

§4. .
24. F2 (R) (r1 )(r1 ) ,
N

(r) =
i=1

(r - ri ),

(r) = (r) - ,

1 = , v

V0 , .
( . [1], .72, [2] .1, 24)

25. F2 (R) |k |2 k (r) (. 24). F2 (R) |k |2
( .[1], . 73 [2] , . 1, 25)

26. , . , , 5


, , . 25 |k |2 F2 (R), R.
( . [1], . 90-92 [2], . 1, 44)

§5. .
27. R 10-4 , = , , .
( . [1], . 115 [2], . 2, 1)

28. , R 10-4 .
( . [1], . 116 [2], . 2, 2)

29. t ( dt ) p(t)p(t +t). t x(t)x(t +t).
( . [1], . 145 [2], . 2, 28)

30. t p x " " t 1/.
( . [1], . 146 [2], . 2, 29)

6


31. , t ( dt ) x(t) .
: x(t) = x(t) - x = F (t - t )F (t) = (t ) = , , t < t x(t)F (t) =
0 0 t

dt

1 - e-

t

F (t - t )

/2 |t | , 0 |t | > ,
t

1, dt 1 - e- 1 2 1 = = , 2 2 m2 2m

t . (. [2], . 2, 30)

32. , , .
( . [1], . 117 [2], . 2, 3)

33. (x |x, t) t 0 , t x = x W (x |x) x = x , .
( . [1], . 123-124 [2], . 2, 8)

34. (. 33) - .
( . [1], . 124-125 [2], . 2, 9)

35. - x > 0, , , x = 0 .
( . [1], . 126 [2], . 2, 11)

36. - U = 7


mg x , t = 0 x = x0 , x (x)2 . , , t, .
( . [1], . 127-128 [2], . 2, 12)

37. 0 (x) a x a +a U (x)

x L j = const, (0) = 0 > (L) U (x) = U0 = 0 a.

( . [1], . 135-136 [2], . 2, 19)

38. , , , ( , ), , x2 . D0 t = 0 .
: , N = 4 R3 n 3 N = -4R2 , , R(t) = R0 - t = R0 (1 - t/T ) , T = /nR0 . n ( . [1], . 136 [2], .2, 21)

§6. .
39. , F (t) , (t), t = 0, J ( ) -- = 0.
( . [1], . 201-202 [2], . 3, 6)

40. , J ( ) (t), , (t), (t - , t) . , 0, 1. 8


( . [1], . 199-201 [2], . 3, 4)

41. (t) F (t) = 2 exp{-|t|} 2 = F (0) . G(t) I (t) (t) = (t).
( . [1], . 203 [2], . 3, 7)

42. (t) F (t) = 2 exp{-|t|}, 2 = F (0) . G(t) I (t) (t), (t) (t)+ (t) = (t) < .
: , i + = i ,
=


2

2 2

+

.

J ( ) (t), (t) I ( ) = J (0)
2 2

2 -

2 2 - 2 +2 2 +

2

.

G(t) = (t) (t) = F (0) (. [2], . 3, 8)
2 2

2 +

e-

|t|

-

- e

|t|

.

43. , (t), (t) (t) Ё (t) = (t) (.. ). Ё
( . [1], . 203-204 [2], . 3, 9)

44. (t) (t +t) (t) J ( ) , (t) .
( . [1], . 211 [2], . 3, 13)

9


45. Fx (t) = x(t)x(0) Fv (t) = v (t)v (0) , 2 U (x) = m0 x2 /2 , .
( . [1], . 212-214 [2], . 3, 15)

46. , R , . , L LI 2 /2 = /2 ( Q2 /2C = /2) R.
( . [1], . 217-219 [2], . 3, 19)

47. Q(t), I (t) R t.
( . [1], . 216-217 [2], . 3, 17)

48. , (0, 0 ).
( . [1], . 188 [2], . 3, §8)

49. (. 15) I , , M = , = I .
: 2 = /I ,
2

=

2 2 -

2

=

2 .

§7. .
50. 10


, .
( . [1], 458-459, [2] .5, 6,7)

51. D D = const = v = const.
( . [1], 472-473, [2] .5, 15)

52. n = const = v = const.
( . [1], 473, [2] .5, 15)

53. ( , , p = n) , p(z ) = const.
( . [1], 475., [2] .5, 16)

54. , D : D n = const. = const = const.
.

55. = const = const .
( . [1], 475, [2] .5, 17)

56. , , , = const, , = const 2 = v = const e = const -.
( . [1], 476-477, [2] .5, 18)

11


57. , ( F ). , -.
( . [1], 478-479, [2] .5, 19)

58. , - .
( . [1], 481, [2] .5, 20)

59. H- ( ).
( . [1], 549, [2] .5, 60)

60. a) ; ) .
( . [1], 549-550, [2] .5, 61)

12


. 1. , . 2. , . 3. , (N = const, ), . 4. , (V = const, ), . 5. , , (N = const, ), . 6. , , (V = const, ), . 7. , . 8. ­ . 9. . ? 10. , . 11. . 12. R . 13. . 14. . . . 15. . ?

13


16. . 17. . . 18. , . 19. . , ( , ). 20. , . H ? 21. . . 22. N1 V1 . , N , V , V/N = v = const, V1 = const . 23. N1 V1 . , N1 = N1 - N1 N N , V, V1 , V1 /V = const N1 / N . 24. , . - . 25. , , . 26. , p(t)p(t +t) , . 27. , x(t)x(t +t) , . 28. ­ . 29. , . , 14


(«-»). 30. . 31. = const . 32. ( ). 33. H - . ? 34. , , .

15


, , , . ( [2]). . 1. ( ). w e
S

2. , : () ; () ; () . () w e
S -E /T

= e-

F /T

;

() w e

S -(E +pV )/T -/T

= e-

G/T

;

() w e

S -(E-µT N )/T

=e

3. , . w exp pV - S - µN 2

4. . ? p +p = F (t); F (t) 0; F (t1 )F (t2 ) = (t1 - t2 ), (t) 1 |t| < , (t) = 0 |t| > dt

1 -

5. t ( ) t 1. (p(t))2 2mt (t = 1), 2 t (t (x(t))2 = m 1)

6. ? (t) n tn Pn ( 1 ,t1 ; ... ; n-1 ,tn-1 | , 3 Pn (n ,n + dn ) n ,tn ) = P2 ( n-1 ,tn-1 | n ,tn ) 16


7. . P2 ( 1 ,t1 | 3 ,t3 ) = P2 ( 1 ,t1 | 2 ,t2 )P2 ( 2 ,t2 | 3 ,t3 ) d2

8. ­ . 1 = div D + U t 9. ­ . (x, t) = x2 exp - 4(/ )t 4 (/ )t 1 (x, t) = x2 1 exp - 4Dt 4Dt

10. ? (t) , wn (1 ,t1 ; ... ; n ,tn ) 11. ( ­). F (t) = F (0) e
-|t|

12. ? (t) , : wn (1 ,t1 + t0 ; ... ; n ,tn + t0 ) = wn (1 ,t1 ; ... ; n ,tn ) t0 13. .
= J ( ) ( - )

14. R . E
2

= 4R

15. N . wN = t
N i=1

H w w H · - ri pi ri pi

= {H, w}

17


16. s- . Fs (t, r1 ,..., rs , p1 ,..., ps ) = V
s

w

N

dq dp dr1 ...drs dp1 ...dps

17. F1 (t, r , p) n(t, r ) j (t, r ), q . n(t, r ) = N V F1 (t, r , p) dp, j (t, r ) = N V q p F1 (t, r , p) dp m

18. . 1 F1 U F1 F1 (t, r , p) + p· - · = t m r r p F1 t



19. , . F1 (t, r , p) 1 F1 U F1 1 + p· - · = t m r r p v (|r - r |) F2 (t, r , r , p, p ) · dr dp r p

20. . (0) F1 (t, r , p) 1 F1 U F1 F1 - F1 + p· - · =- t m r r p 21. . F1 (t, r , p) 1 F1 U + U (t, r ) F1 + p· - · = 0, t m r r p N U (t, r ) = n(t, r )(|r - r |) dr = F1 (t, r , p )(|r - r |) dr dp V 22. . 1 f = (f f1 - ff1 )ud dp1 , t v u = |p1 - p|/m, d = adad, f = F1 (t, p), f ,f1 ,f1 f p p , p1 p1 , p , p1 p, p1 a, 23. , . F (r , p) = exp( + · p + |p|2 ) = n(r ) 18 const |p - p0 (r )|2 exp - (2m(r ))3/2 2m(r )


24. H - H - . H(t) = F (t, r , p) ln F (t, r , p) dr dp ; (2 )3 dH(t) dt 0

F -- .


[1] .., « », . . , 1987 . 560 . [2] .., « », III, . . , ., 2002., 448 . [3] .., .., , II. . , 1981 ., 47 .

19