Στατιστική Φυσική της θερμοδυναμικής ισορροπίας
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Transcript of Στατιστική Φυσική της θερμοδυναμικής ισορροπίας
2004
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ii
, , . . , , . , , . . , , , . . , , . , , , , , . , , , , , . , , , . , , , , , , , , . , , . , . , , , , , , . , , . , , , , , , , , , . . , , ( ), . , , , . , , , .
iii
, . . , , , . , . . . . , , . , ( ) . . . . . (Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein Planck) , . , , Boltzmann. , Fermi-Dirac BoseEinstein, . , , . , , , , . , , , , . , Maxwell , . , , . , virial . BBGYK. , , , . , , , , , , . : , , . : ) , statistical ensemble, . ) , grand canonical distribution grand partition function, , . , iv
. , , , . , , , . , ( , , ), , . , , , Debye , , . IV IV-, IV- IV-, IV-. Fermi Bose. , , , BBGYK. , , , , , , , . , , . ( ), , , , . , , . , . 2004.
v
vi
, 1993 2003, , , . . . . , . , , , ;, . , , , .
vii
viii
.. 1
: .- : . . 5 . 5 . . 6 , . 9 . 11 Kelvin-Planck. 12 . 13 Carnot . 15 . 17 Clausius. KelvinPlanck. 19 Carnot . 21 . 23 . 24 Carnot . 27 Clausius. 29 . 31 . 34 & . 35
- : . . 39 . 40 H . 42 . 43 . 44 . 47 . 48 . 51 . 53 . 54 Maxwell. 56 , . 57 Cp CV. 61 . 63 & . 64
I : .- : . . 69 . 71 . 72 . 74 . 75 . 78 . 80 . 81 . 82 . 83 . 85
ix
. 86 & . 87
- : .. 88 . 89 . 90 . 91 Boltzmann. 92 . 95 . 96 . 98 . 99 . 100 . 102 . 102 . 103 . 104 . 105 & . 107
- : . . 108 . 110 . 112 . 112 . 114 & . 116
- : . . 117 . 118 . 119 . 120 . 122 . 123 . 123 . 124 . 125 . 127 & . 127
- : .. 129 . 129 . 130 . 132 . 132 . 133 . 134 . 136 (,) . 138 Taylor . 140 . 141 & . 143
- : .. 144 . 144 . 145x
. 148 . 149 . 151 . 154 . 155 . 158 . 159 . 161 . 162 & . 163
: .II- : . . 169 . 170 Kelvin-Planck Clausius. 172 . 174 . 175 . 176 & . 179
- : .. 180 . 181 De Broglie, . 184 . 186 . 187 , . 189 . 191 . 192 . 196 . 198 & . 201
III- : .. 204 Einstein . 204 . 209 . 211 . 213 . 214 Debye. 215 H Debye. 217 3. 219 (34). 220 & . 221
III : .. 223 . 223 . 224 . 225 , , . 226 Wien. 228 . 229 . 231 . 232xi
Kirchoff. 235 & . 237
III- : .. 238 . 239 . 243 . 245 . 247 . 249 & . 253
IV : .V- : .. 257 . 257 . 258 . 262 , . 264 . 266 . 267 . 269 . 271 . 275 . 277 . 278 & . 280
IV- : .. 281 . 281 . 286 . 286 . 290 . 292
IV- : .. 296 . 297 , . . 298 Clausius-Clapeyron. 300 Clausius-Clapeyron. 302 . 304 . 306 Andrews. 307 . 309 & . 311
IV- : .. 313 . 314 . Maxwell-Boltzmann. 316 Fermi-Dirac, Bose-Einstein Planck. 318 . 320 . 321 . 324 Boltzmann. 326 & . 329
xii
IV- : Fermi.. 331 Fermi . 332 Fermi. 334 Fermi. 337 Fermi. 339 . 342 Fermi. 344 . 347 & . 350
IV- : Bose.. 352 Bose-Einstein . 352 Bose . 357 Bose-Einstein . 359 . 361 Bose. 364 Bose. 367 Bose. 369 & . 371
V : .V- : .. 375 . 375 . 378 . 380 . 382 . 383 . 386 . 388 & . 390
V- : . . 391 . 392 . 394 Maxwell. 397 . 399 . 402 virial . 408 . 411 & . 413
V- : BGY. virial Clausius. 416 . 418 Boltzmann. 421 BBGYK BGY. 424 & . 428
VI : .A : . 431 : Stirling. 434 : Lagrange. . 435
xiii
. 437
: Gauss.. 439 Gauss. 441
: . . 443 n. 443 n. 444 . 445 . 446 . 447 . 449 Gauss n. 450 . 452 . 452
: . 454 : n- . 456 : . 459 . 460
. 461 . 463
xiv
. , , . . , , . . , . , . , , , . , , 6 1023 . 18gr 6 1023 , 1.7 1021 . , , 1016 . , , 10n , n 20. . , . , , , , , , , , . . , . ( ). . 1016 , , . . , , , , , , , . , , , . , ( ), , , -1-
, , , . . , , . , , , . , . . , , , . , . . , , . , , . , , , , , . , . , , , , 19 . , , . , , , , , , , . , . , , , , , , , .
-2-
I- : . - : .
-4-
- . . , . . . , , . . , , . . . . . , , , . . , , . . . .
. , , . , , , . , , , , , , . . , , , , , , . . , , , , , . : , , , , , -5-
, . . , , , , . , , , , . . , , , , , . , , . , , . V , . , , . , , , p . V V . , , . T Kelvin ( o K ) , 1 T lim ( pV ) (1) R V R = 8.31434 107 erg / o K / mole . , , . , , . , , . . , . , .
. . , , , , , , . . ( . : WAB ) , , , , . . , , , . , , , U A U B , -6-
( , , , WAB ) .
(2) . , UO . , , . . , 1 2, U U1 U 2 , 1 2, ( U = U1 + U 2 ). , 1 2, , = 1+2, , , , , , 1 2. W1 1 W12 , 2, W1 . , , U1 A U1B . , , W1 = W1 + W12 = U1 A U1B (3) , , 2, W2 = W2 + W21 = U 2 A U 2 B W21 2 1, , , W21 = W12 , , , (3), (3)
( U B U A = WAB )
W1 + W2 = (U1 A U1 ) + (U 2 U 2 B ) W , , , W = W1 + W2
(4)
W = W1 + W2 = U A U B U A U B , . , (4),
U U = (U1 U1 A ) + (U 2 B U 2 ) . , . , , . , , , U = U1 + U 2B . , , . , , . , , , -7-
, , ... , . . . , , , ... , , , , , . WAB . , , , . , ,
U B U A WAB , , , . , , , , , , . , , . , Q AB , , , . U B U A = Q AB WAB (5) , , , , . , , , , . . , , , . , , . , (5) , , , , . (5), , . , (5), . . (5) . , , , , , , . , , , , , . . , -
-8-
. . , , . (, , ).
, . , , . , . , . , . , , . , , . , , , : . dQ , , dW , dU , , , dQ = dU + dW (6) , , K 1 , 2 , ..., K . , ,
dW =
dk k =1
K
k
(7)
1 , 2 , ..., K , , . k , (7), k . (6) (7), , ,
dQ = dU +
dk k =1
K
k
(8)
, , . , , , , . dW , dW = pdV , p , dV . , , , . , a . -9-
p a , . ( 1). ' , s , , W = p a s cos . a s cos V a s . W = p V , dW . , ,
dW =
pV
. , , ,
dW = p V = pdV dV . , W12 , V1 V2 V1
W12 =
V1
pdV
, .
, , . p . , V , W W = p V
a . , , , s , ( 2). W W = p a s cos , , , , p a , . a s cos V -
-10-
. W = p V . dW W .
dW = W = p V = p V = p dV . dW , , . dW = dW = p dV
, W
W=
V . , , (.. , Joule ...). , , .
p
dV
= p dV = p V
. , , , . , , . , , , , . , , , , . , , . , , , , . , . : , , , , . . -11-
, , , . , , . , , . , W , , . , . , , . , , , . . . , , , .
KelvinPlanck. . , , . , , . , , , Q =W , ( ) Q W , . , , , . , , , . , , : . KelvinPlanck. , 3, , , q w , , , . , 3, , , , , , . , 4, , , q1 1, q2 2,
-12-
w = q1 q2 . 4 .
3 4, , , , q w , , , . . q w , Q W , . 4, , Q1 Q2 , , 1 2, , Q1 = q1 Q2 = q2 .
. , . , , , , . , , , . , , . , , , . . ( , , -13-
). , , . , . , , , , , . , , . , , , , , , , . , , , . , . , , . . , , , , . , , , . , , . , , , , . , , , . , , . . . , . , , . . , , , , . , : , Q1 , Q2 , ...., QN 1, 2, ....,, , W . , Q1 , Q2 , ...., QN 1, 2, ....,, , W , , , . , , , , Q , W ( 5). W = Q , . , , -14-
, . , W =Q0
, , . , , . , Q = W = 0 , , ( 6). Q AB , WAB . , QBA WBA , . ,
Q = Q AB + QBA W = WAB + WBA
Q = W = 0 , QBA = Q AB WBA = WAB , , . , , . Q = W = 0 . : , , . , , .
Carnot . , 4, w q1 (9) , , w , , q1 . w = q1 q2 , -15-
q2 , q = 1 2 q1 Carnot : , , . , , . ( 7), q A1 qB1 1, q A2 qB 2 2, wA wB , .
, , . ( 7) , q A2 2, q A1 1, wA .
q A1 = qB1 (10) ( , , ). , , +, , , , Q2 = q A2 qB 2 2, W = wB wA 1 , (10). , + , . , . Q2 = W = wB wA 0 wB < wA . , (10), wB wA q1B q1 A , (9), B A . , , -16-
, A B . , , , A = B . , . , , , + . , ,
Q2 = W = wB wA < 0 wB < w A ,, (10), (9), B < A , . Carnot .
. 8, , 4,
Q1 = q1 , Q2 = q2 , W =w (11) Q1 + Q2 = W (12) , . (9), (11) (12), W Q = = 1+ 2 (13) Q1 Q1 , (11), Q1 Q2 . , , . , 1 2. Carnot , , (13), Q1 Q2 . , , Q1 Q2 1 2, , Q1 Q2 . , , , .
-17-
1 2 , 1 2, , 1 Q = 1 (14) Q2 2 Q1 Q2 . , , , Q1 , Q2 W , . (14) , , Q1 Q2 . (14), , , , , , . Carnot. (14) , . , t t, ( , ) t = 273.16 o K . , Carnot 8, , 1 Q = 1 Qt t Q 1 = 273.16 1 o K Qt 1, . , , . (13) (14) W = = 1 2 1 Q1 , Carnot , , 4, (11), w = = 1 2 1 q1 . , , 1 1, , 2 2, . , . Carnot , , , Carnot. , ,
-18-
1
2 1
. 1 .
Clausius. Kelvin-Planck., , , Kelvin-Planck, . . . ( , ). , , , , , . , , : , , . , Clausius , , , 9. , 9, , w , . .
Clausius Kelvin-Planck. , : Kelvin-Planck, Clausius. Clausius Kelvin-Planck. . , , Clausius, Kelvin-Planck. Kelvin-Planck -19-
Clausius. . ( 10) , . Kelvin-Planck . q A1 1, wA = q A1 . , 9, Clausius. , , qB 2 2, 2 , qB1 1 ( ), 1 , 2, wB . ( wB = wA ). , , +, , . , , ( wB = wA ), 2, 1, 9, Clausius. , , Kelvin-Planck, Clausius.
, , . ( 10) , . Clausius . 9, , , q A 2 2. , 4, Kelvin-Planck. qB1 1, wB , qB 2 2. 2 . , , q A = qB 2 . , +, , , wB 1, 3, Kelvin-Planck. Clausius Kelvin-Planck. , , , . , , Kelvin-Planck -20-
. , Clausius . , , , .
Carnot . , 9, q2 , , , 2, w , q2 w (15) w = q1 q2
= q2 ( q1 q2 )
Carnot : , , . , , .
( 11), q A2 qB 2 2, q A1 qB1 1, wA wB , . 2 1. , , . ( 11) , q A2 2, q A1 1, wA . q A 2 = qB 2 (16) +, , , ,
Q1 = q A1 qB1 1, W = wA wB 1 , . , + , . , . -21-
Q2 = W = wA wB 0 wA wB . , (16), q A2 qB 2 wA wB (15) A B . , , , A B . , , , A = B , , . , , , + . , , Q1 = W = wA wB < 0 wA < wB , (16), (15), B < A , , , . Carnot . 9, , 1 2 1 2, . , , Q1 = q1 , Q2 = q2 , W = w (17) , , , W = Q1 + Q2 , Q 1 = 2 = (18) Q W 1+ 1 Q2 Carnot, , 1 2, , Q1 Q2 . , , , , . Carnot, Q1 Q2 , , (14). , (14) (18), , 1 2, 1 2 = = 1 1 1 2 2
-22-
2 1 2
. , , , 2 < 1 .
. , , . , p . p = p ( T ,V ) , T V . , , , , . , , , . ( ), . , , , , . , , , , . , , , , , . , , , , . , , , , , , , , . , , , ( U = U (T ,V ) ). , , . , , (, , ...) . , , , , . , , , , . , . , , . , , , , , . , , , , , , , , , , , -23-
. , , , . (1), . , , , , , . , (1) T , , , , ,
pV = RT (19-) , V V , pV = RT (19-) . , . , ( ) , (19), U = U (T ) ,( ) (19) ( ) . Clapeyron.
. T ( ), V p . , , , , , , . , . -. , , , , (20) dQ = dU + pdV , , dV = 0 ,
dQV = dU (21) V . . dT . CV dQV , , , dQ CV V (22) dT , (21),
-24-
dU (23) dT , , , . , , . , , dU = CV dT . , , , (23), . . CV CV =
dU (24) dT , U . , ( CV ), , , , . , . , , , CV =V2
CV
W=
V1
pdV = p (V V ) ,2 1
( )
. (20)
dQ = dU + d ( pV ) Vdp = d (U + pV ) Vdp , dp = 0 ,
dQ p = d (U + pV ) = d (U + RT )
(25)
Clapeyron. C p dQ p , , , dT dQ p (26) Cp dT , (23) (25), C p = CV + R . , C p = CV + R
(27)
C p C p . (27), , , , , R , . , , . -25-
, . (20), , dQ = pdV . W12 , 1 2 , Q12 , , V2
W12 = Q12 =
V1
V2
pdV = RT
V1
V dV = RT ln 2 V V1
(28)
Clapeyron, . , , . . (20), , 0 = dU + pdV dU = CV dT = CV dT
, , RT dV 0 = CV dT + V dT dV CV +R =0 (29) T V , , . (29) , . , 1: (T1,V1 ) 2: (T2 ,V2 ) , , , (29) T2
T1
dT dV CV +R = T VV1
V2
T2
T1
CV
V dT + R ln 2 = 0 T V1
(30)
, . , ( ), , , 3R 2 . , , , (.. 300), , , . (30), T V CV ln 2 + R ln 2 = 0 T1 V1 (27) R = C p CV . ,
-26-
C p CV T V ln 2 + ( 1) ln 2 = 0 T1 V1 T1V1 1 = T2V2 1 (31-) , . (31-) Poisson. p1V1 = p2V2 (31-) - , , T1 T = 12 (31-) 11 p1 p2 1 -. , (31) , , , . , .
Carnot . , Carnot. , , 1 2, . , , T1 T2 , . , , , . , . , , . , 1 2, T1 T2 , . T1 T2 . , -27-
. , 1 , T1 T2 , 2, . 2, , T2 T1 , 1, . 12 , T V , , , Carnot. , , T1 . (28), Q1 1,
V (32) Q1 = RT1 ln B VA , T2 . , 2 V (33) Q2 = RT2 ln V , , . , , (31). , (31-) VB T2 = V T1 VA T2 = V T1 , VB V = V A V V V V ln B = ln = ln VA V V , (32) (33), Q T 1 = 1 (34) Q2 T2 Carnot, Q1 Q2 Carnot 1 2 , . . (34) 1 2, Carnot. , Q1 Q2 , , 1 2 1 2, , -281 ( 1) 1 ( 1)
Q1 1 = Q2 2 (34) T1 1 = T2 2 . R , (1), Tt = 273.16 o K , . , 2 , 273.16 o K . , , T1 1 = o 273.16 K 273.16 o K
T1 = 1 () 1 . , , . T , , , .
Clausius. , , , Q1 , Q2 , ..., QN 1, 2, ..., , T1 , T2 , ..., TN , , W .
Ti =1
N
Qii
0
, . Clausius Clausius. ( 13), Carnot C1, C2, , C, , Q1 , Q2 , ..., QN 1, 2, ..., , , Q01 , Q02 , ..., Q0 N 0, T0 , W1 , W2 , ..., WN . Carnot Ci , (34), Q T i = i, i = 1, 2,...., N Q0i T0 Qi Q0i = , i = 1, 2,...., N (35) Ti T0 , , *, N
-29-
Carnot. , Carnot, , * , , . * 1, 2, ..., ,
W =W +
*
Wi =10i
N
i
0
Q* =
Qi =1
N
(36)
, , * , , Q* , W * , , , Q* . W * , Q* , . , Q* 0 (35), (36),
(37)
i =1 N i =1
N
Qi 1 = Ti T0 Qii
i =1
N
Q0i =
Q* T0
(38)
, (37),
T
0
(39)
Carnot , , . , Carnot, , , , , ,
-30-
Ti =1 N
N
Qii
0
Ti =1
Qii
0
(40)
, , , 13 (39) (40).
Ti =1
N
Qii
= 0,
( )
(41)
, , , , * . * . , , Q* , * , . (38) , ,
Ti =1
N
Qii
< 0,
( )
(42)
Clausius. N , , , , . , , , , , . , Clausius, dQ 0 (43) T , , . , , (41) (42). T , Clausius, , , . , , , . , . , . , Clausius, , , .
. , , -31-
. , , , , . , , , . ( 14), , , , , . , , , , Clausius,
a+
dQ = T
A( a )
B
dQ + T
B( )
A
dQ =0 T
(44)
+ , , , , , . , , , , ( 14). , , . : .
dQMN = dU MN +
dk k =1 K k k =1
K
k , MN
(45-)
dQNM = dU NM +
d
k , NM
(45-)
dU NM = dU MN d k , NM = d k , MN , , . , , (45), dQNM = dQMN
B( )
A
dQ = T
A( )
B
dQ T
(44) -32-
A( a )
B
dQ = T
A( )
B
dQ T
, , , . , (46), dQ T , . S , S B S A
SB S A
A,
B
dQ T
(46)
, R. Clausius, . (46) dS = dQ T . , , . , , , , , . , S , , , (46), M
S
O ,
dQ T
, , , . , . 1 2 ( 15), , , . =1+2, , 1 2 . , , . . T , , , .
1
-33-
S1B S1 A =
A B
B
dQ1 = T
A
B
B ( dQ1 ) dQ1 + T T
(47)
1 , dQ1 dQ1 2. , 2, , S2 B S2 A =
( )
A
A
dQ2 = T( )
A
B
B ( dQ2 ) dQ2 + T T
A
(48)
, , dQ2 2 , dQ2 1. , , dQ2 = dQ1 , dQ ( ( dQ = dQ1 ) + dQ2 )
, (47) (48)
( S1B S1 A ) + ( S2 B S2 A ) = A
B
B ( ( dQ1 ) dQ2 ) + = T T
A
TA
B
dQ
, 1 2, SB S A =
A
B
dQ T
, ,
S B S A = ( S1B S1 A ) + ( S2 B S2 A ) , . , , 1 2, , , S B = S1B + S2 B , 1 2, , , , .
. . , , , , ( 16).
, , , , , . , Clausius, -34-
dQ T ) . , , Q T . ) T < T < T . ) T < T Q < 0 , , , . ) T < T Q > 0 , . , , a a q Q , q , . ( ).
-38-
- . . . , , dQ = dU +
dk k =1
K
k
(1)
dU , d k K k . , dS , , dQ , , dS = dQ T , (1)
TdS = dU +
dk k =1
K
k
(2)
(1), , , (2) , . (2), , , . (2) 1 dS = dU + T
Tk =1
K
k
d k
(3)
, , , . , . , , K
S = S (U , 1,..., K ) S (U , )
(4)
. , , . , , , . , , , , . , , , , . . , , , K , K + 1 . , , -
-39-
, , . , (4), S dS = dU + U
dk =1
K
k
S k U , k
(5)
. U , k k , k , . ( , , , , , , ( Y z ) x, y ,..., w , Y x, y , z , ...., w , , , z x, y , ...., w . x, y , ...., w , z , Y ). , , (5), (3). , U , 1,..., K , 1 S = U T
(6)
S = k , T k U , k k = 1,..., K
(7)
U , 1,..., K T = T (U , 1,..., K ) (8) k = k (U , 1,..., K ) , k = 1,..., K (9) .
. , . , , , , ( 1). +, . , , U t . -40-
, +, , . , + . St + . , U A U B , . , ,St = S A (U A ) + S B (U B )
(10)
U t + U A + U B = U t = . (11) dU B = dU A (12) St + , U t . (11), , St , (10), . U A . St U A . , (12), S B dU B S A S B dSt S A = = + =0 dU A U A U B dU A U A U B A B A B St U A 2 S B dU B 2 S A 2SB d 2 St 2 S A = = 0 TA > TB TB TA Q , , , . , , ,
St
-42-
. , Q . , . (16) . (16), , , , , , , . . , , (16). , , . , , - , , (8) , , , U = U ( T , 1,..., K ) (17) . , (17), (4) (9), , K , S = S (T , 1,..., K ) (18) k = k (T , 1,..., K ) , k = 1,..., K (19) K (19), K , . , , , K , K + 1 K . (.. , , ,..) . , , . (17), (18) (19) .
. , , dQ , dT . C dQ (20) C dT .
-43-
, , , , dQ = dU , dU , , U dU U ( T + dT , ) U ( T , ) = dT T
, , U dQ = dT T
, (20), U C = T
(21)
. , , . , (21) . , , dQ = TdS . , , , S dQ = TdS T S ( T + dT , ) S ( T , ) = T dT T
S C = T T
(22)
. , , , , , . . (21) (22) , , , . , , .
. , , , . , ,
dU = TdS
dk k =1
K
k
, , -44-
. , , . TdS = d ( TS ) SdT ,
d (U TS ) = SdT
dk k =1
K
k
F , F U TS
(23)
dF = SdT
dk k =1
K
k
(24)
. F Helmholtz, , . (24): ( dT = 0 ) (24)
dk k =1
K
k
= dW = dF
dW , , dF . , . (24) F = S T
(25)
F = k k T , k
,
k = 1,..., K
(26)
, , , , , . , K (26) K . , (25) (26) . , , , . , (23), , , , , , . FA+ B +, , , , , T ( , , +),
FA+ B = U A+ B TS A+ B = U A + U B T ( S A + S B ) = U A TS A + U B TS B = FA + FB , , . , , . -45-
, , T . ( 3). , , , , , . . , , , , , , . + , , . Q . , , , , . , , , , . , S
S = U
S
( Q ) =
Q T
U , . (6), Q .
, S + Q * (27) S * S0 = S + S = + S S ,0 T * S0 +, S , S S ,0 . , , Q = U U ,0 (27)
U U ,0 U F + S S ,0 = + ,0 S ,0 T T T F . * S * S0 =
-46-
F = TS * + A0 (28) * A0 TS0 + (U ,0 TS ,0 ) . +. , , , (28), . , , : , , . . , , , . , 3, , , . , , , , , , . , T , . Q W . + , , (27). Q = U U ,0 + W (27) U U ,0 + W * S * S0 = + S S ,0 T * T S * S0 = W F + F ,0
(
)
(29)
, , . , , , . . * . , S * S0 = 0 W = F + F ,0 = F (24). , , * ( S * S0 > 0 ) (29) W < F : , . .
. , , , . . p = p(T ,V ) . , , -47-
. , , . pdV . , , TdS = dU + pdV (30)
1 S = U V T
(31)
p S (32) = V U T (6) (7). , (24), dF = SdT pdV (33) F S = T V (34)
F p = (35) V T (25) (26). (35) . CV , S U CV = T (36) = T V T V , , (21) (22). , , , , , , .
. (32) (35) , , , , . , , , . , , , ( 4). , . , , , . +, , , , ,-48-
. . , . . U A + U B = . , dU B = dU A (37)
, V A VB . dVB = dV A (38) + . , , . . St + St = S A (U A ,V A ) + S B (U B ,VB ) (39) U A U B , (37). V A VB , (38). , , , U A V A . +, , , , , St (U A ,V A ) U A V A . , (39) (37), S B dU B S A S B St S A = = + =0 U A U A V U B V dU A U A V U B V A B A B , (31), TA = TB , , (39) (38), S B dVB S A S B St S A = = + =0 V A V A T VB T dVA VA T VB T A B A B , (32), p A pB = TA TB , (40), p A = pB-49-
(40)
, , . , , . , , , , , , .
, . , , T . , ( 5). V A VB , , , , Vt + . + . , , . Ft . , , Ft = FA + FB FA FB , . FA T V A . . T , + . . , V A . , , dVB = dV A . , Ft V A FB dVB FA FB dFt FA = = + dVA V A T VB T dV A V A T VB T A B A B 2 FB dVB 2 FA 2 FB d 2 Ft 2 FA = = + 2 2 2 2 2 dV A VA T VB T dV A VA T VB T A B A B
(35), , , dFt = pB p A dVA p p d 2 Ft = A B 2 dV A V A TA VB TB p A pB , ,
-50-
. , Ft , , . , , p A = pB p A pB + 0 .
. . , , , (83), . , (22), (36) (49), , . Nernst (1906) : , . , Nernst. , , . (22). , , C dS = dT T 0 T , T = 0 , S (T , ) =
0
T
C ( T , ) dT T
(84)
T . , , , (T , ) . , , , . . , (84) . , , T +0
lim C ( T , ) = 0
(85)
, , T +0
lim CV ( T ,V ) = 0
(86)
, , (49) T +0
lim C p ( T , p ) = 0
(87)
( , (85), (86) (87),-63-
, , T 0 . , (84), ). T = 0 , , . , , S lim = 0 T 0 p T , Maxwell, V lim =0 T 0 T p
T = 0 lim p = 0 (88)T 0
, , T = 0 . S lim =0 T Maxwell T 0 V T 0 T V
p lim =0
, (59), p lim =0 T 0 T (89)
& .1. S U
A ( ) , SO UO , , , n . , .2. , , . , , . ( ). 3. C1 = C2 C T1 T2 . , -64-
S SO = A ( ) (U U O )
n
, . , . . ;4. . , T2 . C p , . , , , T1 , T2 . . 5. T1 T2 . , , , . , . 6. . , , p T , T . 7. Van der Waalsa p + 2 (V b ) = RT V a b . a U ( T ,V ) = U ( T ) V U ( T ) . CV .
8. C p CV Van der Waals. 9. , , . . 10. V . 1 2, V1 V V1 . , , , , 1, 2 . ,
-65-
V , .
. ) . ) Joule,
J V V ( (-9) ). ) , Van der Waals, Joule . ) . ) ;11. , F , , L F = aT 2 ( L L0 ) , a L0 . L = L0 CL CL = bT , b . ) . ) ( S L )T . ) (T0 , L0 ) (T , L ) . ) (Ti , Li ) T f , L f . ; ) CL ( T , L ) .
T
(
)
12. . 13.
2 p CV =T 2 V T T V
2V C p = T 2 p T T p
-66-
- : . - : . - : . - : . - : . - : .
-68-
- . . , , . . , , , , . , , . , , . , , , , , , , , , , , . , , , , . V. . . , , , , ( ). , , . , , . . N , 3N 3N , , , , 3N . 3N , , f . , , . , . , , , , , , , , , . , , , . -69-
, , , , . , , , , . , , 10. , N , , , 3N , , , N . , , , . , . . , , , , . : , 2,3,..,5,... - , , . , , , , . , (.. , , ..). , , , , , . . , , , . . , , . , , , , . . . , , ' , , . , . , , , . , , , , . , -70-
, . , , , , . (.. , , , , , ...) : (.. , , , , ...) , , , (.. , ...). . , , , , , , , . , , (, ) , , , . , , . , , , , . , , , , . , , . , , . , , , ( ) . , , , ( ) .
.O , , . , , , , , . , , , , , , , . , , , 1/6. . , , ; , ' , , , . , , . , , M , M a a . M -71-
Pa a
Pa a , M , , , M , M a "" a . , , a . , , . . , , M , , . M , , M . , . , , , , . , , , , . , , M 1 1, M 2 2, M 3 3 ... M - , M , M 1 , M 2 , M 3 , ..., M Pr r (1) M Pr r , , , r . .
P = r r r
Mr = M
Mr
r
M
=
M =1 M
, , .
. , , , . , , , , , . . , , , -- , , , : . , , . -
-72-
. , , , , . , , , , , , . ( , , ), , , , , . , , , ( 108 sec ) , . , , , . . Coulomb, . . ' , , , , . Coulomb . , , . ( , , , ). , . , . , , , , - . , , , . , , , , . , , , , , , , . , , , . ' , . , , , , , . . , , , , , -73-
. , , , , , , .
., , . . , , , , . . X . , , ... r X X r , X . Pr r , X
P Xr r
r
(2)
X . (3) r X , X , X , . , , , , ( ) X 1 (4) X Nr r
X
P (X
X
)
2
N . N 1023 X 1011 , - . X , (2), , , X , , X . , , . , , . . , , , , . , , . , , ,
-74-
, , . , . . ' , . , , , , . , , , . , , . , , , , . , , : . , , , , . , , , , , .
. , . , , . (2) (3), . Pr , , , , . , , , , , . Pr , (2), X . , ' , . . , , , , . , , , , , . , , . , , , , , -75-
, , , , , , . , , . . , , . , , : : , , . , , , , , . , , , , . , , , , , . , , , , , . : : , , , . . , . , --. , , . , , . , , , . . , , , 6 . , : 1/6, , 1, 2 ... 1/6 . 1/6 ; , 6. . , , . , , , -76-
, , , , (, ) . , , : , , , . . . . , , . , , . . . . , , , , . , , . . , , , . , , , , , . . --, , , , , , , , , . , , . , , . , , , . . . , . , . . , , . , -77-
, . . , , . . , , , , . , . . , , , , , . , . , , , . , , , , , , , , , . , , , , , , .
. , , , , . , , , , , , : . , : , , , . 1 2 , M 1 M 2 , 1 2, . . M 1 M 2 , , . . , , 1, 2. , 2, , 1. , t , 1 M 1 2 M 2 . 1 2 , , , M1 + M 2 = M (5)
-78-
dt M 1 M 2 dM 1 dM 2 , . (5), dM 2 = dM 1 (6) , , dM (1 2) , dt , 1 2. , , dM (21) , , 2 1. , ,
dM 1 = dM (21) dM (1 2)
(7)
, dp , t , , dt , , dp = dt , , , , . dp , dt , , dM ( ) ( ) , dt dM , t . dp =
dM ( ) = dt
(8)
(8) (7) dM 1 = 21 2 12 1 (9) dt , 1 2, , , 1 2 2 1 12 = 21 (10) (9) dM 1 = ( 2 1 ) (11) dt (6), (11) dM 2 = ( 2 1 ) (12) dt , , (11) (12)
d ( M 2 M1 ) = 2 ( 2 1 ) dt (13) , 2 1 = ( 2 1 )t = 0 e 2 t
(13)
(14)
, t = 0 , 1 2, , . -79-
, . , , t = 0 M 1 M 2 , , , , , (14), M 1 M 2 ( 1 2), , , . . , , (14), , , . , , . , (9), , , .
., , , , 1, 2, 3, M 1 , M 2 , M 3 ,... M , , , 1, 2, 3,..., . M 1 , M 2 , M 3 ,... . ! = (15) 1 ! 2 ! 3 !....... . (15) , , , M r , , , . , (15) ln = ln M !
ln M !r r
(16)
. M , M r r , , . , , , Stirling ( ), (16) ln = ln M M
( Mr
r
ln M r M r )
M=r
M
r
ln = M ln M
Mr
r
ln M r
(17)
, , (1) , M r = Pr M (17), , ,
-80-
ln =
P ln Pr r
r
(18)
M . , , , (18), Pr ln Pr . S kB
P ln Pr r
r
(19)
. k B , , Botzmann. , . . , , . , , , , .
. , , . (19), dS dP 1 dPr Pr = k B ln Pr r k B dt dt Pr dt
r
r r
, . 1 dPr dPr d Pr Pr = 0 = = Pr dt dt dt
r
r
, , . , , dS dP = k B ln Pr r (20) dt dt
r
r , r, , r. , (9), M , r s , r dPr = sr Ps rs Pr dt , , r, . , , r, -81-
. , , r . dPr = ( sr Ps rs Pr ) dts, s r
, , , , rs = sr (21) r dPr = rs ( Ps Pr ) (22) dt
s
r s , s = r . , , r, (22), (20), , , dS = k B ln Pr rs ( Ps Pr ) = k B rs ( Pr Ps ) ln Pr (23) dt
r s
r,s
(23), r s. , , dS = kB sr ( Ps Pr ) ln Ps = k B sr ( Pr Ps ) ln Ps (24) dt
r,s
r,s
, , (23) (24), (21), dS k B = rs ( ln Pr ln Ps )( Pr Ps ) (25) dt 2
rs
, .
. , , , (25). , (19), , (25) . (19), x ln x , x , , , 0 x 1 . x ln x x = 1 . x 0 ( 2). x , x ln x , , 1 e , x = 1 e . , , , . ' , , . , , -82-
(19), . , (19), , , , . ( ). , , (25), ( x y )( ln x ln y ) . , , , x = y , x y . , (25), , , , . . , (25) , ' , (25), , . , . , , . (25), , , . , , . , , . , , , 1 . (19) , 1 1 1 Pr ln Pr = ln = ln
, , , S0 = k B
P ln P = k ln = k1r r B r
B
ln
(26)
, , , (26). Lagrange, .
. Pr -83-
P ln Pr r r
r
(27)
Pr
P =1r
(28)
Pr . . Pr , . , ,d =
ln P dP = 0r r r r
(29)
(28)
dP = 0r
(30)
Pr , (30), (29) ln Pr = 0 Pr . , , (30), Pr . , (30) a , , , (29),
(ln P + a ) dP = 0r r r
(31)
(28), Pr ( ), 1 . . P2 , P3 ,... , P , , , a 1 ln P + a = 0 (32) 1 (31)
(ln P + a ) dP = 0r r r =2
P2 , P3 ,... , ln Pr + a = 0 , r = 2,3,... (33) , (32) (33) Pr = e a C , r = 1, 2,......... , , Pr , , C , (28)
1=
C = CPr =r =1 r =1
C = 1 Pr = 1 ,
-84-
0 =
r =1
Pr ln Pr =
ln = ln = ln 1 1 1r =1
, , , (19), S0 = k B ln (34) , , . Pr Pr 1 , Taylor Pr , 1 , . , , 1 Pr = + Pr
, Taylor, 1 1 Pr ln Pr ln + (1 ln ) Pr + Pr2 2
=
P ln P ln + (1 ln ) r r r =1 r =1
1 Pr + 2
Pr =1
2 r
(28) , , . , k B , 1 S k B ln k B 2
Pr =1
2 r
(35)
Pr (34). , , , .
. (22) (25) . . , , . , , , , , , . (25), , . . , , , , , , . , , , , . -85-
, , , (25), . , , , , . , , . , . . , , , , , (25), , , , . . . , , , , . , , . , (22), . . , , (25), . , , , , . .
. , , J. W. Gibbs, statistical ensemble. , , , , , . , . . , (, , ). t , , , 1 t . Et . , , Et , , , . , , , Pr r ( ), Er , E r = Et 1 t , Pr = Er Et 0, , , . , , . , -86-
. , , , , .
& .1. , . , , , . 2. f ( x ) = x ln x . x , . lim f ( x ) = 0x +0
.3. , 6 , . , 0 . 0 6 . . 4. , N , . . ( N 1023 ) , Stirling, , . 5. , 6 , . , 0, , 2 , ..... , 0, , ....., 6 .
6. N , . E = M . . , M , , , .
-87-
- .. . , , . , , . , , , , . , . , . . , , , , IV, . . , , , . , , . , , . . , , , . , , , . , , , . . , . , , , . . , , . , . , , , . , , . -88-
, , , . , , , . , , .
. , , ( 1). Et +, , , , , , , , , . E A , E B , , , Et = E A + E B + (1)
, . ( 1). E A E B , (1). , , + , Et = E A + E B (2) (2) , , + . , + , , , . (2) , . (2) : ,,,..... Et . -
-89-
1 ,,...,. (2) Et = E A + E1 E1 1. 2, ,...,, , (2), E1 = E B + E2 E2 2. Et = E A + E B + E 2 Et = E A + E B + E + ....... + E X (2) . . ( ) , , . . . . . . , , , , . , , . , , , , , . .
. , , . , , . . . , N j1 , j2 ,..., jN N r { j1, j2 ,...., jN } , . , , , . , , , , , N .
-90-
, , , . , , , . , , , R r1 , r2 ,... R {r1, r2 ,....} . , , L 1, 2,...,L, , . R , , r1 , r2 ,..., rL L 1, 2,...,L, . , PR . 1, 2,...,L. , 1, 2,...,L . 1. Pr1 r1 , , R , 1 r1 , 2,...,L Pr1 =r2 ,.., rL
P
R
, , , PR . , r1 , 1 , , . 1, , , . 2,...,L. , , , , , , . 1, 2,...,L. 1, 2,...,L, , , .
. 1, +. , , . t + , , , Pt Pt = 1 t . r s . , , Prs = 1 t-91-
Prs + r s , . , Prs r , s . PA, r r , . , , Prs s , r . + , Et , s , r E B , s E B , s = Et E A, r E A, r r . B ( EB ) , E B . , , B ( Et E A, r ) , Et E A, r , , , r . , , rs , r . , , +, , r . + 1 t , r PA, r =
B ( Et E A, r ) t
(3)
PB , s sPB , s =
, E A . (3) (4) , + . . , , , , , , , . , , (3) (4) , , . , .
A ( Et E B , s ) (4) t A ( E A )
Boltzmann. , , . -92-
(3) (4). , , . . , , , . , , , . , 1, . , , ( 2), , . , , +. r . (3) ( 1, ) Pr r Pr =
Er r , t + ( Et Er ) , Et Er .
( Et Er ) t
(5)
, , . . (5) ln Pr = ln t + ln ( Et Er ) (6) , , , , Et + . , Er , , Et . , , ln ( Et Er ) ln ( Et ) ln ( Et ) Er Et
-93-
(6) ln ( Et ) Er Et , , f .
ln Pr ln ( ( Et ) t )
ln ( Et ) ln Pr = lim ln ( ( Et ) t ) Er = C Er f Et
(7)
lim
ln ( Et ) f Et
(8)
C lim ln ( ( Et ) t )f
(9)
, , . Pr = Ce Er (7), , . . C r ,
P = Cer r r
Er
=1
, 1 Pr = e Er Z Z
(10)
er
Er
(11)
. (10) (11), , , Boltzmann. . (10) Boltzmann r , Z , Boltzmann , . , , , , , , . K 1 , 2 , ....., K , Er r Er = Er (1, 2 ,....., K ) (12) , Z , (11), , (12), , (8), K -94-
Z = Z ( , 1, 2 ,...., K ) (13) .
. , (8), (10) (11). . Z , (10), . , , (11), , , . . (11) : E0 ( ) , (11) Z=
er
E0 Er
e
= e E0
er
Er
(14)
Er Er E0 . Er . () (14) . , , . , , . , , . . : , , . . , , , , , . . , , . , , . , , . . . , . -95-
. , . . . ( ), . , . , , , , , . , .
. , (8), , , , . , . , , , , , ( 3). PR R PR = e ER Z Z=
(15) (16)
eR
ER
. . , R r s . R , E A, r E B , s r s , . (15) (16) :
Prs =
e
E A, r E B , s
e Z
(17)
-96-
Z=
er,s
E A, r E B , s
e
r s , , . : , , . , , , , , , . , Z=
er
E A,r
es
EB , s
= Z A ( ) ZB ( )
(18)
ZA ( )
er
E A, r
(19-)
ZB ( )
es
EB , s
(19-)
(17)
Prs =
e A, r e B , s Z A ( ) ZB ( )
E
E
(20)
, , PA, r r , . , , Prs , r , s . , , PA, r PA, r =
Ps E
rs
, (20), PA, r = e A, r ZA ( ) e A,r ZA ( ) E
s
e B , s e A,r 1 = ZB ( ) Z A ( ) ZB ( )
E
E
es
EB , s
, , (19-) PA, r = (21-)
s , , PB , s e B ,s = ZB ( ) E
(21-)
, , . , , . , (21) -97-
, , . , ( 2). , , , , , . , , . , , .
. , , ...., . R , , , , , , ...., . r , s , .., q , , E A, r , E B , s , ...., E X , q , E R , R {r, s,...., q} , E R = E A, r + E B , s + .... + E X , q PR R e ER e A,r e B ,s ....e X ,q = (22) Z Z Z . PR r , s , ......., , , q . Prs...q , r , s , ....., q , Prs....q = PA, r PB , s ....PX , q PR = E E E
,, ...., , , (23) ZA ZB ZX PR Prs....q . (22) (23) Z = Z AZ B ......Z X (24) , , . (, , ) . Boltzmann. .Prs....q = e E A, r
e
EB , s
e
E X ,q
-98-
(24) ln Z = ln Z A + ln Z B + ...... + ln Z X (25) , , ...., , . , , , , .
. , . . E , , E =
P Er r
r
(26)
. , , (26) E =
r
Er
e Er 1 = Z Z
E er r
Er
(27)
. e Er E r e Er =
(28)
, , , Er r , . (28), (27) E = 1 Z
r
ln Z 1 Z e Er = = Z
(29)
, . E , 2 E
(E
E
)
2
= E2 E
2
(30)
, , . 1 2 2 E2 = Pr Er = E r e Er (31) Z
r
r
, (28),
-99-
2 e Er 2 E r e Er = 2
(31) E2 = 1 2 Z 2
r
1 2Z e Er = 2 Z
(32)
, (30), (31) (32), 2 E = 2 1 Z 1 2Z 1 Z 2 2 = Z Z Z
2 E =
2 ln Z 2
(33)
, . (13) (29), . , (29), (33) 2 E =
E
(34)
, (30), . , , , , , . , , , , (34). , . . , (34), . .
. , , , , ...., , . , . , , , ...., , ln Z = ln Z A + ln Z B + + ln Z X , , (29)
-100-
E = E A + E B + E X , , , , ...., . ( , ). . , , , , , , , ....., , , (34), (35) , , , . , , . : , , , , , , , . , , , , N , . , N ( E ~ N ). , , , 2 . E ~ N , N ( E ~ N ). , E 1 ~ E N N ~ 1023 , 1011 . , , ( ) , , . , , U , E (U E ), , . (29) ln Z U = 2 2 2 2 = A + B + + X
(36)
(34)2 E =
U
(37)
, ,
-101-
, , , (37), .
S , , S = k B
P ln Pr r r r
r
Pr (10) (11). , (10), S = kB
P ( E + ln Z ) = k P E + kB r r r r
B
ln Z
Pr
r
, , U . , , . S = k BU + k B ln Z (38) , , , (36) . (38) . , , , , ...., , , U = U A + U B + +U X ln Z = ln Z A + ln Z B + + ln Z X U Z , , . , (38) S = k B (U A + U B + + U X ) + k B ( ln Z A + ln Z B + + ln Z X ) = ( k BU A + k B ln Z A ) + ( k BU B + k B ln Z B ) + + ( k BU X + k B ln Z X )
, (38), , , S = S A + S B + + S X , . , .
. , , , . , . . , , , . -102-
, ( ) , , , , . , . , , . ( , , ). . , , . , , IV.
., , , N , , . . , . . , , N . , . , = 1, 2,..., N , , N , j , j , j . j , , , , . , N , . , j , j , r , N , r { j1, j2 ,....., jN } Er Er = j1 + j2 + ....... + jN = (39)
=1
N
j
(40)
N
-103-
Z=
er
Er
(39) (40) N N exp j = e j (41) j1 , j2 ,..., jN =1 j1 , j2 ,..., jN =1 . N . , , j . , (41), N , , , j , (41) , , . , ,
Z=
Z=
e =1j
N
j
(42)
=
ej N
j
(43)
(42) (43) Z=
=1
(44)
, (43), . e j Boltzmann j . (44) , , .
. , . , p1,i1 = 1 i1 , . q N 1 i1 , . q q = {i1, j2 , j3 ,......, jN }
Eq , ,
-104-
Eq = i1 + j2 + j3 + ....... + jN = i1 +
=2
N
j
Pq Pq = e Eq
Z
1 j 1 = e i1 e =2 = e i1 Z Z
N
e =2
N
j
q , p1,i1 , p1,i1 = 1 Z
q
e
Eq
=
1 i1 e Z
j2 , j3 ,..., jN = 2
e
N
j
, (41), 1 p1,i = e i1 Z
ea =2 jaN
N
ja
(43) (44)
p1,i1 =
= e i1 aN 2 a =1
=e
i1
a =2 N
N
1
a =2
=
e
i1
1
1 , , . , , pi i pi = e i
, , , , , N .
. , ln = 2 ln = 2
(45)
2 =
(46)
. (44)
-105-
ln Z =
ln =1
N
(47)
, (47) (36) (45) ln Z ln U = = = =1 =1
N
N
N
N
(48)
, , N . , , (47) (37) (46), 2 E
2 ln Z 2 ln 2 = = = 2 =1 2 =1
(49)
N . , , . , , , , , , . j . , , , j . , ,
=
ej
j
p j j
pj = e
j
(44), N , Z = N . , , ,
ln = 2 ln = 2
2 =
, (48) (49), N ,
U=N 2 E = N 2
E = N
-106-
E 1 = U N , .
& .1. . 2. , , ..., , . , , =++...+. 3. , , ..., , . , , , , (24). 4. . 5. , , . 6. (49) (48).
-107-
- . . , , , . , , . , . . . . , . . S = k B
P ln Pr r
r
(-19)
. dS , , Pr dS = k B
(ln P + 1) dP = k ln P dP k dP = k ln P dPr r B r r B r B r r r r r
r
(1)
Pr . , ,
Pr = e Er Z (1) dS = k Br r r
dP (ln Z + E ) = k
B
ln Z
dP + k E dPr B r r r
r
, , .
Er dPr = k B d Pr Er k B Pr dEr r r r , , U . , , dS = k B
-108-
dS = k B dU k B
P dEr r
r
(2)
Er , r , , , . 1 , 2 , ..., K K ,
dEr =
Er d k k k =1k
K
(3)
(2) (3), r k , dS = k B dU k B
d P k r k =1 r
K
Er k k
(4)
k
P r r
Er , k k
k = 1,...K
(5)
, , k . k
k ,r
Er k
k
, , r . (5), (4) ,
dS = k B dU + k B
dk k =1
K
k
(6)
, , , , . , K
S = S (U , 1, 2 ,....., K )
(7)
, , . . , , . , , . , , , . , , . , K + 1 , , . K , K + 1 . K + 1 U , 1 , ..., K (6) S = kB U
(8)
-109-
S = kB k k U , k
(9)
, , , , .
. (6) 1 dU = dS kB
dk k =1
K
k
(10)
. . dW dW
dk k =1
K
k
(11)
k k . . , , k , . , . k , . . , , , , . , , . . , , dS dQ (12) kB , . , dQ , , , dS , dQ = TdS , T ( , , ). (12) (13) , -110-
= 1 k BT
. , (8), 1 S U T
(14)
(13), , , dQ = TdS . ( (14) , ). , , , , (13), T . , , , . , , . , , T , T . (13) , , , , . , , , , . , , . , . (14), , , . . . . , , , , . , . , , . , , , . . ( , , , , ...) (13), (9) S = k T k U , k
(15)
(6), (13), TdS = dU +
dk k =1
K
k
(16)
, , -, .
-111-
. . U < 0
(17)
(13) U d 1 U U = dT = k BT 2 T
, (17), U >0 T
(18)
, , . , U S C = =T T T
. ( , -, , ). (7), (14), T = T (U , 1, 2 ,....., K )
, , , , , , U = U ( T , 1, 2 ,....., K )
(19)
. (7) (15), k , ,
k = k (U ,1, 2 ,....., K ), , (19), k .
k = k (T ,1, 2 ,....., K )
(20)
(20), . , .
. -112-
, , , , , . . , (16) TdS = d ( TS ) SdT d ( TS ) SdT = dU +
dk k =1 K
K
k
d (U TS ) = SdT
dk k =1
k
, F = U TS , ,dF = SdT
dk k =1
K
k
(21)
F , , Helmhotz , , . (-37), . S = k BU + k B ln Z (-37) , , = 1 k BT (13) , , F = U TS = k BT ln Z (22) , , , . ( ) Z = Z ( T , 1, 2 ,....., K )
(23) (24)
, , FF = F ( T , 1, 2 ,....., K )
, , (21), , F = S T
(25)
F = k , k = 1,..., K k T , k-113-
(26)
, (22), (25) , (26) , , K .
. , . , , . k B ln t , t . Et . Et . , , , , , . . , , . (16), . , dU = 0 d k = 0 , k = 1, 2,.. K . , , , , , . . , , . , , ,S = kB
P ln Pr r
r
(-19)
Pr ,0 = e Er Z
(27)
, , S0 = k BU + k B ln Z , , Pr Pr Pr ,0 , . , , .
P = 0r r
(28)
, . ,
U =
E P = 0r r r
(29)
-114-
f ( x ) x ln x Taylor x = x0 , , 1 f ( x ) x0 ln x0 ( ln x0 + 1) x x2 2 x0 x x x0 . , (-19), , Pr , S S0 k B ( ln Pr,0 + 1) Pr k2B P1 Pr2 r ,0
r
r
, , (27), k 1 S S0 + k B Pr2 ( ln Z + Er + 1) Pr B 2 Pr ,0
r
r r
= S0 + k B ( ln Z + 1)
P + k E P 2 Pr B r
kB
1r ,0
Pr2
r
r
r
, (28) (29), , k 1 S S0 B Pr2 Pr ,0 2
r
, Pr , . , (27), ( , ) . , : , , , . (21). , , , , . , , , . , , , F = U TS =
( E + k T ln P ) Pr B r r
r
(30)
, (27), , F = F0 k BT ln Z Pr Pr . , , . , (28). Taylor (30), Pr , F F0 + ( Er + kBT + kBT ln Pr,0 ) Pr + k BT P1 Pr2 2 r ,0
r
r
, (27) (28), (13), k T 1 F F0 + B Pr2 Pr ,0 2
r
-115-
, , Pr , . : , , () , . , , , . . , , , , (27). , , , . , Lagrange.
& .1.
E = ( k BC )
12
T C .2. . Lagrange . 3. . Lagrange . 4. N , , . . ni i . N! = n1 ! n2 !........ Lagrange, , ni , .
-116-
- . . . . , . . , , . , , , . N , . , , , . n , , n , ,
, . 0 , , . N . , , . , , . , , . N , , .. N ~ 1023 . , , . 23 24 20. . n , , ,
n = 0 + n , n = 0,1, 2,..... (1) 0 , -
n = ( n + 1 2) , n = 0,1, 2,..... , h 2 , h Planck. 0 2 -117-
. . , , , , .
. , , , . , , , . , (1),
=
en =1
( 0 + n )
=e
0
en =1
n
e . , .
e 0 1 e Z = N
=
ln Z = N ln = E0 N ln 1 e
(
)
(2)
E0 N 0 .
ln Z U = U . , N U E0 = (3) 1 e (3) , . . (3), = k BT Tc Tc k B (4) (3) N U E0 = T T (5) e c 1 , , .
-118-
. ( T ~ Tc ) , (5), ( ), N , , . , , Tc T > N (6) , , ( , ) k BT . . , , T Tc 2T Nk B . 1 Tc T . Tc T . , Tc T 10 . , , Tc T = 20 , C 9 107 Nk B 106 Nk B , . T = Tc 20 , , , . , , , , , , , ( N ~ 1023 ) Boltzmann k B .
. (5), , ( T >> k BT , (.. T = Tc 20 ), U E0 4 109 N k BT ( N 1023 ) 1014 . , , , , k BT . , (2), , E0 , , , , Tc T . , Z Z e E0 Z ln Z = E0 + ln Z = N ln 1 e Tc
(
T
)
(8)
. , , ln Z . , , ln Z . T > E (13) 3R 1 , 6 2 12 x sinh( x ) x . , (12), , . ( ). , - Einstein, e x e x 1 x sinh x = e 2 2 (12) 2
CV 3R E e E ,
2
> 1 y y=
(e
ex x4x
1
)
2
x 4e x
"' y x .
T > 1
1 1 2 +1 H ( ) + 1 + 2e ( ) 1 + 2e = e 1 ( 2 + 1) e 2 e 2 2 , , >> 1 , , (27) (30),
(
) (
)
U Ng B B
(
e ,
)
>> 1
(31)
Ng B . , , >> 1 , , (19), , , , , , , Ng B B , , , (31).
-243-
CB . , ,
U CB = T B , (27), (19), CB = Ng B B
( g B B ) dH ( ) dH ( ) =N d T B d k BT 22
( N = N A ), dH ( ) CB = R 2 (32) d . , , , H ( ) .
3 , = 3 2 . , , , . 3 . ( > 1 :
=
e(
+1 2 )
e
2
2 +1 1 e ( ) e 1 e
(1 + e
2 e (
+1)
) e (1 + e )
ln + ln 1 + e + e ,
(
)
>> 1
, (29), (33) -245-
S Nk B (1 + ) e ,
>> 1
(34)
, . , , , N 2 ). , ' , . , , ' . : , = 1 2 , , , , . , . , (40), . . (38) (40). (38), 1 U min = Ng B B 2 n = N . , , . (40), n N . ,
-250-
0. n , , (38), , , . n n = N 2 . . n < N 2 , n = N 2 , n . n 0 , , 1 U max = + Ng B B 2 .
6 = 1 2 , . , , 2U u (43) Ng B B k T 1 B = (44) g B B , , (38), u (-1,1). , , . . , . , , , , . (38), (39) (43),
-251-
1 u 1 u 1 + u 1 + u S = Nk B ln ln + 2 2 2 2 7 u , , (43), . , , , (38), , . , , 1 S = T U B . u = 0 . , , 6. , , u > 0 , , , ' . 7 .
7 , , . , u , (43), , , u = 0.5 . , , , , . , , . , (43), , u = +0.5 . , , 7, u = +0.5 , , , . , ' , Pound, Purcell Ramsey 1951, ' , . , , . : N. F. Ramsey, "Thermodynamics and statistical mechanics at negative absolute temperature.",-252-
Physical Review, 103, 20 (1956). M. J. Klein, "Negative absolute temperature.", Physical Review, 104, 589 (1956). E. M. Purcell and R. V. Pound, Physical Review, 81, 279 (1951). A. Abraham and W. G. Proctor, Physical Review, 106, 160 (1957) 109, 1441 (1958).
& .1. 1018 cm3. spin 1 2 . 30kGauss. 3. , , . 2. N spin 1 2 . B . U . . ; 3. N , B , . 0 spin 1/2. , , . 4. spin 1/2 0 . 50000kGauss. 50% (300) ; ; : ( 0 = 1020 erg/Gauss), ( 0 = 1.4 1023 erg/Gauss). 5. N spin 1 2 . T B . . 6. 1023 erg/Gauss. 0.01 .-253-
7. B . 2 spin 1/2 0 . , , 0 B . , f hf = 2 B . . . T , , 0 B > 1 , + ( x + 1) 2 x ( x e ) 1 + 12 x
(2)
1 1 n n >> 1 (3) + + , n ! 2 n ( n e ) 1 + 12n 288n 2 n , ,
n ! 2 n ( n e )
n
1 n . Stirling. Stirling . (3) 1 1 + ln n ! n ln n n + ln ( 2 n ) + 2 12n n , , ln n ! n ln n n (4) 1 12n , , . , , ln ( 2 n ) ( ). , , n = 1010 ln ( 2 n ) 25 . n ln n n 2.2 1011 . , , n (4) , , . Stirling.
-434-
Lagrange. . u x y ( u = u( x, y ) ). x y u ( ). du u , x y u u u du = dx + dy = 0 (1) x y , x y , (1) dx dy x y . (1), x y u =0 (2-) x u =0 (2-) y , , x y , , , g ( x, y ) = 0 (3) , , (1) (2), dx dy , g g dx + dy = 0 (4) x y (3). u (4) dy g x dx dy = g y dy (1). ( x y (4) u ). du dx , u g x u =0 (5) x g y y u . u u ( x y ) , , ( (3)) . , Lagrange, .
-435-
u x y , x y u . , , (1). , , x y , (3), , , (4), , , x y u . (4) , Lagrange, g g dx + dy = 0 x y . (1), u g g u (6) + dx + + dy = 0 x y x y , , . , , x y , , x , . , . u g + =0 (7) y y (7), (6) g u + dx = 0 x x x , dx . u g (8) + =0 x x , , , x y , u (6). , (3), (7) (8) x y u , . (7) , (8), , (5). , Lagrange , , : u x y , s . u = xy u u du = dx + dy = ydx + xdy = 0 (9) x y x y u . x y . dx + dy = 0 , ,
-436-
, (9),
( y + ) dx + ( x + ) dy = 0
(10)
x , x+ =0 (10) y+ =0 x= y=s 2 , u x y , s , , , , .
. Lagrange N , , < N . u N x1 , x2 , ..., x N u = u( x1, x2 ,..., x N ) u . xn , n = 1, 2,..., N , u , u du =
xn =1
N
un
dxn = 0
(11)
, , xn , gm ( x1, x2 ,..., x N ) = 0, m = 1,..., M < N . , dxn
xn =1
N
g mn
dxn = 0,
m = 1,..., M
(12)
xn , u . , , (12) xn , (12) , , Lagrange, m
m
xn =1
N
g mn
dxn = 0,
m = 1,..., M
(13)
(13) m . , , , , (11) (13)M u g + m m dxn = 0 (14) xn xn n =1 m =1 N xn , N . x1 , x2 ,..., x N M . m . , , -
N
-437-
u + xnN
m =1
M
m
gm = 0, xn
n = N M + 1,..., N
(15)
(14) M u g + m m dxn = 0 xn xn n =1 m =1 n = 1 n = N , x1 ,..., x N . , dxn , . , (15),
u + xn
m =1
M
m
gm = 0, xn
n = 1,..., N
(16)
u . N (16), M xn , N + , xn u , m .
-438-
Gauss.. , . . . x1 , x2 , ..., x N N x pn , n = 1, 2,..., N , x xn . pn , , . x . . x , , x1 , x2 , ..., x N , , , x x1 , x2 , ..., x N . , ,
pn =1
N
n
=1
(1)
pn . ( ) x x x =
p xn =1 N
N
n n
(2)
, y = y ( x ) , x , y = yn y ( xn ) x x , ,
p yn =1
n n
(3)
x
(x x
x
)
2
=
p (xn n =12
N
n
x
)
2
(x
)x
2
= x2 + x
2x x
= x
2
x2
x =
2
x2
(4)
-439-
x2
=
p xn =1
N
2 n n
(5)
, , z ( a, b ) . , dp ( z, dz ) ( a , b ) z z + dz . ( z ) dp ( z , dz ) (6) dz , , z . (6)
( z)
dp ( z, dz ) = ( z ) dz p ( z1, z2 ) z z1 z2 ( z1 < z2 ) ( a , b ) p ( z1, z2 ) =z2
dp ( z, dz ) = ( z ) dzz1 z1
z2
(7)
z , , ( a , b ) ,
( z ) dz = 1a b
b
(8)
( z ) z . z z , (2), z = ( z ) zdza
b
(9)
, u z , (3), u = ( z ) u ( z )dza
(10)
z , ' x .
z z2
(z z )
2
=
( z) ( z a
b
z
)
2
dz =
z
2
z2
= ( z ) z 2dza
b
Gauss.-440-
G ( z ) , z ( , + ) , 2 1 z m 2 2 , >0 (11) G ( z) = e ( ) 2 Gauss z . (11) Gauss . (11) z m = 0 . , z z = m . z + 2 z
=
u ( z m ) 22 z
1 G ( z )( z m ) dz = 22 +
+
2 z m e ( )
2 2
( z m )2 dz
1 = 32 2 2
e u u 2du
2
+
z = .
e u u 2du =
2
1 2
1 G ( z ) z , m = 10 . Gauss z , m , . z , Gauss, . , 1, m +
p =
p z ( m , m + ) . p z
m
G ( z ) dz
-441-
m +
p = 1 p = 1
m
G ( z ) dz
m +
, G ( z ) m , m +
m
+
G ( z ) dz =
m G
( z ) dz
+
G ( z ) dz
m +
G ( z ) dz
+
G ( z ) dz = 1 2
m
m +
G ( z ) dz
p +
p = 2
( z m ) p = 2+
m +
G ( z ) dz =
1
2
+
m +
2 z m e ( )
2 2
dz
eu
2
2
du
(12)
. , , , , (.. 5 ). , , :+
e
2
2
d =
+
e
2 2
2
d =
d e
+
(
2
2
2
e 2
) = e 2
2 2
2
2 e = 2
+
2
d
=
e
2
e 2 2
e
+
2
3
d =
e 2 + 3
2 3e = 4
+
2
d
1 1 1 2 + O 4 (4), , =
1 >> 1 1 2 , , , , 10 p +
2 e
2
2
(13)
e
2 2
d
e 50 10
1 23 1 100 1.9 10
, (13), p10 z m 10 < z < m + 10 p10 1.5 1023 . , , z .
-442-
. . Vt ( 1), N . , , N . . , , , .
. . , N . . , , , ( ) . , , p = V Vt , V . q ( ) , , q = (Vt V ) Vt . ( ) , , , p + q =1 (1) n . , n , n . N n .
n. , -443-
PN ( n ) n 0 n N . n , , , PN ( n ) , . , (2), Bernoulli.
, n n N n , p n q N n . n , . n N n . N ! n ! ( N n ) ! n N . , PN ( n ) , n , N! (2) PN ( n ) = pnq N n n !( N n )!
( x + y)
N
=
n !( N n )! x yN!n =0
N
n N n
(3)
x + y N - . x = p y = q , (1) (2),
Pn =0
N
N
(n) = 1
, n ( ) . PN ( n ) .
n. (3) x x xN ( x + y )N 1
=
n!( N n )! nx yN!n =0
N
n N n
(4)
,