Теории / Садовский М.В. Лекции по статистической физике (2000)

80

¯ã饭­ë¥ ç«¥­ë áâ६ïâáï ª ­ã«î ¯à¨ r ! 0. ¥à¢ë© ç«¥­ ¥áâì ªã«®­®¢áª®¥ ¯®«¥

á ¬®£® ¤ ­­®£® ¨®­

b. â®à®© ç«¥­ ¯®í⮬㠨¬¥¥â á¬ëá« ¯®â¥­æ¨ « , ᮧ¤ ¢ ¥¬®£®

'a, ª®â®àãî ¬ë ¢¢¥«¨ ¢ (4.33): 'a =

;Zae .

ª¨¬ ®¡à §®¬ ¯®«ãç ¥¬ á«¥¤ãî饥 ¢ëà ¦¥­¨¥ ¤«ï ª®à५ï樮­­®© í­¥à£¨¨

¯« §¬ë:

V

Za2na0 = ;V e3

r

3=2

Ecorr =

2 e2

a

T aZa2na0

(4.42)

X

¨«¨, ¢¢®¤ï ¯®«­ë¥ ç¨á«

à §«¨ç­ëå ¨®­®¢ ¢ £ §¥ Na;= na0V :

Ecorr = ;e3r

; aNaZa2

3=2

(4.43)

T V

­â¥£à¨àãï â¥à¬®¤¨­ ¬¨ç¥áª®¥ ᮮ⭮襭¨¥ ¨¡¡á { ¥«ì¬£®«ìæ (2.66), § ¯¨á ­-

­®¥ ¢ ¢¨¤¥

E

= ;

@

F

, ¬®¦­® ­ ©â¨ ¨§ Ecorr

б®®в¢¥вбв¢гойго ¤®¡ ¢ªг ª б¢®¡®¤­®©

T2

@T

T

í­¥à£¨¨:

NaZa2!

3=2

F = Fid ;

2e3

r

Xa

(4.44)

3

T V

®áâ®ï­­ãî ¨­â¥£à¨à®¢ ­¨ï ¬®¦­® ¯®«®¦¨âì à ¢­®© ­ã«î, ¯®áª®«ìªã ¯à¨ T ! 1

¤®«¦­® ¡ëâì F = Fid. âáî¤

¯®«ãç ¥¬ ¤ ¢«¥­¨¥:

NT

e3

NaZa2!

3=2

P = V

;

rT

Xa

(4.45)

3V 3=2

£¤¥ N = Pa Na. ­ «®£¨ç­®, ª ª ¨ ¢ëè¥, ¬®¦­® ­ ©â¨:

3=2

2e3

P

1=2

Xa

!

= id ; 3T N

NaZa2

(4.46)

S =

j

fGj ln Gj ; Nj ln Nj ; (Gj ; Nj) ln(Gj ; Nj)g:

(5.14)

X

¢®¤ï ®¯ïâì á।­¨¥ ç¨á«

§ ¯®«­¥­¨ï < nj >= Nj=Gj ¯®«ã稬 á«¥¤ãî饥 ¢ëà -

¦¥­¨¥ ¤«ï í­âய¨¨ ­¥à ¢­®¢¥á­®£® ä¥à¬¨-£ § :

S = ;

j

Gj[< nj > ln < nj > +(1; < nj >) ln(1; < nj >)]

(5.15)

X

§ ãá«®¢¨ï ¬ ªá¨¬ «ì­®á⨠í⮣® ¢ëà ¦¥­¨ï ¯à¨ ¤®¯®«­¨â¥«ì­ëå ãá«®¢¨ïå:

Nj =

Gj < nj >= N;

"jGj < nj >= E

(5.16)

j

j

j

X

X

X

â.¥. ¯® ¬¥â®¤ã ­¥®¯à¥¤¥«¥­­ëå ¬­®¦¨â¥«¥© £à ­¦ ¨§:

@

[S + N + E] = 0

(5.17)

@ < nj >

j =

(Gj + Nj ; 1)!

(5.18)

(Gj ; 1)!Nj!

S =

X

f(Gj + Nj) ln(Gj + Nj) ; Nj ln Nj ; Gj ln Gjg

(5.19)

j

S =

X

Gj[(1+ < nj >) ln(1+ < nj >); < nj > ln < nj >]:

(5.20)

j

85

ਠNj

Gj (5.15), (5.20) ¥áâ¥á⢥­­® ¯¥à¥å®¤ïâ ¢ ¡®«ìæ¬ ­®¢áªãî ä®à¬ã«ã

(3.23):

X

X

e

Gj[< nj > (1 ; ln < nj >)]; < nj > 1:

S =

Gj < nj > ln

=

(5.21)

j

nj

j

®¡à â­®¬ ¯à¥¤¥«ì­®¬ á«ãç ¥, ª®£¤

Nj Gj, â.¥. < nj > 1, í­âய¨ï ¡®§¥-£ §

(5.20) ᢮¤¨âáï ª:

X

S =

Gj ln eNj ;

(5.22)

j

Gj

Gj ;1

N

áâ â¨áâ¨ç¥áª¨© ¢¥á (5.18) j =

j

.

(Gj;1)!

¡é¨¥ ᢮©áâ¢

ä¥à¬¨{ ¨ ¡®§¥{£ §®¢.

­¥à£¨ï ᢮¡®¤­®© (í«¥¬¥­â à­®©) ç áâ¨æë ¨¬¥¥â ¢¨¤:

"p =

1

(p2x + py2 + pz2) =

p2

:

(5.23)

2m

2m

gd = g dpxdpydpzdV

g = 2s + 1

(5.24)

(2 h)3

£¤¥ s { ᯨ­ ç áâ¨æë. ®í⮬㠨¬¥¥¬:

dNp =

gd

:

(5.25)

e

"p;

1

T

­â¥£à¨àãï ¯® dV ¯®«ãç ¥¬ ¯à®áâ® ¯®«­ë© ®¡ê¥¬ £ § V . ®£¤

¤«ï à á¯à¥¤¥«¥­¨ï

dNp =

gV p2dp

(5.26)

2 2h3 e

"p;

1

¨«¨ à á¯à¥¤¥«¥­¨¥ ¯® í­¥à£¨¨:

T

gV m3=2

p

(")d"

"d"

dN"

=

=

N";

(5.27)

";

p2 2h3

1

1

e

T

e

T

86

£¤¥ ¢¢¥«¨ ®ç¥­ì ¯®«¥§­ãî ¢¥«¨ç¨­ã:

gV m3=2 p

mp"

p

N(") = p

2h3 " = gV

2 2h3

;

£¤¥ p" =

2m"

(5.28)

2

1

(")

gV m3=2

1

p

"

N =

d"

N

=

d"

(5.29)

";

";

Z0

1

p2 2h3 Z0

1

e T

e T

¢®¤ï ¡¥§à §¬¥à­ãî ¯¥à¥¬¥­­ãî "=T = z, § ¯¨è¥¬:

N g(mT )3=2

Z0

1

p

z

V = p2 2h3

dz

(5.30)

ez; T 1

gV T m3=2

1

;"

= p

2h3

Z0

d"p" ln 1 e

T

(5.31)

2

­â¥£à¨àãï ¯® ç áâï¬, ­ 室¨¬:

2 gV m3=2

1

"3=2

=

Z0

d"

(5.32)

";

;3 p2 2h3

1

e T

E =

1

"dN" =

gV m3=2

1

d"

"3=2

:

(5.33)

";

Z0

p2 2h3 Z0

e

T

1

ª ¨§¢¥áâ­® ¨§ â¥à¬®¤¨­ ¬¨ª¨ =

;P V , â ª çâ® ¨§ (5.32), (5.33) ¯®«ãç ¥¬ ®¡®¡-

饭­®¥ ãà ¢­¥­¨¥ á®áâ®ï­¨ï ª¢ ­â®¢®£® ¨¤¥ «ì­®£® £ §

¢ ¢¨¤¥:

P V =

2 E

(5.34)

3

¯à¥¤¥«ì­®¬ á«ãç ¥ £ §

®«ìæ¬ ­

¨¬¥¥¬ E = 3NT=2 (§ ª®­ à ¢­®à á¯à¥¤¥«¥-

­¨ï) ¨ (5.33) ¯¥à¥å®¤¨â ¢ ãà ¢­¥­¨¥ « ¯¥©à®­ : P V = NT .

¥à¥¯¨áë¢ ï ãà ¢­¥­¨¥ (5.32) ¢ ¢¨¤¥ (áà. (5.30)):

gp

m3=2T 5=2

1

z3=2

2

P =

3 2h3

Z0

dz

(5.35)

e

";

1

T

88

¨á«® ª¢ ­â®¢ëå á®áâ®ï­¨© í«¥ªâà®­®¢, ¤¢¨¦ãé¨åáï ¢ ®¡ê¥¬¥ V á ¡á®«îâ­®©

¢¥«¨ç¨­®© ¨¬¯ã«ìá ¢ ¨­â¥à¢ «¥ p; p + dp à ¢­®:

4 p2dpV

2 (2 h)3 :

(5.41)

¯ã«ìá

p = pF (¨¬¯ã«ìá ¥à¬¨). ®«­®¥ ç¨á«® í«¥ªâà®­®¢ ¢ íâ¨å á®áâ®ï­¨ïå ®¯à¥-

¤¥«ï¥âáï ¨§3:

pF

V p3

V

N =

Z0

2

F

2h3

p

dp =

3 2h3

:

(5.42)

®£¤

¤«ï ¨¬¯ã«ìá ¥à¬¨ ¯®«ãç ¥¬:

N

1=3

pF = (3 2)1=3 V

h

(5.43)

¨§ (5.43) ¯®«ãç ¥âáï ¯à®áâ ï ®æ¥­ª

pF h=a, £¤¥ a { á।­¥¥ à ááâ®ï­¨¥ ¬¥¦¤ã

ç áâ¨æ ¬¨ ¢ £ §¥.

®®â¢¥âá⢥­­®, í­¥à£¨ï ¥à¬¨ à ¢­ 4:

p2

h2

N

2=3

h2

"F =

F

= (3 2)1=3

V

(5.44)

2m

2m

ma2

áâ¥á⢥­­®, çâ® ®­

â ª¦¥ à áâ¥â á à®á⮬ ¯«®â­®á⨠£ §

(N=V )2=3.

ã­ªæ¨ï à á¯à¥¤¥«¥­¨ï ¥à¬¨:

np =

1

(5.45)

e

"p;

+ 1

T

¯à¨ T ! 0 ¯à¥¢à é ¥âáï ¢ \áâ㯥­ìªã ¥à¬¨":

np

=

1

¯à¨

p pF

(5.46)

0

¯à¨

p > pF

¨«¨

1

¯à¨

"

= "F

n" =

(5.47)

0

¯à¨

" > = "F

¨¬¯®â¥­æ¨ « £ §

¥à¬¨ ¯à¨ T = 0 ᮢ¯ ¤ ¥â á £à ­¨ç­®© í­¥à£¨¥© ¥à¬¨:

= "F

(T = 0)

(5.48)

4 pF3

3 ªâ¨ç¥áª¨ âãâ ¯à®áâ® ¢ëç¨á«ï¥âáï ®¡ê¥¬

áä¥àë ¥à¬¨ VF

=

, ç¨á«® í«¥ªâà®­®¢

3

VF

®¯à¥¤¥«ï¥âáï ç¨á«®¬ ¤®áâ㯭ëå ¨¬ á®áâ®ï­¨© \¢­ãâà¨" í⮩ áä¥àë ª ª:

N = 2V

, çâ® ¨

(2 h)3

89

ਠª®­¥ç­ëå ⥬¯¥à âãà å T

"F

(ᨫ쭮 ¢ë஦¤¥­­ë© £ §) áâ㯥­ìª

¥à¬¨ \à §¬ë¢ ¥âáï" ¢ ®¡« áâ¨ í­¥à£¨© T (á¬. ¨á.5-1). ¥âà㤭® ¯®­ïâì, çâ®

á à®á⮬ ⥬¯¥à âãàë, ¯à¨ T "F à á¯à¥¤¥«¥­¨¥ ¥à¬¨ ¯¥à¥å®¤¨â ¢ à á¯à¥¤¥-

«¥­¨¥ ®«ìæ¬ ­ . ®®â¢¥âá⢥­­®, á à®á⮬ ⥬¯¥à âãàë 娬¯®â¥­æ¨ « ­ 稭 ¥â

㬥­ìè âìáï ®â ¯®«®¦¨â¥«ì­®© ¢¥«¨ç¨­ë ¯®à浪

"F ¨ áâ ­®¢¨âáï ®âà¨æ ⥫ì­ë¬

¢ ¡®«ìæ¬ ­®¢áª®© ®¡« á⨠T

"F .

p2=2m ¨ ¨­â¥-

®«­ ï í­¥à£¨ï £ § ¯à¨ T = 0 ¯®«ãç ¥âáï 㬭®¦¥­¨¥¬ (5.41) ­

£à¨à®¢ ­¨¥¬ ¯® ¢á¥¬ ¨¬¯ã«ìá ¬ ¤® p = pF :

V

pF

V p5

4

F

E = 2m 2h3 Z0

dpp

= 10m 2h3

(5.49)

¨«¨, á ãç¥â®¬ (5.43)

3(3 2)2=3 h2

N

2=3

E =

10

m

V

N

(5.50)

¢ë஦¤¥­­®£® £ § :

5

m

V

5=3

P =

(3 2)2=3

h2

N

(5.51)

â ª çâ® ¯à¨ T = 0 ¤ ¢«¥­¨¥ ä¥à¬¨-£ §

(N=V )5=3.

ªâ¨ç¥áª¨, ¢á¥ í⨠ä®à¬ã«ë ¯à¨¬¥­¨¬ë ¨ ¯à¨ ¤®áâ â®ç­® ­¨§ª¨å ⥬¯¥à -

âãà å T

"F . ®¯à ¢ª¨ ª ­¨¬ ¨¬¥îâ ¯®à冷ª ¢¥«¨ç¨­ë (T="F )2. ¥¬¯¥à âãà

¥à¬¨ (⥬¯¥à âãà ¢ë஦¤¥­¨ï) TF

"F

¤«ï í«¥ªâà®­­®£® £ § á ¯«®â­®áâìî

N=V 10

22

3

á¬;

, ⨯¨ç­®©, ­ ¯à¨¬¥à ¤«ï ¬¥â ««®¢, ¨ ¬ áᮩ ç áâ¨æ m me, £¤¥