Chapitre 5 Ensemble grand canonique .pdf



Nom original: Chapitre 5 - Ensemble grand-canonique.pdfTitre: Chapitre 5 - Ensemble grand-canonique.pdfAuteur: Julien

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Ω1 (E1 , δE, V1 , N1 ) +" Σ1 8 Ω2 (E2 , δE, V2 , N2 ) +" Σ2 "3 Ω(E, V, N ) +" Σ 01/
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PQ

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Σ2
Σ1
E1, V1 , N1

E1 , N1
E2 = E − E1, V2 , N2 = N − N1

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,039)' -0-.7' 6' (.)-/157'+ N '- 7'+ B0753'+ V1 '- V2 #
(.)-/157'+ ED*+ <

Ω1 (E1 , δE, V1 , N1 ) = F039)' 68*-.-+ 6' Σ1
.B'1 *,'):/' 6.,+ [E1 , E1 + δE]
'- N1 (.)-/157'+
Ω2 (E2 , δE, V2 , N2 ) = F039)' 68*-.-+ 6' Σ2
.B'1 *,'):/' 6.,+ [E2 , E2 + δE]
'- N2 (.)-/157'+
Ω(E, δE, V, N ) = F039)' 68*-.-+ 6' Σ
.B'1 *,'):/' 6.,+ [E, E + δE]
'- N (.)-/157'+
G/ δE ≪ dE1 ≪ E1 ? 7' ,039)' 68*-.-+ 6' Σ1 .=.,- *,'):/' 6.,+ 78/,-');
B.77' [E1 , E1 + dE1 ] '- N1 (.)-/157'+ '+-

Ω1 (E1 , δE, V1 , N1 )

dE1
.
δE

H"#$I

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N (E1 , dE1 , N1 ) = Ω1 (E1 , δE, V1 , N1 ) Ω2 (E − E1 , δE, V2 , N − N1 )

dE1
. @D9EB
δE

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.' N (E1 , dE1 , N1 ) G

Ω(E, δE, V, N ) =

N Z
X

N1 =0



−∞

"

Ω1 (E1 , δE, V1 , N1 )

#
Ω2 (E − E1 , δE, V2 , N − N1 )
dE1 .
×
δE

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'3 >$' *' &#01%' .' 2)%345$*'6 .' Σ1 6#43 N1 / '63 )*#%6

P1 (E1 , dE1 , N1 ) = p1 (E1 , N1 ) dE1 =

N (E1 , dE1 , N1 )
,
Ω(E, δE, V, N )

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6#43/ 5#023' 3'&$ .' @D9EB '3 @D9HB/

p1 (E1 , N1 ) =

N Z
X

N1 =0

Ω1 (E1 , δE, V1 , N1 ) Ω2 (E − E1 , δE, V2 , N − N1 )

,



−∞

Ω1 (E1 , δE, V1 , N1 ) Ω2 (E − E1 , δE, V2 , N − N1 ) dE1

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Ω2 (E − E1 , δE, V2 , N − N1 ) = eS2 (E−E1 ,V2 ,N −N1 )/kB

@D9NB

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!

"# $% &'(")$**" +"##" &",%-.," /0#$0, &" E1 = 01 N1 = 01


∂S2
∂S2
E1 −
N1 + . . . .
S2 (E − E1 , V2 , N − N1 ) = S2 (E, V2 , N ) −
∂E2 E1 =0
∂N2 E1 =0
N1 =0

N1 =0

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1
∂S2
=
,
234@5

∂E2 E1 =0
T
N1 =0

∂S2
−µ
=
,
234A5

∂N2 E1 =0
T
N1 =0

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p1 (E1 , N1 ) =

Ω1 (E1 , δE, V1 , N1 ) e−β(E1 −µN1 )
N Z
X

N1 =0

.


−β(E1 −µN1 )

Ω1 (E1 , δE, V1 , N1 )e

234?I5

dE1

−∞

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p(E, N ) =

e−β(E−µN ) ρ(E, V, N )
,
Ξ

234??5

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Ξ(T, V, µ) =

∞ Z
X

N =0



−∞

e−β(E−µN ) ρ(E, V, N ) dE .

234?!5

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#$./1& )& 7*1%43(,&+ Nℓ 8

Pℓ =

e−β(Eℓ −µNℓ )
,
Ξ

=B:?!A

$H ,* C$#3%4$# )& 7*1%4%4$# 01*#)23*#$#4'(& &+% )$##6& 7*1

Ξ=

X

e−β(Eℓ −µNℓ ) .

=B:?>A



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¯
#$./1& )& 7*1%43(,&+ L(3%(& *(%$(1 )-(#& 5*,&(1 .$F&##& N
X
¯=
N
Nℓ Pℓ .
=B:?BA


;&,$# ,-&I71&++4$# =B:?!A8 $# 5$4% '(&

Nℓ Pℓ =

1 ∂ −β(Eℓ −µNℓ )
e
.
β Ξ ∂µ

=B:?MA

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¯ = 1 ∂ log Ξ .
N
β ∂µ

:;<= >

?#), +.-+)-', -@9(',A/' $#5'(('

E¯ =

X


#( (#&' 3)' 0

Eℓ e

−β(Eℓ −µNℓ )

'& *#(+0 *@.%,D2 :;<=E>0

Eℓ Pℓ ,


∂ −β(Eℓ −µNℓ )
=−
e
,
∂β βµ




log
Ξ
.
E¯ = −
∂β βµ

:;<=B>

:;<=C>

:;<FG>

H( (#&' .)22/ 3)'0 2/ #( +.-+)-' -'2 *9,/19'2 %., ,.%%#,& I β .1'+ µ +#(2&.(&0
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*' :;<=C>0
∂ −β(Eℓ −µNℓ )
:;<F=>

e
= (Eℓ − µNℓ ) e−β(Eℓ −µNℓ ) ,
∂β
'& .-#,2

¯ = − ∂ log Ξ .
E¯ − µN
∂β

:;<FF>

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?#), +.,.+&9,/2', -. J)+&).&/#( *) (#$4,' *' %.,&/+)-'20 #( +.-+)-' -.
1.-'), $#5'((' *) +.,,9 *) (#$4,' *' %.,&/+)-'2

N2 =

X


Nℓ 2 Pℓ .

:;<FE>

!"# $%$&'"$# () *+#,-+$ .)# #)..!#' / β )-$& βµ = cte0 !1 .$"' .$12$# *3$%$&'"$# ($
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! ! "#$%%&'(

!

"#$#%&#'($()* + %,-./&*#0) 1!23456 0) &

Nℓ 2 Pℓ =
9%0': 0) ;(/* -<'#'(

N2

∂ 2 −β(Eℓ −µNℓ )
e
.
β 2 Ξ ∂µ2
1




1 ∂2Ξ
1 ∂
1 ∂Ξ
1 ∂Ξ
1 ∂
=
− 2
= 2
2
2
β Ξ ∂µ
β ∂µ Ξ ∂µ
β ∂µ Ξ ∂µ


2
1 ∂ 2 log Ξ
1 ∂ log Ξ
=
,
+
2
2
β ∂µ
β ∂µ

1!2785

1!27!5

:0#*6 <0$;*( *()/ =( %,-./&*#0) 1!23 5
2
¯2 =
σN
= N2 − N

¯
1 ∂N
1 ∂ 2 log Ξ
=
.
2
2
β ∂µ
β ∂µ

1!2745

2
¯
>& =(')#?'( '(%&*#0) $0)*'(
√ ./( σN = O(N ) (* ./( =0)< %& @/<*/&*#0) '(%&A
¯ )2
¯ = O(1/ N
*#B( B&/* σN /N
C( $D$(6 () ;'0<-=&)* <0$$( %,0) &B&#* E&#* =&): %& :(<*#0) 18285 ;0/'
%,():($F%( <&)0)#./(6 0) &6 ;0/' %(: @/<*/&*#0): =,-)('G#(


2
¯


E

log
Ξ
2
2
=−
.
σE = E 2 − E¯ =
1!27 5
∂β 2 βµ
∂β βµ


¯ 2
¯ (* =0)< σE /E¯ = O(1/ E)
H<# ()<0'( 0) & σE2 = O(E)

!

"#$%%&'(

I0$$( =&): %& :(<*#0) 8246 :# %(: -)('G#(: Eℓ =(: -*&*: |ℓi 1./# 0)* Nℓ ;&'A
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=( %& B&'#&*#0) =(: ;&'&$?*'(: (J*(')(:6 :0#*
X
Xn dΛn ,
1!27M5
d¯W =
n

&B(<6

Xn =

X ∂Eℓ
∂Eℓ
=
Pℓ .
∂Λn
∂Λ
n


1!27N5

!"#$%&' () '*+',-.' /&"*01 "*2*$34'

!

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9 '*7:$)%&/# 0;2415

1 ∂ −β(Eℓ −µNℓ )
∂Eℓ −β(Eℓ −µNℓ )
e
=−
e
,
∂Λn
β ∂Λn

0124<5

/# ) )'/.( :$+ '+( -).)=>%.+( Xn ?/#@$8$7( )$, A).&)6'+( Λn (/#% B/##7( -).

Xn = −

1 ∂ log Ξ
,
β ∂Λn

012435

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9 '*7:$&'&6.+E +(% B/##7 -). d¯W = −P dV E ') -.+((&/# 0=/F+##+5 B$ (F(%>=+
+(% B/##7+ -).
1 ∂ log Ξ
P =
,
0124G5
β ∂V
+# )#)'/8&+ )A+? '*7:$)%&/# 0;24H52

!" #$%&'()*+ ,&-$./('%*$%)*0
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"# F .+=-')L)#% ') D/.=$'+ 012345 -/$. ') -./6)6&'&%7 8.)#BM?)#/#&:$+E /#
%./$A+

1 ¯
¯ .
S = kB log Ξ +
E − µN
012445
T
N& /# B7O#&% '+ -/%+#%&+' 8.)#BM?)#/#&:$+ 0/$ 8.)#BM-/%+#%&+'5

¯ ,
J = E¯ − T S − µN

0124;5

/# -+$% 7?.&.+ ') .+')%&/# 012445 (/$( ') D/.=+

(/&%

J = −kB T log Ξ ,

012415

Ξ = e−β J .

0124!5

I+ 8.)#B -/%+#%&+' J B7-+#B B+( A).&)6'+( &#B7-+#B)#%+( T E V +% µ P

J = J(T, V, µ) .

0124 5

!"! #$%&'()#* +&,$-.('%#$%)#/
!" #$%&'()*$(++( ,"-* "+.'/



∂J
∂J
∂J
dT +
dV +
dµ .
dJ =
∂T V,µ
∂V T,µ
∂µ T,V

!(+.) +" #&6)$*$.) 0123157


∂ log Ξ
∂J
= −kB log Ξ − kB T
.
∂T V,µ
∂T V,µ

012345

012385

9'7 #:";'</ 012==57



1 ∂ log Ξ
∂ log Ξ
1
¯ ,
=−
=
E¯ − µN

2
2
∂T V,µ
kB T
∂β
kB T

012>?5

(* #.)@7 #:";'</ +:(A;'(//$.) 012335 #( +:()*'.;$(7

∂J
= −S .
∂T V,µ

012>B5

I)6)7 @.D;*( *()- #( 012315 (* #( +" '(+"*$.) 012B 57 .) "

∂J
¯ .
= −N
∂µ T,V

012>35

C( DED(7 #( +:(A;'(//$.) 012315 #- F'")#G;.*()*$(+ (* #( +" '(+"*$.) 0123=57
.) .H*$()*

∂J
= −P .
012>=5
∂V T,µ

J" #$%&'()*$(++( 012345 /:&@'$* "+.'/

¯ dµ .
dJ = −S dT − P dV − N

012>>5

K(**( '(+"*$.) *'"#-$* H$() +" '(+"*$.) *L('D.#M)"D$N-( 0B2=357 N-( +:.) ;(-*
&@'$'( /.-/ +" O.'D(

d(E − T S − µN ) = −S dT − P dV − N dµ,

012>15

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!

!"#$%&' () '*+',-.' /&"*01 "*2*$34'

!" #$%&'()*+,* -*. *+.*/0)*.
¯ → ∞2 #3*4 E/
¯ N
¯
¯
"#$% &# &'(')* )+*,(-./$#('01*
N

√ *) V /N 5$'%2

¯ ) *) σE /E¯ = O(1/ E)
¯ = O(1/ N
¯ )2 &*% 714)1#8
¯ = O(1/ N
61'%01* σN /N
)'-$% δE .* &9:$*,;'* *) .1 $-(<,* .* 6#,)'41&*% %-$) $:;&';*#<&*%= >,2 61'%01*
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.*3'*$) #&-,%

¯ E¯ + δE] -1 Nℓ 6= N
¯
0
%' Eℓ ∈
/ [E,



,
@A=BCD
Pℓ =
1

¯
¯
¯

%'
E

[
E,
E
+
δE]
*)
N
=
N
 ¯


¯)
Ω(E, δE, V, N

¯ δE, V, N
¯ ) *%) &* $-(<,* .9:)#)% #3*4 :$*,;'* .#$% &9'$)*,3#&&* [E,
¯ E¯ +
-E Ω(E,
¯
δE] *) $-(<,* .* 6#,)'41&*% :;#& F N = "9#6,G% @B=AHD2 #&-,%2 &9*$),-6'* ;,#$.8
¯ V, N
¯ )2 6-1, )-1)*
4#$-$'01* 4-I$4'.* #3*4 &9*$),-6'* ('4,-4#$-$'01* S(E,
¯
¯
3#&*1, .* E 2 V *) N = J*4' '(6&'01*2 61'%01* )-1)*% &*% #1),*% ;,#$.*1,%
)+*,(-./$#('01*% 6*13*$) ?),* -<)*$1*% 6#, .:,'3#)'-$ .* &9*$),-6'*2 &9:01'8
3#&*$4* .*% .*1K *$%*(<&*%=

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)*,3#&&* [y, y + dy] 3#1)


X
X
dy
1
1
e−β(Eℓ −µNℓ ) 
,
e−β(Eℓ −µNℓ ) = 
P(y, dy) =
Ξ
Ξ
δy
ℓ ayant [y,y+dy]

ℓ ayant [y,y+δy]

@A=B D
-E -$ # '$),-.1') 1$* '$4*,)')1.* δy 2 5K:* 1$* L-'% 6-1, )-1)*%2 )*&&* 01*
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6#,)'*&&*
X
Ξ(y, δy; T, V, µ) =
e−β(Eℓ −µNℓ ) ,
@A=B!D
ℓ ayant [y,y+δy]

*) &* ;,#$.86-)*$)'*& 6#,)'*&

J ′ (y; T, V, µ) = −kB T log Ξ(y, δy; T, V, µ) ,

@A=BOD

!"! #$%&'()*+,&+-,. )%$+-,.

!

"# $%&'#'("()* +,-. / $01) 23*4%(%0 4&550


1 e−β J (y;T,V,µ)
P(y, dy) = p(y)dy =
dy .
Ξ
δy

+,-,6/

78 9&() #"&%2 :10; 2(5("#(%0508) #1 4#2 <0 "308205'"0 4#8&8(:10; "0 5#=(515
<0 "# $%&'#'("()* <0 y 4&>84(<0 #904 "0 5(8(515 <1 ?%#8<@$&)08)(0" $#%)(0"A( &8 <*90"&$$0 "0 ?%#8<@$&)08)(0" $#%)(0" J ′ (y; T, V, µ) #1)&1% <1 5(8(@
515 y = y ⋆ ; &8 #

1 ∂ 2 J ′

′ ⋆
(y − y ⋆ )2 + . . . ,
!"!#$
J (y) = J (y ) +
2 ∂y 2 y=y⋆
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(%*


∂ 2 J ′
> 0,
∂y 2 y=y⋆

y ⋆ '+, -' ./0/.1. 2' J ′ (y)" 3'-40

!"!)$

!"56$7 40 48,/'0, %-4*+ -9%::*4;/.%,/40

<%1++/'00' :41* -% =40(,/40 2' :%*,/,/40 :%*,/'--'




Ξ(y, δy; T, V, µ) ≃ e−βJ (y ) e−
%&'(

σ −2

(y−y ⋆ )2
2σ 2

,


∂ 2 J ′
> 0.

∂y 2 y=y⋆

!"!>$

!"!5$

σ " @*7 J ′ (y ⋆ ) = O(N ) (%* (9'+, 10' &%*/%8-'
10' &%*/%8-' .%(*4+(4:/A1'7 y ⋆ = O(N ) ',



∂ 2 J ′
N
1
=O
=O
.
!"!!$

2
2
∂y y=y⋆
N
N

?% -%*<'1* 2' -% <%1++/'00' &%1,
';,'0+/&' ', +/

y

'+,

B-4*+7 29%:*C+ !"!5$7

σ 2 = O(N )

',7 :1/+A1' -% 2'0+/,D 2' :*48%8/-/,D

p(y)

'+, :*4:4*,/400'--' E -% =40(,/40 2' :%*,/,/40 :%*,/'--'7 -% F1(,1%,/40 *'-%,/&'
2910' &%*/%8-' .%(*4+(4:/A1'

y

&%1,

∆y
σ
= ⋆ =O

y
y



1

N



.

!"!G$

J = −kB T log Ξ(T, V, µ) ,

!"!K$

H0I07 -' <*%02J:4,'0,/'- ,4,%- &%1,

!

!"#$%&' () '*+',-.' /&"*01 "*2*$34'

"# $% &"'()*"' +, -%.)*)*"' /.%'+0(%'"'*12, )")%$, ,3) +"''4, -%.
Z ∞ ′
Ξ (y, δy; T, V, µ)
dy .
Ξ(T, V, µ) =
δy
−∞

5676 8

9' 2)*$*3%') $:%--.";*<%)*"' /%233*,'', 5676=8 "' % %$".3




J ≃ J (y ) − kB T log





σ

δy



;

5676>8

-"2. σ/δy = O(N α )? $, +,2;*@<, ),.<, ,3) O (log N α ) = O (log N ) ,) +"'(
-"2. 2' 3A3)@<, <%(."3("-*12, 5 N ≫ 18 B
567C!8

J = J ′ (y ⋆ ) .

D' E"*) +"'( 12, 3* $% E%.*%F$, y ,3) <%(."3("-*12,? )"2) ("<<, $, 3A3)@<,?
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!" #$%&'()*$&)'+ ,- .$ /'+%&)'+ ,- 0$(&)&)'+
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ℓ = (ℓ1 , ℓ2 , ℓ3 , . . .) ,

567CK8

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+%'),3 -"2. (J%12, +,/.4 +, $*F,.)4
(1)

(2)

(3)

Eℓ = Eℓ1 + Eℓ2 + Eℓ3 + . . . ,

567CL8

,) 12, $, '"<F., )")%$ +, -%.)*(2$,3 ("..,3-"'+%') 3"*) 4/%$ G $% 3"<<, +,
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(2)

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Nℓ = Nℓ1 + Nℓ2 + Nℓ3 + . . . .

567C=8

!"! #$%&'()*$&)'+ ,- .$ #'+%&)'+ ,- /$(&)&)'+

!

"#$% &'% &#()*+*#(%, -. /#(&+*#( )' 0.1+*+*#( 21.()3&.(#(*4$' )'5*'(+

Ξ=

X

e



(1)
(1)
(2)
(2)
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1

1

2

2

3

3

,

6789:;

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%#*+

Ξ=

Y

67897;

ξn ,

n

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ξn =

X

e



(n)
(n)
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.

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%#$%3<+.+ ℓn 5.$+

Pℓn =

e



(n)
(n)
−β Eℓn −µNℓn

ξn

.

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X −β “E (r) −µN (r) ”
ℓr
ℓr
e
.
>?7@ A
ξr =
ℓr

B-6 &3 '.#+ %& -"'+$& $ '+&0+*+3# %+&C "#$#'6 =.%+ >5&123 $//+00+ ℓr = 0A 2&
2,,&/" >5&123 $//+00+ ℓr = 1A6 $=+,
(r)

(r)

E0

= E1 = 0 ,

>?7@DA

(r)
N0
(r)
N1

= 0,

>?7EFA

= 1.

>?7EGA

8$ ;23,#.23 %+ /$-#.#.23 %1&3 '.#+ %& -"'+$& =$&# $02-'

ξr = 1 + eβµ .
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X −β “E (i) −µN (i) ”
ℓi
ℓi
.
e
ξi =

>?7E!A

>?7EIA

ℓi

J. 23 /2'+ +3,2-+ ℓi = 0 /2&- &3 '.#+ =.%+ +# ℓi = 1 /2&- &3 '.#+ 2,,&/"6 23 $

E0

(i)

= 0,

>?7EKA

(i)
E1
(i)
N0
(i)
N1

= ǫ,

>?7E?A

= 0,

>?7E@A

= 1.

>?7EEA

H02-'

ξi = 1 + e−β(ǫ−µ) .

>?7E A

L&.'5&+ .0 ( $ N '.#+' %& -"'+$& +# N '.#+' .3#+-'#.#.+0'6 0$ ;23,#.23 %+ /$-#.#.23
9-$3%<,$323.5&+ %& '('#)*+ =$&# +3M3


N
N
Ξ = ξrN ξiN = 1 + eβµ
1 + e−β(ǫ−µ)
.

>?7EDA

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!

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J = −kB T log Ξ = −N kB T log 1 + eµ/(kB T ) + log 1 + e−(ǫ−µ)/(kB T ) .
567 98
:# '+,&,2%#+ &% .(&%+,*# 567;<8= *# )('+ $%&$'&(. &3(#+.*),( /' 2>2+?@(
S=−





∂J
= N kB log 1 + eµ/(kB T ) + log 1 + e−(ǫ−µ)/(kB T )
∂T
µN
(µ − ǫ) N


.
567 <8
−µ/(k
T
)
B
T [1 + e
] T [1 + e−(µ−ǫ)/(kB T ) ]

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N
N
¯ = − ∂J =
+
.
N
−µ/(k
T
)
−(µ−ǫ)/(k
B
BT )
∂µ
1+e
1+e

567 F8

¯ − T S − µN
¯ = &3H#(.-,( @*>(##( /' 2>2+?@( E%'+
G',21'( J = E
¯=
E¯ = J + T S + µN


1+

e−(µ−ǫ)/(kB T )

.

567 !8

G',21'( &3H#(.-,( /' 2>2+?@( E%'+ E = nǫ= *I n (2+ &( #*@B.( /3%+*@(2
,#+(.2+,+,(&2= &( #*@B.( @*>(# /3%+*@(2 ,#+(.2+,+,(&2 E%'+

n
¯=

N
1+

e−(µ−ǫ)/(kB T )

.

567 ;8

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(i)

Pℓi =

e



(i)
(i)
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i

i

ξi

567 68

(+ %#%&*-'(@(#+ )*'. &(2 2,+(2 /' .H2(%'7 A( #*@B.( @*>(# /3%+*@(2 /3'# 2,+(
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n
¯i =

1 ∂ log ξi
.
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567 L8

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1
n
¯i =
.
012 34
−(µ−ǫ)/(k
BT )
1+e
¯ = Nn
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E = nǫ ,
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2

N!
n 2
.
012 O4
Ω(n) = [CN ] =
n!(N − n)!
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<&6./6)#/#&=$+ :)$% )'/.(

S = 2 kB [N log N − n log n − (N − n) log (N − n)] .
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∂S
1 ∂S
2 kB
N
1
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T
∂E
ǫ ∂n
ǫ
n

012OR4

012OT4

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n=

N
1+

eǫ/(2 kB T )

.

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!

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¯ =N
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ǫ
@!AFGC
µ= .
2
E1 %*'(-)=)1. &* %,+$-.). 3)1+ @!A HC6 #1 #;.0*1. )-#%+

n
¯=

N
1+

eǫ/(2 kB T )

,

@!AFHC

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)-#%+

S = 2 kB [N log N − n
¯ log n
¯ − (N − n
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