V
Ni(i = 1, 2, ..., N)
U V
N1, N2, ..
� t �
t �
dW = -P dV,
d W d W
W
dQ = dU - dW = dU + P dV.
U, V, Ni S
U
U1, U2
S h
1, h
2 S
S =
S(U, V, Ni)
S = S(U, V, Ni)
S(U, V, Ni) = S(U, V, Ni),
"S
> 0,
"U
V,Ni
S
U, V, Ni
U = U(S, V, Ni),
U
("U/"S)V,N = 0.
i
n
"U "U "U
dU = dS + dV + dNj.
"S "V "Nj V,S
V,Ni S,Ni j=1
"U
a" T,
"S
V,Ni
"U
- a" P,
"V
S,Ni
"U
a" �i.
"Ni V,S
T P �
1 "V
a" ą,
V "T
P
1 "V
- a" �T ,
V "P
T
T "S 1 d Q
= a" CP ,
N "T N dT
P P
T "S 1 d Q
= a" CV .
N "T N dT
V V
ą �T
CP CV
U = -U(S, V, Ni)
T = T (S, V, Ni),
P = P (S, V, Ni),
�i = �i(S, V, Ni).
(T, P, �i)
T (S, V, Ni) = T (S, V, Ni).
dU = T dS - P dV, dQ = T dS,
U S
U
S N (U, S, V, N)
� �
(h
, S, V , Ń)
N N , N Ń,
�
V = V + V = const
�
S = S + S = const,
� �
dV = dV + dV = 0 i dS = dS + dS = 0 dU = dU + dh
.
"U "U "h
"h

� �
dU + dh
= dS + dV + dS + dV
� �
"S "V
"S "V
�dS � �
�
= T dS - P dV + T - pdV .
�
S = S + S = const
� �
dS = -dS, dV = -dV ,
�)dS �)dV
(T - T - (P - P = 0.
S, V
�, �.
T = T P = P
d2U > 0.
d2U = USS(dS)2 + 2USV dSdV + UV V (dV )2
"2U "T
USS = =
"S2 "S
"P "T
USV = UV S = - = ,
"S "V
"P
UV V = -
"V
f(x, y) = A(dx)2 + B(dy)2.
A B
dT = USSdS + USV dV.
2
USV
-1
d2U = USS (dT )2 + (UV V - )(dV )2
USS
"2U "2U
= (dT )2 + (dV )2 a"
2 2
"T "V
V T
UT T (dT )2 + UV V (dV )2.
"2U " "U "P 1
= = - = .
2
"V "V "V "V V �T
T T T
dV
dT UT T
USS
"2U "T T
USS = = = .
"S2 V "S CV
V
CV 1
d2U = (dT )2 + (dV )2.
T V �T
 dT dV
CV e" 0, �T e" 0,
"T "P
e" 0, d" 0.
"S "V
V T
L(q1, .., qn, q1, ..qn, t)
Ł Ł
H(q1, ..., qn, p1, ...pn, t)
n
H(q1, ..., qn, p1, ...pn, t) = L(q1, .., qn, q1, ..qn, t) - qipi
Ł Ł Ł
i=1
"L
pi = , (i = 1, 2, ..n)
"qi
Ł
U U(S, V, N1, ..., Nn)
F = F (T, V, Ni)
"U "F
F = F (T, V, Ni), T = , -S =
"S "T
F = U - T S, dF = -SdT - P dV + �idNi
U = U(S, V, Ni) V P E
"U "E
E = E(S, P, Ni), -P = , V = ,
"V "P
E = U + P V, dE = T dS + V dP + �idNi.
U S T V P G =
G(T, P, Ni)
"U "G
G = G(T, P, Ni), T = , -S =
"S "T
"U "G
-P = , V =
"V "P
G = U - T S + P V, dG = -SdT + V dP + �idNi.
 ś
S T N �
"U "ś
ś = ś(T, V, �), T = , -S =
"S "T
"U "ś
�i = , -N =
"N "�
ś = U - T S - �N, dś = -SdT - P dV - Nd�.
M
H T V, P, T
V -M, P H.
P H
"M
� = ,
"H
dU = T dS + HdM, dE = T dS - MdH,
dF = -SdT + HdM, dG = -SdT - MdH.
"U "E "G "F
T = = , S = - = -
"S "S "T "T
M H H M
"U "F "E "G
H = = , M = - = -
"M "M "H "H
S T S T
U F, E G
"2F "2F
CV = -T , (�T )-1 = V
2 2
"T "V
V T
"2F "2G "M
CH = -T , �T = - = .
2
"T "H2 T "H
H T
 �(x1, ..xf ) f
1
S =
2
m OZ
1
m ą
2
OX
1
En = (n + ) �
2
�
n = 1, 2, .. N N n1, ..., nN
(q, p)
�h = �p�q
�h
�h (�p�q e" )
 r
1
S =
2
ą�B �B H
OZ
1
mi = ą , (i = 1, 2, 3)
2
1
m1, m2, m3 ei = �BH mi = - ei = -�BH
2
1
mi = E
2
M
m1 m2 m3
�B -3H�B
�B �B
�B �B
�B �B
-�B �B
-�B �B
-�B �B
-3�B �B
-�BH
-�BH (+ + -), (+ - +), (- + +)
-�BH
(+ + -) (+ - +) (- + +)
 (E, E + �E)
&!(E)
-�BH y
yk
&!(E, yk) +1
&!(-�BH, -1) = 2 &!(E, yk)
y
yk
&!(E, yk)
P (yk) = .
&!(E)
y
yk&!(E, yk)
< y >= = P (yk)yk.
&!(E)
k
yk
-�BH
(+ + -) (+ - +) (- - +) P+
OZ
2 1 1
< �B > a" < y > = �B Pk(yk)yk = �B - �B = �B.
3 3 3
k=ą
A A
Ć
A = A + A
A (E, E +�E) &!(E) A
(E , E + �E ) &! (E )
Ę = E + E = const.
f
&!(E) <" Ef .
N
 N
N
N(N - 1)/2
1/2 OZ
Ą Ą/2
Ć
&!(E) A
(E, E + �E) E E
E
P (E)
A (E, E + �E)
Ć
&!(Ę)
P (E) =
Ć
&!
Ć
&!
Ć
P (E) = C&!(Ę),
Ć
C E &!(E) A
(E, E + �E) A A
Ć
&!(E) = &!(E)&! (E ) = &!(E)&! (Ę - E),
P (E) = C&!(E)&! (Ę - E).
&!(E) <" Ef E &! (Ę -E)
E &! (E ) E
ź
ln P P ln P
"P
= 0.
"E
ln P (E) = ln C + ln &!(E) + ln &! (Ę - E) =
= ln C + ln &!(E) + ln &! (E ).
" ln &!(E) " ln &! (E )
- = 0,
"E "E
"&! (E ) "&! (E) "E
= - .
"E "E "E
" ln &!(E)
a" �(E).
"E
ln P
�(E) = � (E ).
E E A A lnP (E) �
T
(kBT )-1 a" �.
T kB kB =
1.38054 � 10-16erg/st
S = kB ln &!.
&!(E)
T = T
1 "S
= ,
T "E
E U
&!(E)
y
P (yk) P (y)
V N (E, E +�E)
P (Er)
Er
C : E < Er < E + �E
P (Er) =
0 :
C
P (Er) = 1.
r
Ć
A A
A A
A A A Er E Er
A A
Pr A r Er
Er + E = Ę = const.
Ć
A r A
A E = Ę - Er
Pr = C &! (Ę - Er).
C A A
Er E ln Pr E = Ę 0
" ln &! (E )
ln &! (Ę - Er) = ln &! (Ę) - Er + ...
"E 0
ln P P ln
ln P P
Er E
" ln &!
= � = (kBT )-1,
"E 0
Er
A A A
ln &! (Ę - Er) = ln &! (E ) - �Er
&! (Ę - Er) = &! (E ) exp(-�Er).
C &! (Ę) Er
Pr = C exp(-�Er).
C
exp(-�Er)
Pr = .
exp(-�Er)
r
A
Er
Pr exp(-�Er)
A
y
r
yre-�E
< y >= .
r
e-�E
 H(p, q)
y
< y >= C ye-�H(p,q)d�
a" C y (p, q)d�
d� N
d� = h-3N (2Ą)-3N dp1dp2, ..., dpN dq1dq2, ..., dqN .
h3N (p, q)d�
(p, p+dp, r, rdr)
(q, p) = exp(-�H(q, p))
T
V
A
p2
E = .
2m
(r, r + dr)
(p, p + dp) P (p, r)dp dr
(dp dr/h3)
exp(-�E)
d3p d3r 2
P (p, r)d3p d3r = C e-�p /2m.
h3
P r r
P (p)d3p (p, p + dp) r
2
P (p)d3p = P (p, r)d3pd3r = Ce-�p /2md3p.
r
(v, v + dv)
2
P (v)d3v = Ce-�mv /2d3v.
N S = 1/2
�
H
< � > �i = ą�
(+) H
 = -�H (-) = �H
+ -
(+) (-)
+
P+ = Ce-� = Ce��H, P- = Ce-��H.
�H
y = ��H = ,
kBT
H
T
�iP (�i) �P+ - �P-
< � > = = =
P (�i) P+ + P-
e��H - e-��H
� = � th(��H).
e��H + e-��H
H
T < � >, H, T
M = N < � >
kBT �H y 1
<"
ey <" 1 + y, e-y <" 1 - y, �! th y y 1.
= = =
< � >H" y H" 0,
N�2
M = H = �H,
kBT
N�2 C
� = <" .
kBT T
kBT �BH �! y 1
ey e-y �! th y H" 1.
< � >= th (��H) H" �.
r
Z a" ZN (V, T ) a" e-�E .
r
r
Pr = Z-1e-�E .
r
Ere-�E 1 "Z "lnZ
< E > = = - = - .
r
e-�E Z "� "�
< ("E)2 > = < (E- < E >)2 > = < E2 > - < E >2 .
1 "2Z
2
r
< E2 > = Z-1 Er e-�E = =
Z "�2
r
2
" 1 "Z 1 "Z " < E >
= + = - + < E >2 .
"� Z "� Z2 "� "�
" < E > "2lnZ
< ("E)2 > = - = .
"� "�2
x
x d W
"Er "Er
r
dW = - < "xEr > = - < dx > = Z-1 - dx e-�E .
"x "x
r
� x
"Er 1 "
r r
- e-�E = e-�E
"x � "x
"Z " ln Z
dW = Z-1�-1 dx = �-1 dx.
"x "x
ln Z
" ln Z " ln Z
d ln Z = d� + dx = - < E > d� + �dW,
"� "x
d(< E > �) = �d < E > + < E > d�
d ln Z = �dW + �d < E > -d(< E > �)
d(ln Z + � < E >) = �(dW + d < E >) = �dQ
d Q
d Q
dS = .
T
-1 -1
d(ln Z + � < E >) = �dQ = �T dS = kB dS = d(kB S),
-1
ln Z + � < E >= kB S,
kBT ln Z+ < E >= T S kBT ln Z = T S- < E >= -F.
< E > U
F = -kBT ln Z.
F Z
F Z
A A
A
Ć
A = A + A
Pr A Er Nr
A r
Ć
Pr(Er, Nr) = C&! (Ę - Er, N - Nr).
A &!
Ć Ć
ln &! (Ę - Er, N - Nr) ln &! (Ę, N) -
" ln &! " ln &!
Er - Nr
"E 0 "N 0
A
" ln &!
a" �.
"E 0
S = kB ln &!
" ln &! "S
= (kB)-1 .
"N 0 "N
0
dU p �
dU = T dS - pdV + �dN �! dS = + dV - dN
T T T
 S = S(U, V, N)
"S "S "S
dS = dU + dV + dN,
"U "V "N
"S
� = -T .
"N
r
Ć Ć
&! (Ę - Er, N - Nr) = &! (Ę, N)e-�E +��Nr,
r r r
Pr = Ce-�E +��Nr = Ce-�E N ,
 = exp(��) C
" "
r r r
C-1 = e-�E N = ZN N a" ZG(, V, T ).
r
Nr=0 r Nr=0
ZG(, V, T )
Er Nr
-1
r r
Pr = ZG e-�E N ,
r r
N e-�E Er " ln ZG
< E > = = - ,
r r
e-�E N "�
V,N
r r
N e-�E Nr " ln ZG
< N > = = .
r r
e-�E N "
T,V
ZG
ś
ś = -kBT ln ZG(, V, T )
Ń
Ń
f
(q, p) E = E(q1, ...qf , p1, ...pf )
pi
E = (pi) + E (q, p)
i
E pi
 pi
i
(pi) = bp2
i
i
b
pi
p2 p
exp(-�E) dp dq
i
< >= .
i
exp(-�E)dp dq
exp (-�( + E )) dp dq
i i
< >= =
i
exp (-�( + E )) dp dq
i
exp(-� ) exp(-�E )dq dp exp(-� )dpi
i i i i
= =
exp(-� ) exp(-�E )dq dp exp(-� )dpi
i i
"
- exp(-� )dpi " "
i
"�
i
= = - ln e-� dpi .
exp(-� )dpi "�
i
-"
" "
2
i
i
W a" e-� dpi = e-�bp dpi.
-" -"
y = �1//2pi
2 2
W = �-1/2 e-by dy �! ln W = -1/2 ln � + ln e-by dy.
T �
" 1 1
< >= - - ln � =
i
"� 2 2�
1
< >= kBT.
i
2
kBT
 N V
N
p2
i
E = + U(r1...rN ),
2m
i=1
ri, pi i U
1
Z = h-3N exp -� (p2 + ... + p2) + U(r1...rN ) d3r1...d3rN d3p1...d3pn,
1 1
2m
2 2
1
1 N
Z = h-3N e-(�/2m)p d3p1... e-(�/2m)p d3pN e-(�U(r ...rN )d3r1...d3rN .
"
2
e-(�/2m)p d3p,
-"
U
N
"
V 2
Z = śN a" e-(�/2m)p d3p .
h3 -"
"
2
ś = e-(�/2m)p d3p =
-"
3/2
"
2 2mĄ
x y z
= e-(�/2m)(p +p2 +p2)dpxdpydpz = .
�
-"
3 3 2Ąm
ln Z = N ln V - ln � + ln .
2 2 h2
" ln Z N
< p >= kBT = kBT ,
"V V
< p > V = NkBT.
" ln Z 3N 3 3
< E >= - = = 3N < e > = kBT N = < p > V,
"� 2� 2 2
3
< e >= kBT
2
" < E > 3
CV = = NkB.
"T 2
V
3 3 2Ąm 3
S = kB(ln Z + � < E >) = NkB ln V - ln � + ln + ,
2 2 h2 2
3
S = NkB ln V + ln T + � ,
2
3 2ĄmkB 3
� = ln +
2 h2 2
V
3
S = S = N kB(ln V + ln T + �),
2
3
S = 2N kB(ln 2V + ln T + �),
2
S - 2S = 2N kB(ln(2V ) - ln V ) = 2N kB ln 2.
N
Z
<"
Z = �! ln Z = ln Z - ln N! ln Z - N ln N + N.
=
N!
T V
< E >
S = kB(ln Z + � < E >) =
3
= kBN(ln V + ln T + �) + kB(-N ln N + N) =
2
V 3
= kBN(ln + ln T + � ),
N 2
� = �
 N �
T H
e = -�mH,
m
-S, ... + S
S
Z = e��mH,
m=-S
� = �m
S
< � >= Z-1 �me��mH.
m=-S
S
1 "Z
�me��mH = .
� "H
m=-S
1 1 "Z 1 " ln Z
< � >= = .
� Z "H � "H
�H
kT
�H
y = = ��H.
kT
S S -1
Z = eym = eym + eym =
m=-S m=0 m=-S
S S
1 - ey(S+1) e-y(1 - e-yS) e-y(S+1) - e-yS
= eym + e-ym = + = .
1 - ey 1 - e-y 1 - ey
m=0 m=1
e-yS - ey(S+1) e-y(S+1/2) - ey(S+1/2) sinh(y(S + 1/2))
Z = = = .
1 - ey e-y/2 - ey/2 sinh(y/2)
ln Z = ln sinh(S + 1/2)y - ln sinh y/2.
1 " ln Z "y " ln Z
< � >= = � .
� "y "H "y
� (2S + 1) cosh(2S + 1)y/2 cosh y/2
< � >= -
2 sinh(2S + 1)y/2 sinh y/2
< � >= �SBS(y),
BS(y)
BS(y) = (2S)-1 [(2S + 1) coth(2S + 1)y/2 - coth y/2] .
S
N < M >
< M >= N < � >= N�SBS(y).
y 1
1 y
coth y <" + .
y 3
1 2(2S + 1) (2S + 1)2y 2 y
BS(y) <" + - - =
2S (2S + 1)y 6 y 6
1 y (S + 1)y
= (2S + 1)2 - 1 = .
6 2S 3
y 0
y 1
s + 1 S + 1
< M >= �H <" N�S , y = NS ��H
3 3
�2S(S + 1) C
� = N <" ,
3kBT T
S
y 1
coth y <" 1,
S + 1 1
BS(y) <" S-1 - = 1.
2 2
M <" �SN.
 N
x
xi xk
�s(x1, , ..xi, ..xk, ..xN ) = �s(x1, , ..xk, ..xi, ..xN ),
�a(x1, , ..xi, ..xk, ..xN ) = -�a(x1, , ..xk, ..xi, ..xN ).
xi = xk
�a(...xk, ..xk, ..) = -�a(...xk, ..xk, ..).
� = 0
F (q, p)
< F >= Z-1 F (q, p) (q, p)dq dp
(q, p) q p
t |�k(t) >
Ć
L |Lk >
|� >
|Lk >
|�j(t) >= |Lk >< Lk|�j(t) >a" aj(Lk; t)|Lk >,
k k
aj(Lk; t) a"< Lk|�j(t) > .
P (k, t)
t |�k >
Ć = |�k(t) > P (k; t) < �k(t)|.
k
Ć
Ć
L
< Lj| Ć|Li >= < Lj|�k > P (k; t) < �k|Li >=
k
= ak(Lj; t)a"(Li; t)P (k; t).
k
k
i = j
< Li| Ć|Li >= |ak(Li; t)|2P (k; t).
k
|ak(Li; t)|2 t
|�k > |Li > P (k; t) |�k >
< Lk| Ć|Lk >a" T r Ć = 1,
k
(q, p; t)d� = 1.
Ć
F
|�k >
P (k)
Ć Ć|�k
< F >= < �k|F > P (k).
k
|Li >
Ć Ć|�k
< F >= < �k|Li >< Li|F > P (k) =
k i
Ć|�k
= < Li|F > P (k) < �k|Li > .
i k
|�k > P (k) < �k| = Ć,
k
Ć Ć Ć|Li Ć Ć) Ć).
< F >= < Li|F >= T r(F = T r( ĆF
i
< F >= F (q, p) (q, p)dq dp.
Ej (E, E + �E)
&!(Ej)
= �i,jCj,
i,j
&!-1(Ej) : E < Ej < E + �E
Cj =
0 :
|�i >< �i|.
Ć = &!-1(E)
E |�i > Ei
-1
i
P (i) = ZN e-�E .
-1 -1 -1
i
Ć = ZN e-�E |�i >< �i| = ZN e-�$
|�i >< �i| = ZN e-�$
.
i i
i
ZN = e-�E = T r e-�$
.
i
e-�$

.
Ć =
T r e-�$

T r xe-�$
-1
Ć
< x >= T r (x�) = = ZN T r xe-�$
.
Ć ĆĆ Ć
T r e-�$

"
ZG(, T, V ) = N ZN (T, V ).
N=0
 T
En = �(n + 1/2), � = k0/m, (n = 1, 2, ...)
n
T r $
e-�$
n Ene-�E "
< E >= = = - ln ZN .
n
e-�E "�
T r e-�$

n
"
n
ZN = e-�E = e-� �(n+1/2) =
n=0 n
n -1
= e-1/2� � e-� � = e-1/2� � 1 - e-� � .
n
1
ln ZN = - � � - ln 1 - e-� � .
2
" ln ZN 1 e-� � 1 1
< E >= - = � + = � + .
"� 2 1 - e-� � 2 e� � - 1
�
1 �! e� � 1.
kBT
1
< E >= �( + e-� �).
2
T 0
1
E0 = �.
2
�
1 � � 1.
kBT
e� �
1 1
< E >= � + H"
2 (1 + � � + ...) - 1
1 1 1
H" �( + ) H" � .
2 � � � �
 (� �)-1 2
< E >= kBT.
p2 1
E = + k0x2.
2m 2
1 1
< Ek > = kBT, < Ep > = kBT.
2 2
< E > = kBT.
N V T
r s r r nr
r
R R
ER = n1 + n2 + ... = nr,
1 2 r
r
R r
r
ZN = e-�E = e-� nr.
R R
s
r
r
nse-� nr
R
< ns >= ,
r
r
e-� nr
R
r
r
nse-� nr
n1n2,..
< ns >= =
r
r
e-� nr
n1n2,..
s r
r
nse-� ns (s) e-� nr
ns n1n2,..
= .
s r
r
e-� ns (s) e-� nr
ns n1n2,..
(s)
s
(s)
s
nse-� ns
ns
< ns >= =
s
e-� ns
ns
1 "
s
= - ln e-� ns .
� " s ns
" "
ns -1
s s s
e-� ns = e-� = 1 - e-�
ns=0 ns=0
1 "
s
< ns >= ln 1 - e-� .
� " s
-1
s
< ns >fot = e� - 1 .
s
(s)
s
0 + e-� ZN-1
1
< ns >= = .
(s) (s)
(s) (s)
s
s
ZN + e-� ZN-1 ZN /ZN-1 e� + 1
(s)
1 1 2
ZN a" e-�(n +n2 ..)
n1,n2,..
(s)
s N ZN-1 s
(s)
s ZN s
"N "N = 1
"N ln ZN-"N
" ln Z(s)
(s) (s) (s)
ln ZN-"N ln ZN - "N = ln ZN - ąs"N,
"N
(s) (s)
s
ZN-"N = ZN e-ą "N
" ln Z(s)
ąs a" ,
"N
 s Z(s)
ą
" ln Z
ąs = ą �! ą = .
"N
Z
"N
(s)
ZN
= e-ą,
(s)
ZN-1
s
-1
s
< ns >F D = e� +ą + 1 .
s
ą N
-1
r
e� +ą + 1 = N.
r
" ln ZN "F
F = -kBT ln ZN �! ą = = -� = -��.
"N "N
ą = -�� �
-1
s-�)
< ns >F D = e�( + 1 .
(s)
nr = N ZN-"N
r
ą
(s) (s)
s s
0 + e-� ZN-1 + 2e-2� ZN-2 + ...
< ns > = =
(s) (s) (s)
s s
ZN + e-� ZN-1 + e-2� ZN-2 + ...
(s)
s s
ZN 0 + e-� e-ą + 2e-2� e-2ą + ...
= .
(s)
s s
ZN [1 + e-� e-ą + e-2� e-2ą + ...]
s s
nse-n (ą+� )
s
< ns > = .
s s
e-n (ą+� )
s
� � + ą
s s
-1
s
< ns >BE = eą+� - 1 .
 ą
-1
s-�)
< ns >BE = e�( - 1 .
< ns > N
ą
r r
nse�n (�- )
r
< ns >BE = .
r r
e�n (�- )
r
exp(��) < 1
s
� < 0
N T V
"F
= 0
"N
"F
= �,
"N
V,T
� = 0
�
(n1, n2, ...) N!//(n1!n2!...)
N n1 1 n2 2
nr = N
r
MB
ZN = N!(n1!n2!..)-1 exp -� nr =
r
n1,n2,.. r
N! n1 2 n2
1
= e-� e-� ...
n1!n2!...
n1,n2,..
nr = N N
r
N
MB
1 2
ZN = e-� + e-� + ... .
MB
r
ln ZN = N ln e-� .
r
s
" ln ZN e-�
< ns >MB= -kBT = N .
r
" s e-�
r
ZN = exp(- � nr).
r
R r
-1 -1
n1 2 n2
1 1 2
ZN = e-� e-� ... = 1 - e(-� ) 1 - e(-� ) ...
n1,n2,..
f
r
ln ZNot = - ln 1 - e-� .
r
N N N
N ZN N H" N
N = N
exp(-ąN )
N = N
ZN e-ąN ZN e-ąN "N .
N
"N
"N N ZG
ZG = ZN e-ąN �! ln ZG = ln ZN - ąN + ln "N .
N
ln "N
ln ZN = ąN + ln ZG.
ZG
ZG = exp -� nr exp -ą nr =
r
R r r
"
n1 " n2
1 2
= e-(� +ą) e-(� +ą) ... =
n1=0 n2=0
-1 -1
1 2
= 1 - e-(ą+� ) 1 - e-(ą+� ) ...
r
R
, , ..
1 2
r r
ln ZG = - ln 1 - e-ą-� �! ln ZN = ąN - ln 1 - e-ą-� .
r r
ą = -��
BE
r
ln ZN = -��N - ln 1 - e�(�- ) .
r
1 1
ZG = exp (-(ą + � )n1) exp (-(ą + � )n2) ... =
1 2
n1=0 n2=0
1 2
= 1 + e-ą-� 1 + e-ą-� ...
r
ln ZG = ln 1 + e-ą-� .
r
F
r
ln ZND = -��N + ln 1 + e�(�- ) .
r
V
T
-1
s
< ns > = e� - 1 .
= �, p = k,
� k k = �/c c
V
Lx, Ly, Lz
2Ą
kx = nx,
Lx
nx ky kz
nx kx (kx, kx + dkx)
Lx
"nx = dkx,
2Ą
( k, k + d k)
( k) d3k = "nx"ny"nz =
LxLyLz V
= d3k = d3k.
(2Ą)3 (2Ą)3
Nk = f( k)d3k
( k, k + d k)
V d3k
Nk = f( k)d3k = n( k) ( k)d3k = .
(2Ą)3 e�� - 1
n( k) k f( k)
k
4Ą k2 dk 1
Nkdk = f(k)(4Ą k2 dk) = V =
(2Ą)3 e� � - 1
4Ą V �2 d�
= = N�d�.
(2Ąc)3 e� � - 1
k = �/c
(�, � + d�) �
8Ą
< e(�, T ) > d� = 2N� �d� = f(k)�3 d�,
c3
�3 d�
< e(�, T ) > d� = .
Ą2c3 e� � - 1
� T
kBT
< e(�, T ) > d� = �2 d�,
Ą2c3
e� � 1
< e(�, T ) > d� = �3e-� �d�.
Ą2c3
 �
�
� = .
kBT
4
kBT �3 d�
< e(�, T ) > d� = .
Ą2c3 e� - 1
� �max H" 2, 822
T1 �1 T2 �2
�1 �2 �1 �2
= = �max �! = .
kBT1 kBT2 T1 T2
4
" "
kBT �3 d� Ą2
< e(T ) >= < e(T, �) > d� = = (kBT )4,
Ą2c3 e� - 1 15(c )3
0 0
"
x3 dx Ą4
= .
ex - 1 15
0
4
Ą2kB
� =
3
60 c2
4� 4�
4 4
< e(T ) > = T �! < E(T ) > = V T .
c c
" < E > 16�
3
CV = = V T .
"T c
V
-1
s-�)
< ns >= e�( + 1 .
F ( ) =< ns >
T = 0 > � < n >= 0. < � < n >= 1.
�
= � T > 0
F
F ( ) 1 0 2kBT
1.4
1.2
1
0.8 T = 0
< n >
0.6 T > 0
0.4
0.2
F
0
T = 0
pF = kF ,
2m�F
2
kF = .
2
ep = p2/2m
p
ln ZG = ąN - ln 1 - śe-�e .
p
ś = e��
r
" śe-�e
N = ś ln ZG = .
r
"ś 1 - śe-�e
r
"
2ĄV
pdp.
h2 0
r
2
"
" "
2ĄV śe-p /2m 2ĄV 2 2
N = p = śe-p /2m śne-p n/2mpdp =
2
h2 0 1 - śe-p /2m h2 0
n=0
"
"
2ĄV 2
= śn e-p n/2mpdp.
h2
0
n=0
�p2 �
= x �! pdp = dx
2m m
" "
"
2ĄV m 2ĄV m 1
"
N = śn e-nxdx = śn - e-kx 0 =
h2 � h2 � n
0
n=0 n=0
"
2ĄV śn V mkBT
= mkBT = ln(1 - ś).
h2 n 2Ą 2
n=1
2
2Ą N
- ln(1 - ś) =
V mkBT
ś
2
2Ą N
ś = 1 - exp - .
mkBT V
pV
P
= ln ZG = - ln 1 - śe-�e .
kBT
p
"
�p2
pV 2ĄV
2m
= p ln 1 - śe- dp =
kBT h2 0
" "
"
�p2 �p2
2ĄV śn 2ĄV śn "
2m 2m
= p e- dp = pe- dp =
h2 0 n=1 n h2 n
0
n=1
"
2ĄV śn
= mkBT .
h2 n
n=1
ś
" n
2
pV mkBT V 1 2Ą N
= 1 - exp - .
2
kBT 2Ą n2 mkBT V
n=1
t "
2 2 2
2Ą N 2Ą N 2Ą N 1
1 - exp - <" 1 - 1 - <" -
mkBT V mkBT V mV kBT
2
pV mkBT V 2Ą N 1
H" <" N.
2
kBT 2Ą mV kBT
pV = NkBT.
T 0
2
2Ą N
exp - <" 0,
mkBT V
"
m 1 mĄ
p <" (kBT )2 = (kBT )2.
2Ą 2 n2 12 2
n=1
E(x1, x2) = E1(x1) + E2(x2) + Eodd(x1, x2).
N
mi
xą i ą
Ći
ą
xą �i
i
ą
�i = xą - xą.
Ći
i
N 3 N 3
2
1 1
ą
Ek = mi (�ą)2 = mi �Łi ,
i
2 2
i=1 ą=1 i=1 ą=1
xi �
Ć
V (x1, x2, ..., xN )
"V 1 "2V
ą ą �
V = V0 + �i + �i �j + ...
"xą (0) 2
"xą"x�
i
i j
i ą i,j ą,�
(0)
V0 V V
1
ą ł
V = V0 + Aął�i �j ,
ij
2
i,j ą,ł
1 1
ą ą ł
H = V0 + mi(�Łi )2 + Aął�i �j .
ij
2 2
i,ą i,j ą,ł
A
3N
ą ą
�i = Bi,rqr.
r=1
3N
1
H = V0 + qr + �r qr .
Ł2 2 2
2
r=1
qr
qr qr
Ł
3N
 �r qr
nr = 0, 1, 2..
= �r (nr + 1/2) 3N
r
3N 3N
1
E = V0 + nr + �r = -N� + nr �r,
2
r=1 r=1
3N
1
-N� = V0 + �r
2
r=1
� T = 0
3N
ZN = exp -� -N� + ni �i =
n1n2,.. 1
" "
= e�N� e-� �1n1 ... e-� �3N n3N =
n1=0 n3N =0
-1 -1
= e�N� 1 - e-� �1 ... 1 - e-� �3N .
3N
ln ZN = �N� - ln 1 - e-� �r .
r=1
N �r
�(�)d�
(�, � + d�)
"
ln ZN = N�� - ln 1 - e-� � �(�)d�.
0
"
" ln ZN �
< E > = - = -N� + �(�)d�,
"� e� � - 1
0
" < E > " < E >
CV = = -kB�2 =
"T "�
V V
"
e� �
= kB (� �)2 �(�)d�.
0 (e� � - 1)2
�(�)
�(�) �max
�(�) = 0 � > �max
 � �max 1 � �max
e� � = 1 + � � + ...
kBT �max
"
CV = kB �(�)d� = 3NkB,
0
�(�)d�
3N
CV = 3NkB
S = 1/2
OZ
OZ
N
H = -J �i�i+1,
i=1
J
�z a" � i i+1
�i = ą1.
N
N N
N-1
ZN = T re-�H = T r exp K �i�i+1 =
i=1
N-1
= ... exp K �i�i+1 =
�1=ą1 �N =ą1 i=1
N-2
N-1
= ... exp K �i�i+1 eK� �N .
�1=ą1 �N-1
=ą1 i=1 �N =ą1
K = �J
N-1 N-1 N-1
eK� �N = eK� + e-K� = 2ch K
�N =ą1
�N-1
ZN = (2ch K)) ZN-1 = (2ch K))2 ZN-2 = ... = (2ch K))N-2 Z2.
Z2
1 1 1
Z2 = eK� �2 = eK� + e-K� = 4 ch K.
�1=ą1 �2=ą1 �1=ą1
N-1
J
ZN = 2(2ch K)N-1 = 2 2ch .
kBT
" ln ZN
< E >= - = -Jth K,
"T
" < E > kBK2
CV = -T = .
"T ch2K
V
3
ą ą ą
HHeis = - JijSi � Sj
i,j ą=1
2
ą ą ą
HXY = - JijSi � Sj .
ij ą=1
 z
ą
z �
H
Sz
OZ
z
HHeis = - JijSi � Sj - �H Si .
i,j i
�
H
z
H = H0 - �H Si ,
i
H0
�ł �ł
z -1 z
�ł
M(T, H) = � < Sj >= �ZN T r Sj e-�Hłł ,
j j
�ł �ł �ł �ł �ł�ł
"M(T, H) "ZN -1 �ł z "
-2 z
�łZN �ł
�T = = = � T r Sj e-�Hłł + ZN T r Sj e-�Hłłłł .
"H "H "H
j j
N
"H
z
= -� Sk,
"H
k=1
N
"
z
e-�H = �� Ske-�H,
"H
k
N
"ZN
z
= ��T r Si e-�H .
"H
i
�T =
N N
-1 z -1 z
= �2� -ZN T r Ske-�H ZN T r Si e-�H +
k i
N N
-1 z z
+ZN T r Sk Si e-�H =
k i
N
-1 z z -1 z z
= �2� ZN T r SkSi e-�H - ZN T r Ske-�H Si e-�H =
i=1 k k
N N N
z z z
= �2� < SkSi > - < Sk >2 .
i=1 k=1 k=1
N N
z z
�T = �2� < SkSi > -�B�M2(T, H).
i=1 k=1
i k
z z z z z
< SkSi > = < Sk >< Si > = < Sk >2 .
 P V
T
(P, V, T )
(P, T )
I II III x a
b c
a
(Tc, Pc) c
III
II c
(P, V )
pc
Tc
(P, T )
T = Tc
T < Tc
I II III
�c - �g
Tc
Tc
Tc
T = 0
(P, V ) (M, H) Tc
Tc
"P "2P
= 0, = 0.
2
"V "V
T T
 T < Tc
F = -kBT ln Z(T, V, N),
Z(T, V, N)
d3N pd3N q
Z(T, V, N) = e-�H(p,q)d�, d� = ,
N!h3N
Z(T, V, N) = T re-�H.
V Z(T, V, N)
� F
V
N ", V "
� = N/V
&! V
ZN (�) = eNK(�)ch (N (� - �")) ,
� , �" ZN (�)
iĄ
�r = �" ą (2r + 1) , (r = 0, 1, 2...)
N
� � �"
N < " N "
ln ZN
N-1 ln ZN (�) = K(�) + N-1 ln ch (N (� - �")) ,
"
u(�) = - N-1 ln ZN (�) = -K (�) - th (N (� - �")) .
"�
N u(�) �
N "
U
N < "
�
N " � = �"
lim u(�) = -K (�) - , � > �",
N"
lim u(�) = -K (�) + , � < �",
N"
� = �" 2
N 1023
 N
s ą1/2
ą ą �
ą � ł
H = G0 + Gą(j)Sj + Gą�(j, k)Sj Sk + Gą�ł(j, k, l)Sj Sk Sl + ...
ą, �, ł = x, y, z G
S
G
ą � ł
�
H = G0 + G(j, k)SjSk + O(Sj Sk Sl Sm)
J(j, k) = -G(j.k) - G(k, j).
1
H = G0 - J(j.k)SjSk.
2
 J(j, k) = J(|j - k|).
H
1
H = G0 - J(j, k)SjSk + HSj.
2
Hef
Hef = łM
ł
HMF A = Hi,
Hi = (H + Hef ).
E = ą1/2H
�H �H �H
Z = exp( ) + exp(- ) = 2 cosh( ).
2 2 2
M = N < Sz >
m = M/N
MF A
T rSze-�H 1 �łH 1 �m
m =< Sz > = = tanh( ) = tanh( ).
MF A
T re-�H 2 2 2 2
1
0.8
0.6
m
0.4
0.2
0
0 0.25 0.5 0.75 1 1.25 1.5
temperatura