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a r X i v : 0 7 1 0 . 5 0 9 4 v 1 [ c o n d - m a t . s t a t -
m
e c h ] 2 6 O c t 2 0 0 7
D
t
+ v = 0 ; v = e z ; =
v
= 0
D D
S [] = D d2x d log , E d
S (E 0, d) = sup
{ | N [ ]=1 }{S [] | E [] = E
0 , D [] = d }
(x , ) x
d
d d
C (E 0 , s ) = inf
C s [] = D s()d2 x | E [] = E 0 C s s
D (G) = inf D d2x
12
| |2 + G ()
d s G
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G (C ) = inf x
{g(x ) |c(x ) = C } H ( ) = inf x
{h (x ) = g(x ) c (x )}
G H
C
H ( ) = inf C {G (C ) C } G(C ) sup {C + H ( )}
xm h xm G(C ) C = c(xm )
xm G(C ) xm h xm
h
xm
h
G(C ) =sup {H ( ) + C } G(C )
G C
H ( ) = inf C ninf x {g(x ) c (x ) |c(x ) = C }o = inf C ninf x
{g(x ) |c(x ) = C }
H G
G (C ) = inf x {g(x ) |c(x ) = C } = inf x {g(x ) c (x ) |c(x )
= C } + C
inf x {g(x ) c (x )} + C = H ( ) + C.
xm h x c(x) =c(xm ) g(xm ) = h (xm ) + c(xm ) h (x) +c(xm ) =
g(x)
xm h
G H
xm
E [] = 12 D d2x ( )2 =
12 D d2x = E 0
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s
C s [] = D d2x s (). A () D
d ()
d () = 1|D|
dAd
A () = D d2
x { ( x ) } ,
B B |D| D
d () A ()
d ()
D
(, x ) (x ) = x
N [] (x ) +
d (, x ) = 1 .
D [] () D dx (, x ) = d () .
(x ) = +
d (, x ) .
=
E [] 12 D dx E 0 .
S [] D d2x +
d log .
eq
S (E 0 , d)
()
eq (x , ) = 1
z ( eq ) exp [ eq ()] ,
z (u) = +
d exp[u ()] f (u) =
ddu
log z .
z logz
f
eq = f ( eq ) g (eq ) = eq ,
g f eq
G(E 0 , ) = inf inf { | N [ ]=1 }
G [] E [] = E 0 ,
G [] S [] + D d2x +
d () (x , ) .
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s
dsd
(eq ) = eq ,
s
s log z
s () = supu
{u log z (u)} .
s logz
s () = g () log(z (g ()))
ds/d = g s = s
s = s
eq C (E 0 , s ) =G(E 0 , )
eq eq = deq E [eq ] = E 0 G(E 0 , ) = G [eq ]
eq eq log eq + eq =
exp( eq ()) [ log z ( eq ) + eq ] /z ( eq ) eq = deq
+
d (eq log eq + eq ) = log z ( eq )+ eq eq = s (eq ) .
G C G(E 0 , ) =G [eq ] = C s [eq ] C C s [eq ] C (E 0 , s )
G(E 0 , ) C (E 0 , s ).
eq, 2 s = s 2
ds /d = g eq, 2 exp[ 2 eq, 2 ()] /z ( 2eq, 2 )
G [eq, 2] = C s [eq, 2 ] = C (E 0 , s ) G G(E 0 , ) C (E 0 , s
)
G(E 0 , ) = C (E 0 , s ).
C s [eq ] = C (E
0, s ) = G(E
0, ) = G [eq,
2]
eq eq, 2
eq eq, 2 2
C s [] = inf { | N [ ]=1 }
G [] +
d = (x ) .
G (E 0 , ) = inf
inf { | N [ ]=1 }
G [] Z + d = (x )ffE [] = E 0ff = C (E
eq
2 J J = G + E
D eq = 0 = d 2Ds Ds = C s + E
2G [] = 2S [] = D dx d 1 eq ()
2 2C s [] =
D dx s
(eq ) ()2
= + = f
z + z
z 2 !exp [ eq ( )] . f
z
z
eq f z f
= z 2 + z z
/z 2
= d
d / eq = 0 = +
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s
= g (g ) 1 = f
s
= f
1
d 2
/ eq = s
(eq ) ()2
2J [] = D d2x +
d
1eq
2
+ 2Ds [].
J Ds
F (, s ) = inf
{F s [] = C s [] + E [] } .
G
s G(z) = sup y {zy s(y)} G
D (, G ) = inf
DG [] = D d2 x 2
| |2 G ( ) .
F (, s ) = D(, G )
eq = eq F s eq DG
eq F s Ds
c = c F s c DG F s [c ] = DG [c ]
= F s [] D G []
DG F s F s
DG s (c ) = c c = G ( c ) G s s
(s ) 1 = G
F s [] = D d2x [ s () + ] + D d2x 2
D d2x G( ) + 2
= DG []
G s
G(x) = x (s ) 1 (x) s (s ) 1 (x) G( c ) = c c s (c )
F s [c ] = DG [c ]
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