Méthodes de Monte-Carlo en physique statistique

Sz=±~
2
~2π
mz=−gµBSz
g µB
E=−mzBz
i j
E=−JSi
zSj z
J

0
N-1 1 2 3 4 5 N-1
N-2
N-4
N-5 0
1 1 -1 1-1 -1 1 1 -1 11
1...........
i i −1i+1
0N−1
E=−
N−1
X
i=0
BSi+SiSi+1
B
T µ Eµ
pµ=αexp(−Eµ
kT )
k
β=1
kT
α
pµ=e−βEµ
Pµe−βEµ

X Xµ
µ
X=E(X) = PµXµe−βEµ
Pµe−βEµ
mµ=X
i
Si
2N
N= 100
M
moy(X, M) = 1
M
M
X
k=1
pkXk
pkk
pµ
pµ
moy(X, M) = 1
M
M
X
k=1
Xk

pµ
µ
qµ=e−βEµ
x
[0, qmax]
qmax = 1 x qµ
x<qµ
.
. Eµqµ
. x [0,1]
. x < qµ
M

Configuration μNouvelle configuration
Sélection Configuration μ'
Acceptation
Refus
µ µ0
pµ0
pµ
=e−β(Eµ0−Eµ)=e−β∆E
∆E=Eµ0−Eµ
[0,1]
x < e−β∆E
.
.∆E
.∆E <= 0
. x
[0,1] x<e−β∆E
Q
M