Distribution de Boltzmann

N
N
M0M−1m
m
M
m
m= (m+1
2)~ω

Etat Energie
0
1
2
3
4
5
ε1
ε2
ε3
ε4
ε5
ε0
Particule 1 Particule 2 Particule 3 Particule 4 Particule 5 Particule N
..........
ε
m1=1 m2=0 m3=3 m4=1 m5=2
N0M−1
µ={m1, m2, m3, ..., mN}
E=
N
X
i=1
ei=
N
X
i=1
mi=
N
X
i=1
mi
eii
E=Q

N
X
i=1
mi=Q
Ω
Q N
Q
N−1
Q = 15, N = 6
Particule 1
Particule 2
Particule 3 Particule 4
Particule 5
Particule 6
Q+N−1
Q+N−1Q+N−2
Q+N−3
(Q+N−1)!
Q!
(N−1)!
Ω = (Q+N−1)!
Q!(N−1)!
ln(N!) 'Nln(N)−N
1N
ln Ω = (Q+N) ln(Q+N)−Qln(Q)−Nln(N)
 q =Q/N
ln Ω = N((1 + q) ln(1 + q)−qln(q))

20 20N
4NN
N= 100
QN1
Q

m1
m2
Q
Q
0
2 particules
m1+m2=Q±1
Q
N N −1N
1
Ω'QN
E
N
E