Notes for Thermal Physics II
Summary: Number of States and Probabilities
a) Notation
• Ω(E, N)
. . . number of all microstates consistent with E, N
• wN (N1 )
. . . number of all microstates consistent with E, N that have
N1 particles/events in the state E1 (2 state system)
• wN (n1 , n2 , ..., nj ) . . . number of all microstates consistent with E, N that have
n1 particles in the state E1 , n2 in state E2 ,
. . ., nj in state Ej (general case)
1
Ω
• pν
. . . probability of microstate ν; pν =
• pN (N1 )
. . . probability to find N1 particles/events in special state,
e.g., E1 (2 state system)
• pN (n1 , n2 , ..., nj ) . . . probability to find n1 particles in the state E1 ,
n2 in state E2 , . . ., nj in state Ej (general case)
b) Counting the States and Calculating Probabilities
• independent particles that can occupy M states:
Ω = M × M × M × . . . × M = MN
• systems with 2 states [N1 particles/events in state E1 ; (N − N1 ) in state E2 ]
N!
N1 ! (N − N1 )!
N!
1 (N −N1 )
pN
pN (N1 ) =
1 p2
N1 ! (N − N1 )!
wN (N1 ) =
• general case [n1 particles/events in state E1 with probability p1 etc]
wN (n1 , n2 , ..., nj ) =
N!
N!
=Q
nk !
n1 !n2 ! . . . nj !
k
N!
N! Y nk
n
pN (n1 , n2 , ..., nj ) =
p
pn1 1 pn2 2 . . . pj j = Q
n1 !n2 ! . . . nj !
nk ! k k
k
b) Conditions to be Hold
•
•
X
wN (n1 , n2 , . . . , nj ) = Ω
all possible
X nk
all possible nk
with
X
and
X
nk = N
k
pN (n1 , n2 , . . . , nj ) = 1
k
pk = 1