Principles of Equal a Priori Probability
 All distributions of energy are equally probable
 If E = 5 and N = 5 then
5
5
5
4
3
2
1
0
4
3
2
1
0
4
3
2
1
0
All configurations have equal probability, but
possible number of way (weight) is different.
A Dominating Configuration
 For large number of molecules and large number of energy
levels, there is a dominating configuration.
 The weight of the dominating configuration is much more
larger than the other configurations.
Wi
Configurations
{ni}
Dominating Configuration
W = 1 (5!/5!)
5
5
5
4
3
2
1
0
4
3
2
1
0
4
3
2
1
0
W = 20 (5!/3!)
W = 5 (5!/4!)
Difference in W becomes larger when N is increased !
In molecular systems (N~1023) considering the
most dominant configuration is enough for average
8.4 The most probable distribution, equilibrium
distribution, and Boltzmann distribution
 8.4.1 The principle of equal a priori probabilities
Statistical thermodynamics is based on the
fundamental assumption that all possible
configurations of a given system, which satisfy the
given boundary conditions such as temperature,
volume and number of particles, are equally likely
to occur.
   ( N ,U ,V )  WD
D
 Example Consider the orientations of three
unconstrained and distinguishable spin-1/2
particles. What is the probability that two are spin
up and one spin down at any instant?
 Solution Of the eight possible spin configurations
for the system,
 ↑↑↑ ↑↑↓ ↑↓↑ ↓↑↑ ↑↓↓ ↓↑↓ ↓↓↑ ↓↓↓
 The second, third, and fourth comprise the subset
"two up and one down". Therefore, the probability
for this particular configuration is
 P = 3/8