1. Singel particle energy from Bolztman
Boltzmann number of 1-particle state >>number of particles
How many 1-particle states?
Remember the sphere used to explain degeneracy?
number of 1-particle states with an energy lower than e = number of lattice points enclosed by the sphere in the positive octant:

2. Many particles Boltzman energy
The average energy of N particles is N times the average energy of one particle

3. NON Bolztmann ensembles
yi1
yi3
yi2
Fermions: 2 particles cannot occupy the same quantum stateoccupation number nk can only be 0 or 1
P1,2y=-y antisymmetric
yi2
yi3
Bosons: 2 particles can occupy the same quantum stateoccupation number nk can have any value
yi4
yi6
yi5
P1,2y=+y symmetric

4. Partition functions for F or B
There are many ways of counting distributions,
choose Grand-Canonical Ensemble

5. Summing with occupation numbers

6. Fermions
Fermi-Dirac

7. Bosons
from our outstanding math background …
we recognize the series
Bose-Einstein
Combining Bose-Einstein with the Fermi-Dirac we obtain a gral. Eq.

8. Grand-Canonical equation

9. G-canonical equations II
Keep in mind that even though we started with non-interacting
particles, there wavefunctions are symmetrized they are
NOT INDEPENDENT

10. High T low r limit for n
Boltzmann number of 1-particle state >>number of particles 

11. High T low r limit for E

12. Example
Examples: Photon gas, an electromagnetic field in thermal equilibrium with its container
To describe the state of the field, we need to know how many n are in each oscillator
Photons are bosons  n=0,1,2,3…..

13. Cont.

14. cont
The average number of photons in a state j…
Planck
distribution