1. 6. The Theory of Simple Gases
An Ideal Gas in a Quantum Mechanical Microcanonical Ensemble
An Ideal Gas in Other Quantum Mechanical Ensembles
Statistics of the Occupation Numbers
Kinetic Considerations
Gaseous Systems Composed of Molecules with Internal Motion
Chemical Equilibrium

2. 6.1. An Ideal Gas in a Quantum Mechanical Microcanonical Ensemble
N non-interacting, indistinguishable particles in V with E.
( N, V, E ) = # of distinct microstates
Let  be the average energy of a group of g >> 1 unresolved levels.
Let n be the # of particles in level . 
Let W { n } = # of distinct microstates associated with a given set of { n }. 
Let w(n ) = # of distinct microstates associated with level  when it contains n particles. 

3. Bosons ( Bose-Einstein statistics) :
Fermions ( Fermi-Dirac statistics ) :
w(n ) = distinct ways to divide g levels into 2 groups;
n of them with 1 particle, and g  n with none.

4. Classical particles ( Maxwell-Boltzmann statistics ) :
w(n ) = distinct ways to put n distinguishable particles into g levels.
Gibbs corrected

5. Method of most probable value
( also see Prob 3.4 )
Lagrange multipliers
n* extremize

6. BE FD

7.   BE FD 
Most probable occupation per level MB

8. BE FD  
MB: 

9. 6.2. An Ideal Gas in Other Quantum Mechanical Ensembles
Canonical ensemble :
Ideal gas,  = 1-p’cle energy :
= statistical weight factor for { n }.
Actual g absorbed in  ( here is treated as non-degenerate: g = 1).

10. Maxwell-Boltzmann : 
multinomial theorem

11. partition function (MB)
grand partition function (MB)

12. Bose-Einstein / Fermi-Dirac :
Difficult to evaluate
(constraint on N ) 

13. B.E.
F.D. BE FD
Grand potential :
q potential :

14. BE FD
 MB :
c.f. §4.4
Alternatively 

15. Mean Occupation Number
For free particles : BE FD  
see §6.1

16. 6.3. Statistics of the Occupation NumbersFD
Mean occupation number :
BE :
B.E. condensation
FD :
 MB :
Classical : high T
  must be negative & large
From §4.4 : 
same as §5.5

17. Statistical Fluctuations of nBE FD  

18. BE FD
above normal
below normal
Einstein on black-body radiation :
+1 ~ wave character
 n 1 ~ particle character
see Kittel, “Thermal Phys.”
Statistical correlations in photon beams : see refs on pp.151-2

19. Probability Distributions of n
Let p (n) = probability of having n particles in a state of energy  .  BE FD 

20. BE FD
BE :   
FD : 

21. MB :
Gibbs’ correction 
Alternatively  
Poisson distribution 
“normal” behavior of un-correlated events
 prob of occupying state 

22. “normal” behavior of un-correlated events
Geometric ( indep of n )
> MB for large n :
Positive correlation 
FD : 
< MB for large n :
Negative correlation

23. n - Representation
Let n = number of particles in 1-particle state  .
State of system in the n- representation :
Non-interacting particles :

24. Mean Occupation Number
Let F be an operator of the form e.g.,

25. 6.4. Kinetic ConsiderationsBE FD
From § 6.1
Free particles :

26. BE FD
Let p( ) be the probability of a particle in state  . Then    
s = 1 : phonons
s = 2 : free p’cles
All statistics

27.  pressure is due to particle motion (kinetics)
Let n f(u) d3u = density of particles with velocity between u & u+du. 
# of particles to strike wall area dA in time dt
= # of particles with u dA >0 within volume udA dt
Each particle imparts on dA a normal impluse =
Total impulse imparted on dA =  

28. Rate of Effusion # of particles to strike wall area dA in time dt
 Rate of gas effusion per unit area through a hole in the wall is  
All statistics
R   u   Effused particles more energetic.
u > 0  Effused particles carry net momentum (vessel recoils)
Prob.6.14