1. Connection between partition functions
Looking at the natural variables

2. From Q to X

3. Equivalency of ensembles
An example on the equivalency among ensembles
N distinguishable particles, 2 possible states (DE=e with E1=0)
 {aj}={a1,a2,…aN} where aj =0, or 1 and therefore
Microcanonical ensemble: degeneracy of the mth level  number of ways to distribute m objects in a pool of N (i.e. distribute m quanta to obtain E total energy)

4. Equivalency of ensembles II

5. Fundamental thermodynamics
Summary of Ensembles
Ensemble
Constants
Fundamental thermodynamics
Total differentials
Microcanonical
N,V,E
S=kblnW
Canonical
N,V,T
A=-kbTlnQ(N,V,T)
Grand-canonical
m,V,T
pV=kbTlnX(m,V,T)
Isothermal-isobaric
N,p,T
G=-kbTlnD(N,p,T)
Useful ensembles at least one extensive variable N,V,or E
Generalized ensemble with only intensive properties, (m,p,T)
but –kbTlnZ(m,p,T)=0  no fundamental function

6. Fluctuations
Fluctuation:spontaneous deviation of a mechanical variable from its mean… How much it deviates?
Ergodic hypothesis <time>  <ensemble>
rms fluctuation of X=X(t) is equivalent to sX

7. Fluctuations in Canonical Ensemble
What are the fluctuations in the canonical ensemble?

8. Fluctuations in Canonical Ensemble II
The spread of the fluctuations corresponds to the rate at which the energy changes with T
For an ideal gas,
Distribution of energies is like a delta function centered at <E>

9. Fluctuations in Grand canonical

10. Isothermal compressibility

11. Fluctuations in N
For a canonical ensemble, even thought there are fluctuations,
The energy is distributed uniformly. Each system is most likely to be found with energy <E> canonical ensemble equivalent to microcanonical (where E is constant)
Fluctuations in N show that a grand canonical ensemble is most likely to be found with <N> particles  grand canonical canonical ensemble equivalent to canonical (where N is constant)

12. What is the probability of finding a particular value of E?
P(E)
What is the probability of finding a particular value of E?
P(E) W E
E*=<E>
P(E)

13. How do we count states
Let’s count…

14. Distinguishable particles
Canonical ensemble of DISTIGUISHABLE particles/quasi-particles:
a,b,…n.

15. Example for distinguishable particles
Imagine a system with N=1000 degrees of freedom
(1000 quasi particles)
Each particle can be in one of 5 microstates
 There are states to be sampled (and counted!!!)
Using the factorization due to equal-but-distinguishable particles,
we only need to enumerate 5 states to evaluate q
 conversion of one N-body problem to N, 1-body problems

16. Indistinguishable particles
If the particles are INDISTINGUISHABLE
FERMIONS : All indices j; k,…, l must be different. Hence
summations over indices depend on each other.
BOSONS : Indices j;k;…;l need not all be different. Permutations like j; k;…;l and k;j;…;l refer to identical states and must occur only once in the summation.

17. Boltzman Statistics INDISTINGUISHABLE particles 
A particular (and common) case:
T,d number of available energy states >> N
 each an every particle is in a different state
Boltzman Statistics
we have to consider those distribution that are equivalent, that is
There are N! of these combinations which can be subtracted from the pool of microstates by dividing by N!

18. Singel particle energy from Bolztman
Boltzman 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: