1. Principle of equal a priori probability Principle of equal a priori probabilities: An isolated system, with N,V,E, has equal probability to be in any of the  (N,V,E) quantum states or Each and every one of the  (N,V,E) quantum states is represented with equally probability

2. Temperature

3. Canonical ensembles Canonical N,V,T IIIIII A There are A identical replicas with same N,V, and T V=AxVV=AxV N=AxNN=AxN E Each microsystem has an energy value E j (N,V). Each E j is  (E j ) times degenerate a l number of systems in state l, occupation number heat bath(T) Set of {a l } = distribution = a describes the state of the Ensemble A=  j a j E=  j a j E j One possible state of ensemble { 1 2 3 l E l E 1 E 2 … E l a l a 1 a 2 …a l

4. Canonical as subsystem of microcanocal heat bath(T)  E B E j (N,V) N,V,E E=E j +E B

5. Distributions Principle of equal priori probability  every {a} is equally probable Since there are A systems (each with energy E j )  there are A number distinguishable particles that can be distributed according to their a l value  a 1 in group 1, a 2 in group 2, a l in group l … How many times a particular distribution can be found in the ENSEMBLE? We know the answer to this one, right?

6. Example (Nash) 3 systems

7. Example (Nash)II

8. Example (Nash)III systems For E =1000 and A =1000  W=10 600 there are 10 70 atoms in the galaxy For larger and larger A values, the ratio W n / W max = A n

9. Probability Probability of finding a system in E j is obtained by

10. Distribution for max W(a)

11. Maximazing lnW(a)

12. Evaluating the multiplier From here we can calculate all other mechanical thermodynamic properties Probability of finding the quantum state with E j at a given N,V