1. Quantic vs Classic world What defines the granular nature of our Universe? Our classical “behavior” vs the atomic “quantum” characteristics are a consequence of the absolute size of h So far, we “solved” the Q.M. problem and then count all states to get the partition function Q. While trying to count states, we invoke the classical limit and  Can we obtain Q assuming classical behavior for the H ? What do you expect to be “different” between the 2 answers?

2. Going classical…. So far, we “solved” the Q.M. problem and then count all states to get the partition function Q. While trying to count states, we invoke the classical limit and  Can we obtain Q assuming classical behavior for the H ? We can obtain the states energies by solving the classical Hamiltonian equations We count the states by analogy to the quantum treatment…

3. Example I: Harmonic Oscillator

4. Example II: translations

5. Example III: rotations

6. Generalization of the classical treatment Comparing the high T limit of the QM Stat.Mech. with classical Stat. Mech. we infer the constants We are establishing a “correspondence principle” between QM and Classical Statistical Mechanics 

7. N particle systems Consider N interacting particles: Each particle has s degrees of freedom

8. Ensembles

9. Density number

10. Liouville equation N N’ q1q1 q1q1 q1q1 q1+q1q1+q1

11. Liouville eq II

12. Liouville eq III

13. Net flow in all directions

14. Full derivative of f(p,q,t)

15. Constant density concept Volume around p,q NN NN Volume around p o,q o with  N in the surface

16. No crossing of trajectories