1. Thermo & Stat Mech - Spring 2006 Class 18 1 Thermodynamics and Statistical Mechanics Statistical Distributions

2. Thermo & Stat Mech - Spring 2006 Class 182 Multiple Outcomes Distinguishable particles

3. Thermo & Stat Mech - Spring 2006 Class 183 Degenerate States Suppose there are g j states that have the same energy.

4. Thermo & Stat Mech - Spring 2006 Class 184 Boltzmann Statistics (Classical)

5. Thermo & Stat Mech - Spring 2006 Class 185 Most Probable Distribution

6. Thermo & Stat Mech - Spring 2006 Class 186 Most Probable Distribution

7. Thermo & Stat Mech - Spring 2006 Class 187 Constraints (Lagrange Multipliers)

8. Thermo & Stat Mech - Spring 2006 Class 188 Most Probable Distribution

9. Thermo & Stat Mech - Spring 2006 Class 189 Boltzmann Distribution

10. Thermo & Stat Mech - Spring 2006 Class 1810 Quantum Statistics Indistinguishable particles. 1.Bose-Einstein – Any number of particles per state. Particles with integer spin:0,1,2, etc 2.Fermi-Dirac – Only one particle per state: Particles with integer plus ½ spin: 1/2, 3/2, etc

11. Thermo & Stat Mech - Spring 2006 Class 1811 Bose-Einstein At energy  i there are N i particles divided among g i states. How many ways can they be distributed? Consider N i particles and g i – 1 barriers between states, a total of N i + g i – 1 objects to be arranged. How many arrangements?

12. Thermo & Stat Mech - Spring 2006 Class 1812 Bose-Einstein

13. Thermo & Stat Mech - Spring 2006 Class 1813 Bose-Einstein

14. Thermo & Stat Mech - Spring 2006 Class 1814 Bose-Einstein

15. Thermo & Stat Mech - Spring 2006 Class 1815 Constraints (Lagrange Multipliers)

16. Thermo & Stat Mech - Spring 2006 Class 1816 Bose-Einstein

17. Thermo & Stat Mech - Spring 2006 Class 1817 Boltzmann Distribution

18. Thermo & Stat Mech - Spring 2006 Class 1818 Fermi-Dirac At energy  i there are N i particles divided among g i states, but only one per state. g i  N i. How many ways can the N i occupied states be selected from the g i states?

19. Thermo & Stat Mech - Spring 2006 Class 1819 Fermi-Dirac

20. Thermo & Stat Mech - Spring 2006 Class 1820 Fermi-Dirac

21. Thermo & Stat Mech - Spring 2006 Class 1821 Fermi-Dirac

22. Thermo & Stat Mech - Spring 2006 Class 1822 Constraints (Lagrange Multipliers)

23. Thermo & Stat Mech - Spring 2006 Class 1823 Fermi-Dirac

24. Thermo & Stat Mech - Spring 2006 Class 1824 Distributions

25. Thermo & Stat Mech - Spring 2006 Class 1825 Boltzmann Distribution

26. Thermo & Stat Mech - Spring 2006 Class 1826 Boltzmann Distribution

27. Thermo & Stat Mech - Spring 2006 Class 1827 Partition Function

28. Thermo & Stat Mech - Spring 2006 Class 1828 Boltzmann Distribution

29. Thermo & Stat Mech - Spring 2006 Class 1829 Ideal Gas

30. Thermo & Stat Mech - Spring 2006 Class 1830 Ideal Gas

31. Thermo & Stat Mech - Spring 2006 Class 1831 Gamma Function

32. Thermo & Stat Mech - Spring 2006 Class 1832 Partition Function for Ideal Gas

33. Thermo & Stat Mech - Spring 2006 Class 1833 Boltzmann Distribution

34. Thermo & Stat Mech - Spring 2006 Class 1834 Ideal Gas

35. Thermo & Stat Mech - Spring 2006 Class 1835 Quantum Statistics When taken to classical limit quantum results must agree with classical. B-E and F-D must approach Boltzmann in classical limit. What is that limit? Low particle density! Then distinguishability is not a factor.

36. Thermo & Stat Mech - Spring 2006 Class 1836 Classical limit

37. Thermo & Stat Mech - Spring 2006 Class 1837 Quantum Results

38. Thermo & Stat Mech - Spring 2006 Class 1838 Chemical Potential

39. Thermo & Stat Mech - Spring 2006 Class 1839 Three Distributions