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Lecture 10—Ideas of Statistical Mechanics Chapter 4, Wednesday January 30 th

Lecture 10—Ideas of Statistical Mechanics Chapter 4, Wednesday January 30 th. Finish Ch. 3 - Statistical distributions Statistical mechanics - ideas and definitions Quantum states, classical probability, ensembles, macrostates... Entropy Definition of a quantum state.

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Lecture 10—Ideas of Statistical Mechanics Chapter 4, Wednesday January 30 th

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  1. Lecture 10—Ideas of Statistical Mechanics Chapter 4, Wednesday January 30th • Finish Ch. 3 - Statistical distributions • Statistical mechanics - ideas and definitions • Quantum states, classical probability, ensembles, macrostates... • Entropy • Definition of a quantum state Reading: All of chapter 4 (pages 67 - 88) ***Homework 3 due Fri. Feb. 1st**** Assigned problems, Ch. 3: 8, 10, 16, 18, 20 Homework 4 due next Thu. Feb. 7th Assigned problems, Ch. 4: 2, 8, 10, 12, 14 Exam 1: Fri. Feb. 8th (in class), chapters 1-4

  2. Statistical distributions ni xi 16 Mean:

  3. Statistical distributions ni xi 16 Mean:

  4. Statistical distributions ni xi 16 Mean:

  5. Statistical distributions ni xi 16 Standard deviation

  6. Statistical distributions 64 Gaussian distribution (Bell curve)

  7. Statistical Mechanics (Chapter 4) Statistical Mechanics Thermal Properties • What is the physical basis for the 2nd law? • What is the microscopic basis for entropy? Boltzmann hypothesis: the entropy of a system is related to the probability of its state; the basis of entropy is statistical. Statistics + Mechanics

  8. Statistical Mechanics • Use classical probability to make predictions. • Use statistical probability to test predictions. Note: statistical probability has no basis if a system is out of equilibrium (repeat tests, get different results). How on earth is this possible? • How do we define simple events? • How do we count them? • How can we be sure they have equal probabilities? REQUIRES AN IMMENSE LEAP OF FAITH

  9. Statistical Mechanics – ideas and definitions e.g. Determine: Position Momentum Energy Spin of every particle, all at once!!!!! ............ A quantum state, or microstate • A unique configuration. • To know that it is unique, we must specify it as completely as possible... THIS IS ACTUALLY IMPOSSIBLE FOR ANY REAL SYSTEM

  10. Statistical Mechanics – ideas and definitions An ensemble • A collection of separate systems prepared in precisely the same way. A quantum state, or microstate • A unique configuration. • To know that it is unique, we must specify it as completely as possible... Classical probability • Cannot use statistical probability. • Thus, we are forced to use classical probability.

  11. Statistical Mechanics – ideas and definitions The microcanonical ensemble: Each system has same: # of particles Total energy Volume Shape Magnetic field Electric field and so on.... ............ These variables (parameters) specify the ‘macrostate’ of the ensemble. A macrostate is specified by ‘an equation of state’. Many, many different microstates might correspond to the same macrostate.

  12. Statistical Mechanics – ideas and definitions 64 An example: Coin toss again!! width

  13. Ensembles and quantum states (microstates) 10 particles, 36 cells Volume V Cell volume, DV

  14. Ensembles and quantum states (microstates) 10 particles, 36 cells Volume V Cell volume, DV

  15. Ensembles and quantum states (microstates) 10 particles, 36 cells Volume V Cell volume, DV

  16. Ensembles and quantum states (microstates) 10 particles, 36 cells Volume V Cell volume, DV

  17. Ensembles and quantum states (microstates) 10 particles, 36 cells Volume V Cell volume, DV

  18. Ensembles and quantum states (microstates) 10 particles, 36 cells Volume V Cell volume, DV

  19. Ensembles and quantum states (microstates) 10 particles, 36 cells Volume V Cell volume, DV

  20. Ensembles and quantum states (microstates) Cell volume, DV 10 particles, 36 cells Volume V

  21. Ensembles and quantum states (microstates) Cell volume, DV Many more states look like this, but no more probable than the last one Volume V There’s a major flaw in this calculation. Can anyone see it? It turns out that we get away with it.

  22. Entropy Boltzmann hypothesis: the entropy of a system is related to the probability of its being in a state.

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