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Properties of a little system in a big world

Properties of a little system in a big world. Part 1. Part 2. Part 3. System. Bath. Properties of a little system in a big world. Hypothesis. Notation. Ä. X. Where is the energy?. Properties of the little system. System. Bath. Energy levels in a system.

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Properties of a little system in a big world

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  1. Properties of a little system in a big world Part 1 Part 2 Part 3 System Bath

  2. Properties of a little system in a big world Hypothesis Notation Ä X Where is the energy? Properties of the little system System Bath

  3. Energy levels in a system Hamiltonian is part of a (set of) differential equations describing time-evolution of a state Hamiltonian applied to state specifies the energy for that state 4e0 F G H 3e0 D E 2e0 B C e0 State A

  4. Hypothesis: All accessible microstates equally probable Hamiltonian is part of a (set of) differential equations describing time-evolution of a state Hamiltonian applied to state specifies the energy for that state Time-independent Hamiltonian conserves total energy System Part 1 System Part 1 More total energy than the other configurations System Part 1 System Part 1

  5. Properties of a little system in a big world Hypothesis Notation Ä X Where is the energy? Properties of the little system System Bath

  6. Notating states for system and bath Part 1 System e = 3e0 2e0 Ä Ä Ä Ä

  7. Notating states for system and bath Part 1 System e = 3e0 2e0 Ä Ä Ä Ä

  8. Notating states for system and bath Part 1 System e = 3e0 2e0 Ä Ä Ä Ä

  9. Notating states for system and bath Part 1 System e = 3e0 2e0 Ä Ä Ä Ä

  10. Notating states for system and bath [ ] [ ] = Ä States of combination of system and part 1 List of states for system List of states for part 1 = × Number of states for system Number of states for part 1 Number of ways to combine System and part 1 WSYS+1 = WSYS(3e0) W1(2e0)

  11. Notating states for system and bath [ ] [ ] [ ] = Ä Ä States of combination of system and parts 1, 2, ..., N Ä . . . List of states for part N List of states for system List of states for part 1 = × × × Number of states for system Number of states for part 1 Number of states for part N Number of ways to combine System and parts 1, 2, . . . N . . . WSYS+N = WSYS(3e0) W1(2e0)W2(2e0) . . . WN(2e0)

  12. Properties of a little system in a big world Hypothesis Notation Ä X Where is the energy? Properties of the little system System Bath

  13. States for system and bath System Bath System Bath System System Bath Bath

  14. How many states if transfer an energy unit to bath? WSYS+N = WSYS(3e0)W1(2e0)W2(2e0) . . . WN(2e0) [ ] [ ] [ ] = Ä Ä Ä States of combination of system and parts 1, 2, ..., N . . . List of states for part N List of states for system List of states for part 1

  15. Transferring an energy unit: updating lists of states WSYS+N =WSYS(3e0)W1(2e0)W2(2e0) . . . WN(2e0) [ ] [ ] [ ] = Ä Ä Ä States of combination of system and parts 1, 2, ..., N . . . List of states for part N List of states for system List of states for part 1

  16. Transferring an energy unit: updating lists of states WSYS+N = WSYS(3e0) W1(2e0)W2(2e0) . . . WN(2e0) →WSYS(2e0) W1(2e0) W2(2e0) . . . WN(2e0) [ ] [ ] [ ] = Ä Ä Ä States of combination of system and parts 1, 2, ..., N . . . List of states for part N List of states for system List of states for part 1

  17. Transferring an energy unit: updating lists of states WSYS+N = WSYS(3e0) W1(2e0)W2(2e0) . . . WN(2e0) →WSYS(2e0) W1(2e0) W2(2e0) . . . WN(2e0) [ ] [ ] [ ] [ ] = Ä Ä Ä States of combination of system and parts 1, 2, ..., N . . . List of states for part N List of states for system List of states for part 1

  18. Transferring an energy unit: updating lists of states WSYS+N = WSYS(3e0)W1(2e0)W2(2e0) . . . WN(2e0) →WSYS(2e0) W1(3e0) W2(2e0) . . . WN(2e0) [ ] [ ] [ ] = Ä Ä Ä States of combination of system and parts 1, 2, ..., N . . . List of states for part N List of states for system List of states for part 1

  19. Transferring an energy unit: updating lists of states WSYS+N = WSYS(3e0) W1(2e0)W2(2e0) . . . WN(2e0) →WSYS(2e0) W1(3e0)W1(2e0) W2(2e0). . . W1(2e0) [ ] [ ] [ ] = Ä Ä Ä States of combination of system and parts 1, 2, ..., N . . . List of states for part N List of states for system List of states for part 1

  20. Transferring an energy unit: updating lists of states WSYS+N = WSYS(3e0) W1(2e0)W2(2e0) . . . WN(2e0) →WSYS(2e0) W1(3e0)W1(2e0) W2(2e0). . . W1(2e0) [ ] [ ] [ ] = Ä Ä Ä States of combination of system and parts 1, 2, ..., N . . . List of states for part N List of states for system List of states for part 1

  21. Transferring an energy unit like multiplying states WSYS+N = WSYS(3e0) W1(2e0)W2(2e0) . . . WN(2e0) →WSYS(2e0) W1(3e0)W1(2e0) W2(2e0). . . W1(2e0) +WSYS(2e0) . . . WN-1(2e0) WN(3e0)WN(2e0) +WSYS(2e0) W1(2e0) W2(3e0)W2(2e0). . . W2(2e0) WN(2e0) + . . .

  22. Transferring an energy unit like multiplying states WSYS+N = WSYS(3e0)W1(2e0)W2(2e0) . . . WN(2e0) →NWSYS(2e0) W1(3e0) W1(2e0) W2(2e0). . . →WSYS(2e0) W1(3e0) W1(2e0) W2(2e0). . . W1(2e0) +WSYS(2e0) . . . WN-1(2e0) WN(3e0) WN(2e0) +WSYS(2e0) W1(2e0) W2(3e0) W2(2e0). . . W2(2e0) WN(2e0) + . . .

  23. Transferring an energy unit like multiplying states WSYS+N = WSYS(3e0)W1(2e0)W2(2e0) . . . WN(2e0) →NWSYS(2e0) W1(3e0) W1(2e0) W2(2e0). . . W1(2e0) =WSYS(2e0) N W1(3e0)W1(2e0) W2(2e0). . . W1(2e0) =WSYS(2e0) N W1(3e0)W1THROUGHN(@2e0) W1(2e0)

  24. Next transfer of energy looks like previous WSYS+N(2e0)= WSYS(2e0) N W1(3e0)W1THROUGHN(@2e0) W1(2e0) [ ] [ ] [ ] = Ä Ä Ä States of combination of system and parts 1, 2, ..., N . . . List of states for part N List of states for system List of states for part 1 + + + . . . []Ä [] Ä[] Ä. . . Ä [] []Ä [] Ä . . . Ä[]

  25. Next transfer of energy looks like previous WSYS+N(e0)= WSYS(2e0)N W1(3e0)W1THROUGHN(@2e0) W1(2e0) [ ] [ ] [ ] = Ä Ä Ä States of combination of system and parts 1, 2, ..., N . . . List of states for part N List of states for system List of states for part 1 + + + . . . []Ä [] Ä[]Ä. . . Ä [] []Ä [] Ä . . . Ä[]

  26. Next transfer of energy looks like previous WSYS+N(e0)= WSYS(e0) N W1(3e0)W1THROUGHN(@2e0) W1(2e0) Heat bath: N is “very” big !!! [ ] [ ] [ ] = Ä Ä Ä States of combination of system and parts 1, 2, ..., N . . . List of states for part N List of states for system List of states for part 1 + + + . . . []Ä [] Ä[]Ä. . . Ä [] []Ä [] Ä . . . Ä[]

  27. Next transfer of energy looks like previous WSYS+N(e0)= 2 copies of same factor WSYS(e0) N W1(3e0) N W1(3e0)W1THROUGHN(@2e0) W1(2e0)W1(2e0) 1) N receptacles 2) Energy “always” deposited into receptacle with initial energy 2e0 [ ] [ ] [ ] Pretend green is gray = Ä Ä Ä States of combination of system and parts 1, 2, ..., N . . . List of states for part N List of states for system List of states for part 1 + + + . . . []Ä [] Ä[]Ä. . . Ä [] []Ä [] Ä . . . Ä[]

  28. Boltzmann distribution, temperature, and entropy WSYS+N(e0)= WSYS(e0) N W1(3e0) N W1(3e0)W1THROUGHN(@2e0) W1(2e0)W1(2e0)

  29. Properties of a little system in a big world Hypothesis Notation Ä X Where is the energy? Properties of the little system System Bath

  30. Most prevalent macrostate, max entropy, min free energy . . . . . . . . .

  31. Most prevalent macrostate, max entropy, min free energy

  32. Most prevalent macrostate, max entropy, min free energy . . . . . . . . .

  33. Probability function and expectation values

  34. Probabilities and partition function

  35. Probabilities and partition function

  36. Properties of a little system in a big world Hypothesis Notation Ä X Where is the energy? Properties of the little system System Bath

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