Lecture 10—Ideas of Statistical Mechanics Chapter 4,
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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 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





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    Statistical distributions

    ni

    xi

    16

    Standard

    deviation


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    Statistical distributions

    64

    Gaussian distribution

    (Bell curve)


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    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


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    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


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    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


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    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.


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    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.


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    Statistical Mechanics – ideas and definitions

    64

    An example:

    Coin toss again!!

    width


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    Ensembles and quantum states (microstates)

    10 particles, 36 cells

    Volume V

    Cell volume, DV


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    Ensembles and quantum states (microstates)

    10 particles, 36 cells

    Volume V

    Cell volume, DV


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    Ensembles and quantum states (microstates)

    10 particles, 36 cells

    Volume V

    Cell volume, DV


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    Ensembles and quantum states (microstates)

    10 particles, 36 cells

    Volume V

    Cell volume, DV


    Slide17 l.jpg

    Ensembles and quantum states (microstates)

    10 particles, 36 cells

    Volume V

    Cell volume, DV


    Slide18 l.jpg

    Ensembles and quantum states (microstates)

    10 particles, 36 cells

    Volume V

    Cell volume, DV


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    Ensembles and quantum states (microstates)

    10 particles, 36 cells

    Volume V

    Cell volume, DV


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    Ensembles and quantum states (microstates)

    Cell volume, DV

    10 particles, 36 cells

    Volume V


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    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.


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    Entropy

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


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