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Statistical mechanics 6.1

  1. Mar 12, 2008 #1
    [SOLVED] statistical mechanics 6.1

    1. The problem statement, all variables and given/known data
    http://ocw.mit.edu/NR/rdonlyres/Phy...0E63135-AD4E-4F76-9917-349D5439ABF4/0/ps6.pdf
    The answer to Problem 1 part c is 2N. I disagree. I think it should be N because if you specify the z component of the system then you know both of the macroscopic quantities M and E.


    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Mar 13, 2008
  2. jcsd
  3. Mar 13, 2008 #2
    anyone?
     
  4. Mar 13, 2008 #3

    Mapes

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    If you specify only the z component, you have lost information about the number of equivalent states that would give the same M and E. This is key stat mech information. There is only one state with z = [itex]\mu[/itex], but many with z = 0.5[itex]\mu[/itex], and you need to differentiate them. Providing another distance coordinate (or alternatively, an angle) accomplishes this.

    Looking at it another way: two variables are sufficient to describe a line's orientation. With N lines/dipoles, you need 2N variables to describe all the states.
     
  5. Mar 13, 2008 #4
    What line are you talking about?
     
  6. Mar 13, 2008 #5

    Mapes

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    A dipole is like a line segment, in that rotation around the axis of its length is undetectable and does not constitute a degree of freedom.
     
  7. Mar 13, 2008 #6
    I don't understand. What "key stat mech information" can you not get if you have all of the z components. If you have all of the z-components, then you know M, E, and can calculate [itex]\Omega[/itex] with the given equation. What else do you want?
     
  8. Mar 13, 2008 #7

    Mapes

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    You want the number of possible microstates that would result in those macrostate values of M and E. This is the fundamental idea of stat mech: We want to know the probability distribution of microstates that are compatible with our macrostate constraints.

    This doesn't seem to be sinking in, so let's go back to the original question: "How many microscopic variables are necessary to completely specify the state of the system?" You give me a z value for each of the N dipoles. But you can't quit there. You haven't completely specified the microstate yet, since the dipoles can rotate in three dimensions and I don't know any of the x values.
     
  9. Mar 13, 2008 #8
    I understand that you cannot completely specify the microstate of the system by providing only the z components. It is clear that there are 2N variables needed to specify the microstate of the system.

    I think I see the flaw in my thinking now. By "state" in the question they really mean "microscopic state" not "thermodynamic/macroscopic state". Specifying only the z components of the system WILL determine the thermodynamic/macroscopic state of the system by the equations they provide, however, the microscopic state will still be ambiguous. Please confirm that this is correct.
     
  10. Mar 14, 2008 #9

    Mapes

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    Sounds good.
     
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