Statistical mechanics 6.1

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[SOLVED] statistical mechanics 6.1

Homework Statement


http://ocw.mit.edu/NR/rdonlyres/Physics/8-044Spring-2004/00E63135-AD4E-4F76-9917-349D5439ABF4/0/ps6.pdf [Broken]
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.


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The Attempt at a Solution

 
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  • #2
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anyone?
 
  • #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.
 
  • #4
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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.
What line are you talking about?
 
  • #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.
 
  • #6
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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?
 
  • #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.
 
  • #8
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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.
 
  • #9
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Sounds good.
 

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