What is the answer to Problem 1 part c in Statistical Mechanics 6.1?

[ehrenfest]

[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
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.

Homework Equations

The Attempt at a Solution

 

[ehrenfest]

anyone?

 

[Mapes]

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 = $\mu$, but many with z = 0.5$\mu$, 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.

 

[ehrenfest]

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?

 

[Mapes]

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.

 

[ehrenfest]

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 $\Omega$ with the given equation. What else do you want?

 

[Mapes]

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.

 

[ehrenfest]

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.

 

[Mapes]

Sounds good.