Macrostates in Einstein Model of a Solid

[McCoy13]

Homework Statement

Sketch all the possible microstates for an Einstein solid using dots to represents units of energy and lines to separate oscillators. In addition, identify all possible macrostates of the system. Let N=3 (oscillators) and q=3 (units of energy).

Homework Equations

$$\Omega (N,q) = \frac{(q+N-1)!}{q!(N-1)!}$$

The Attempt at a Solution

I got the first part just fine. I sketched out a total of ten configurations of dots distributed among three bins. What I'm not sure about is what qualifies as a macrostate for the Einstein solid. The book (Schroeder's Introduction to Thermal Physics) seems to suggest that the state N=3, q=3 is the macrostate. According to the formula given for multiplicity of a macrostate in an Einstein solid takes N and q as parameters, and for N=3, q=3 it yields 10, the number of microstates I drew. However, the lowest level of definition of state at which degeneracy occurs is a specification such as "one oscillator has 2 units and another has 1" or "one oscillator gets all 3 units of energy".

However, if we interpret the macrostate as N=3, q=3, then there is only one possible macrostate as specified by the problem. Therefore, asking for all the possible macrostates is kind of confusing (or intentionally misleading).

I emailed my professor, but he hasn't emailed me back yet, so I thought I'd ask here in the meantime.

 

[mark726]

Thank you for your post. It seems like you have a good understanding of the microstates of an Einstein solid. As for the macrostates, they are defined by the total number of oscillators and the total energy of the system. In this case, there is only one possible macrostate, which is N=3, q=3. This means that all three oscillators have one unit of energy each.

However, as you mentioned, there can be different configurations within this macrostate. For example, one oscillator can have 2 units of energy and the other two can have 1 unit each. This would still be considered the same macrostate, as the total number of oscillators and energy remains the same.

In summary, the macrostate N=3, q=3 is the only possible macrostate for this system. The different configurations within this macrostate are just different ways of distributing the energy among the oscillators, but they all fall under the same macrostate.

I hope this helps clarify any confusion. Feel free to ask any further questions if needed.