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## Homework Statement

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Strap in, this one's kind of long. (This problem is from 'Six Ideas That Shaped Physics, Unit T' by Thomas A Moore, 2nd edition. Problem T6R2.)

Imagine that aliens deliver into your hands two identical objects made of substances whose multiplicities increase linearly with thermal energy, something like

$$\Omega = \frac{aNU}{\epsilon}$$

where ε is some energy unit and a is some constant. Answer the following questions about these objects.

a) Do they have a well-defined temperature? If so, how does this temperature depend on the objects' thermal energy?

b) If these objects are placed in thermal contact, will energy spontaneously flow from hot to cold? Will the objects eventually come into equilibrium at a certain common temperature? (Hint: I suggest drawing a graph of Ω

_{AB}versus macropartition. This will also help with the next part.)

c) How will the size of the random fluctuations in the energies of these objects compare to those for two normal objects placed in thermal contact?

## Homework Equations

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$$\Omega = \frac{aNU}{\epsilon}$$ From the problem statement

$$\frac{1}{T} = \frac{\partial{S}}{\partial{U}}$$ (definition of temperature)

$$\Omega(N,U) = \frac{(3N+q-1)!}{(3N-1)!q!}$$ (q = U/ε) (definition of multiplicity given in text)

3. The Attempt at a Solution

3. The Attempt at a Solution

Using the equation for the definition of temperature gives me T = U/k. (I can reproduce this if needed but don't want to clutter things too much. I want to focus on the information in the table.) In a 'normal' object, the temperature and thermal energy are related by T = U/(3Nk). (The book uses 'Einstein solids' as its model for these things. So that's what I'm comparing these 'alien' objects to.)

Let's just say N = 1 for simplicity from here on. So this just tells me that the temperature increase for these 'alien' objects, in response to a given increase of thermal energy, is different from that of an Einstein solid by a factor of 1/3. So fundamentally, these objects should behave much like normal ones. I.e. they would have well-defined temperatures, they would come to equilibrium by exchanging heat energy, energy would spontaneously flow from the warmer one to the colder one, etc.

The author hints that I should make a graph or table of macropartition vs Ω

_{AB}. Unfortunately I have no idea how to present a table in this post, so that will make things difficult. But the basic idea is that if I assume each object consists of 1 atom (N = 1, again for simplicity) and there are 6 'atoms' of energy ε, so that U = 6ε, then for each macro-distribution of those 'atoms' (in the left column), I would list the multiplicity Ω

_{AB}of the macrostate as a whole. And this will be the product of the individual multiplicities of objects A and B, as computed from the formula for multiplicity given in the problem statement. So for example, for the distribution 1:5 (where 1 bit of energy is in object A and 5 are in object B), I would compute the multiplicity of A, the multiplicity of B, then multiply them to get the multiplicity of this macrostate.

(Hopefully that's clear. I wish I could just make a table, but I don't know how.)

But right away I run into a problem. Consider the macropartition 0:6, where object A has none of the energy atoms and object B has all 6 of them. If I use the expression for multiplicity given in the problem, then Ω

_{A}is zero, thus making the multiplicity of the whole macropartition zero. This would mean there is no way for one of the objects to have all the energy and the other one to have none. But I don't see any reason why that should be the case here any more than for an Einstein solid.

So I figure maybe I'm using the wrong multiplicity equation, but if I use the other one (the one given in the text, for an Einstein solid), then I just get the same table I would for an Einstein solid, which doesn't seem right either.

Is my problem maybe that I'm only using 1 atom, and the difference between the two kinds of object ('alien' vs 'normal') only comes out when using multiple atoms? But that seems wrong, because even with only one atom, there is still the factor of 1/3 that I mentioned above.

So I guess my question is:

Why is it that when I calculate the temperature/thermal energy relationship, I'm getting a straightforward answer that seems physically plausible (U = kT, versus U = 3kT for a 'normal' solid), but when I try to calculate the multiplicity of the 0:6 macropartition, I get an answer that makes no physical sense? Am I just using the wrong multiplicity equation? Or is this just to show that the given multiplicity equation makes no sense physically? Or something else?

UPDATE: I'm now seeing that I need to use the multiplicity equation given in the text, for both objects, in each macropartition. But now my trouble is that this simply reproduces the macropartition vs Ω

_{AB}table of an Einstein solid, in which case the problem becomes profoundly uninteresting. But I presume there's something I'm supposed to learn here. I don't see how the multiplicity expression given in the problem statement

$$\Omega = \frac{aNU}{\epsilon}$$

comes into play viz a viz the table. I see that I can derive an expression for temperature from it. But I don't see how temperature and the table are related, but the author seems to want us to spot that connection.

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