Non-sensical negative entropy? (grand canonical ensemble)

[Ken G]

I'm not sure what you're saying, why are you claiming "else we cannot solve the problem". mu determines the expectation value of N. Do you have any backing of the statement "I'm saying epsilon has to be relative to that ground state energy, or else we cannot solve the problem, we won't have enough information."? It seems wrong to me, but I don't know how to prove that before I know why you think it's true.

If there is a particle reservoir, it will keep cranking out particles as long as the number of configurations the system+reservoir can reach is increased by adding particles. That's what determines the expectation value of N, physically. The chemical potential may give us a clever mathematical way to calculate this, but we should not lose sight of the basic physics. If we have a negative energy state, meaning that every time a particle is created from the particle reservoir and placed into that state, it releases energy, then that's just what the particle reservoir is going to do, and that is an impossible situation-- it will never reach equilibrium. We cannot derive both mu and N, we need one to tell us the other, but presumably there is some physical truth that sets mu, and so N is what we cannot control unless we have access to the mechanism that controls mu.

So what I'm saying is, it makes sense to me that if we observe the expectation value of N, we can infer mu, and epsilon can be negative or can have anything added to it, and it will just change our meaning of mu. Or, if we know something about what goes into mu, we can infer N, but what we know about mu had better make sure that energy is not liberated by -epsilon (so the value of -epsilon becomes physically important in an absolute way). Finally, we can take the limit as N goes to infinity, as Dickfore did, and get a result that does not depend on N so would seem to be able to infer mu. But this means epsilon is placing a constraint on the physics of mu such that we'll end up with a large but not infinite N, and a different epsilon puts a different physically meaningful limit on mu. What is physically meaningful is the energy that appears in the system when a particle appears in the system, and that is how I interpreted the value of epsilon. This in turn means that Dickfore's analysis can be used, not to derive N, but to find the physically necessary mu, relative to the energy of the state, that creates a large number of particles without creating an infinite number of particles. In other words, there is a physical meaning to the energy that appears in the system when a particle does, and that had better not be a negative number, because the particle didn't exist before and there's no place but the reservoir to conserve that energy.