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one course in electronics left). This is my first post in Physicsforums.

I was thinking about a problem in statistical mechanics. As you all know, if we have a one-

particle system where the particle is subject to some potential V(x), we can solve the particle's

energy spectrum from the Schrodinger equation. When we know the energy spectrum, we can form

the statistical mechanical partition function Z. Then we can calculate E(T), the expectation value

of the systems energy by differentiating log(Z) with respect to temperature. From this we finally

get the heat capacity C(T) by differentiating again.

If a system has a first-order phase transition at some particular temperature, the function E(T)

has a discontinuity at that temperature. Phase transitions are generally only thought to occur

at the thermodynamic limit (i.e. in a system of very many particles). But to me, it seems to be

only a matter of imagining a system with an appropriate energy spectrum to get such behavior to

occur in a one-particle system.

Denote the energy spectrum function with E(n), that specifies an energy for every integer n.

In the case of continuous energy spectrum, we would use function g(E), density of states, instead.

What kind of a function E(n) would lead to a partition function that would have discontinuous

first derivative (or 'almost' discontinuous, jumping very sharply but continuously at some value

of T)? Is it possible to imagine a potential energy function V(x) that would cause a single

particle bound by that potential to have that kind of an energy eigenvalue spectrum?