Hi, I'm Teemu from Finland, and I'm just getting my master's degree in physics completed (only 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?