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I am struggling to grasp the sense of the partition function.

First of all, I had a look at a couple of derivations (which the relevant Wikipedia page follows) in which the concept of heat"energy of a thermal bath" is invoked. Well this is already confusing me: if the thermal bath has an infinite heat capacity I cannot see how it could have a finite energy (or a finite size).

But then I somehow follow the derivation until I am presented with the partition function of a single harmonic oscillator. The partition assigns probabilities to the HO having a certain energy. Well, I do not follow. The HO will have an energy, at a certain moment in time, given by its Hamiltonian. If such energy starys constant, the concept of partition function loses meaning. So I am led to think the thermal bath varies the energy of the oscillator, but how? In derivation bases on gases I see the link between tempertaure and pressure (related to the particles kinetic energy) via the state equation, but I do not see such link for the harmonic oscillator. Finally I do not understand how statistical mechanics could be applied to such a simple system, the harmonic oscillator. For example, what is the meaning of the entropy of a harmonic oscillator?

Any help would be so appreciated.

Thank you lots