# Meaning of the Partition function

Hi all,
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

anorlunda
Staff Emeritus
May I recommend Professor Susskind's video lectures. A 10 lecture course on Statistical Mechanics starts here.
Believe me, it will be well worth your time.

DrClaude
Mentor
Hi all,
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).
It's an idealization. Just imagine a system big enough that any heat exchange you have with it will not change its temperature. In a real-life situation, for a small system the air in a room can be considered big enough.

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?
But what is the Hamiltonian? I guess you are imagining the Hamiltonian of an isolated h.o. What you have now is a h.o. coupled to a termal bath. The full Hamitonian would consist of the Hamiltonina for the isolated h.o., plus that for the isolated bath, plus the coupling between the two. If you trace over the degrees of freedom of the bath, you are left with a system, the h.o., where the energy fluctuates.

Finally I do not understand how statistical mechanics could be applied to such a simple system, the harmonic oscillator.
You can see it through an ensemble average, considering that you have many copies of an identical system, or though a time average of a single system. In both cases, since you don't know the exact state of the bath or its evolution, it is through statistical mechanics that you gain information.