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No matter what the real system or subject is, if there is a partition function $Z$, then these kind of identities hold

$$\langle X\rangle=\frac{\partial}{\partial Y}\left(-\log Z(Y)\right).$$

If one knows the partition function $Z$, than one essentially knows everything. Very much of the system is in this single expression.

Now I see the technical details, the function contains the relevant weights and a sum/integral, which end up producing expectation values if you derive the thing with respect to conjugate parameters. But how can I really understand what's going on? Intuitively. Morally. What makes these kind of systems so that there is one special function, which does this? (Here and here two relevant examples I'm interested the most in.)

What is the requirement for a system, to contain this conjugate variables business in the first place? They are usually related via an equation of motion or an equation of state...however, these model relations are then eventually also just derived. They are identities involving $Z$. Are there maybe statements about the functional dependencies a $Z$ of it's a priori free model parameters?

A more direct formulation: For an initially microscopic system (fields, counting of states and so on) what are the requirements for it and its set of quantities/functions/observables like $X$, such that there is a single valued function/potential, which determines all the $\langle X\rangle$? The point being that you don't have to compute $\langle X\rangle$ like expectation values where $\langle\ \ \rangle$ denotes a complicated functional involving microscopic features.

Also, is there a relationship with generating functions and are there reductionists views/mathematical generalizations of the partition function? I have to add that I really understand how phase transitions and renormalization work, maybe that gives an insight.

I also know that there is an important "macroscopic function", which turns out to generate chern classes in algebraic topology, however I don't know if there are any conceptual relations to the quantity generating objects like partition functions.

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Doens't the Hamiltonian completely describe the time evolution of a classical particle as well? One can also infer everything one wants to know from this formula right? –  Thomas Rot Jan 12 '12 at 16:04
    
@Thomas Rot: Well yes, $f'(x)=cf(x)$ and $f(0)=1$ also describes the whole system and I understand why. If I'm about to describe my system via a symplectic structure $\omega$ and I specify an energy and the systems variable dependencies by the Hamiltonian $H\overset{!}{=}E$, then I understand how the differential equations $i_X\omega=dH$ fix the flow $X$. I guess I have an understanding of geometry, but the soul of the partition function is beyond me. –  NikolajK Jan 12 '12 at 16:10
    
@NickKidman, To me, the partition function "encoding everything" is synonymous with a function "encoding everything" at a point. If you can evaluate a function at a point and take all of it's derivatives then you know everything there is about a function. I realize this is not quite what you're looking for and I bet you would get a much more satisfactory answer if you were to ask the same question over on the physics SE site. You might want to try migrating the question there. –  user4143 Jan 31 '12 at 16:17
    
@user4143: The difference to a function is that in that example, there only is the function and nothing behind it. Here the function values would correspond to the quantities $\langle X \rangle$, but these actually come from some theory below. And I'm looking for the mathematical perspective, on physics stackexchange the qestion might be interpreted as a question, which can be answered by "Yes, there is fundamental physics in nature, but we are interested in macroscopic effects. The partition function is a way to compute averages". –  NikolajK Jan 31 '12 at 16:23
    
I have more information now, namely that the partition function is effentially the cumulant of the system. –  NikolajK Nov 6 '13 at 10:03
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