On a mathematical level, the statistical mechanical partition function is just a Laplace transform of the microcanonical probability distribution, i.e. it's moment generating function. Understanding the mathematical and physical motivation and meaning behind this is something many have unfortunately been unable to help me with, hence I ask here:

What is the intuitive meaning of the n'th moment of a function or probability distribution?

  • I understand that "Broadly speaking a moment can be considered how a sample diverges from the mean value of a signal - the first moment is actually the mean, the second is the variance etc... ", but using this interpretation I can't make sense of what the 5'th moment of $f(x) = x^2$ is, why it's useful, what it says about anything, and how in the world such an interpretation falls out of the mathematical derivation.

Why would one ever say to oneself that they should decompose a function into it's constituent moments via a moment generating function?

  • A Fourier series/transform of a function of, say, time is just an equivalent way to characterize the same function in terms of it's amplitude, phase and frequency instead of time (as a sine wave is uniquely characterized by these quantities, and to a physicist a Fourier series is more or less just a big sine wave where these numbers vary at each point). A Geometric interpretation of every term in the expansion of the Fourier series of a function can be given. Intuitively, just looking at, say, light or sound there is an immediate physical motivation for using a Fourier transform because it decomposes them into their constituent frequencies, which exist physically. A m.g.f. is a bilateral Laplace transform, and it decomposes a function/distribution into it's moments. Thus the Laplace transform should have an almost similar story behind using it, one that should explain the reason for wanting the n'th moment of $x^2$. (An answer saying that it uniquely characterizes "nice" functions/distributions is to miss the point of my question).

Why is it that the Fourier series of an exponentially damped distribution (the moment generating function or Bilateral Laplace Transform) decomposes a function into it's moments?

  • I understand the mathematical derivation, but not the motivation for doing any of this, or why it should work, i.e. why wouldn't the regular Fourier transform just be enough? There has to be meaning behind it, so taking the fact we're using a Fourier series/transform literally, the question arises as to why it is that the constituent waves of an exponentially damped version of a function/distribution allow for an interpretation in terms of moments? (Note "exponentially damped" means using a minus sign in the m.g.f.)

Why does a Wick rotation of the Moment generating function give the Characteristic function?

  • The characteristic function is the Fourier transform of a probability distribution, but it is also a wick rotation of the Fourier transform of an exponentially damped distribution. I understand we are analytically continuing the argument to a new axis, but is there any meaning behind this, does it say anything about the relationship between the Laplace transform's moments and the composite waves that a Fourier series decomposes a function into, and does it help explain why there is a mathematical relationship between the Feynman Path integral and the statistical mechanical Partition function?

In other words, regarding the partition function: What are we doing, why are we doing it, why does a particular method for doing it work & does it explain an interesting connection between two other things? Thanks


closed as too broad by Danu, Brandon Enright, Kyle Oman, Colin McFaul, user10851 Jun 18 '14 at 2:27

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Although this seems like a number of good questions, the way you present them right now is not very suitable for this site. Perhaps you should consider splitting your subquestions off and posting several separate questions instead! $\endgroup$ – Danu Jun 17 '14 at 18:05
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    $\begingroup$ They are all parts of the same single question though: understanding the partition function mathematically. Q1: Meaning of terms in m.g.f., Q2: reason for using m.g.f., Q3: Why method X gives us the m.g.f. Q4: Curious issue of complexifying a m.g.f. giving us something interesting. I can't see how one can understand the partition function without understanding these intimately related issues, an answer to one part is no help without understanding the rest :( $\endgroup$ – bolbteppa Jun 17 '14 at 18:17
  • $\begingroup$ I understand your problem. One way to solve this problem would be to refer back to the other questions when asking each of them. I do not think each of those questions are essentially the same, although they each address an aspect of a broader question (too broad for this site). I also think an answer to even just one part is useful in its own right. One has to keep in mind that the answers here are not just for the asker to learn from, but from anyone who visits, or will visit, this site. $\endgroup$ – Danu Jun 17 '14 at 18:19
  • $\begingroup$ They are all addressed separately in all the links I've provided, and answered on their own out of the context of my question they make little sense, hence the merging of them together as presented - I've put a lot of work into trying to relate these things and trying to figure out how they link together, and as it stands my professor's can't answer these questions as it's not an easy one, giving me answers like those in each of the links separately (in a non-unified manner) so asking me to break it all up is basically to ruin the possibility for an answer quite frankly. $\endgroup$ – bolbteppa Jun 17 '14 at 18:30
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    $\begingroup$ Separating them really makes no sense, anybody answering them would have to read all the other threads anyway and answer in the context of 3 other posts, whereas answering here should be possible in one paragraph addressing all 4 at the same time, if understood properly (my end-goal is to be able to do that!). $\endgroup$ – bolbteppa Jun 17 '14 at 18:36