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At least if $\vec v$ is really only a one dimensional parameter, measuring all the moments $\langle v^n \rangle_f$ seems to give me all the information to compute $\langle A \rangle_f$ with $A(v)$ being a function of $v$. So integer powers of the generator of the function algebra $v$ are special functions.

I wonder if there are other special functions whos expectation are equally useful and if so, how they depend of the system you consider. For example, let $\langle \dots \rangle_f$ be some prescrition to calculate a mean or expectation and

$$\langle A \rangle_f:=\int_\Gamma A(\vec v)\ f(\vec v)\ \text d \vec v,$$

then is there e.g. a statistical significance to $\langle f \rangle_f$?

Maybe this question has a "trivial" yes as answer, because there are more options than taylor series to give a basis for some function algebra of elements $f(v_1,v_2,\dots)$. However, I don't know how this changes if one considers noncomutative generators $[v_1,v_2]\ne0$.

The question comes in part because I'm intimidated by relations like

$$\langle T^n\rangle \equiv \int_0^\infty E^{n-1}\, e^{ - \left( {E /T } \right)^C }{\rm d}E\ = \tfrac{\Gamma \left(\tfrac{n }{ C }\right)}{C} \cdot T^n,$$

i.e. $\langle A(E) \rangle$ expectation values are apprearently trivial to compute for weights which are polynomial in functions $f_{\ T,C}(E)=e^{ - \left( {E /T } \right)^C}$.

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The moments of a probability distribution are generally enough to reconstruct the distribution, under certain conditions (these are explored heavily in the math literature, I don't know the main themes). The other expectation values people use are those of exp(ikV). The thing that is intimidating you is just the definition of the gamma function with a change of scale by T in the integral (it's easy and intuitive), so I don't know what the question is exactly. –  Ron Maimon Jul 12 '12 at 9:26
    
@RonMaimon: In the Gamma case, i.e. Laplace transform, the property of getting simple powers of the temperature and the fact that taylor series happen to use these as basis makes me think that for the exponential functions, basically any function space basis is a valid one to study and thereby know the whole system at one. Here the question is if there are rules to deduce that from the property of f. Then there is the second point if these kind of "basis for measruement space" property gets lost in a noncommutative system. I mean knowning <X> and <P> will in general not help you with <H(P,X)>. –  NikolajK Jul 12 '12 at 9:36
    
In the noncommutative case <H(X,P)> with CCR, the analogue of the Fourier transfrom is the expectation of the exponential of a linear compbination of X and P. –  Arnold Neumaier Jul 12 '12 at 10:56

2 Answers 2

up vote 2 down vote accepted

In addition to the polynomial moments, people often consider the Fourier transform of a probability distribution. This is the expected value

$$ \int \rho(V) e^{ikV} dV $$

These exponential moments are clearly enough to reconstruct the distribution completely.

The polynomial moments are not always enough, but the countexamples are badly behaved. The moment

$$ \int x^n \rho(x) dx $$

is also the n-th derivative of the fourier transform $\rho(k)$ at k=0. If you have two distributions $\rho(1)$ and $\rho(2)$ with the same moments, their difference is a function with all moments equal to zero. In Fourier space, such a function has all derivatives zero at the origin. To make such a function, you can use

$$ f(k) = e^{-{1\over k^2}}$$

And Fourier transform, and then take the positive and negative parts of f(x) as separate probability distributions. These two distributions have all their moments equal.

As for the gamma function identity that is bothering you, you rescale the integration variable by T, and this gives the scaling (this is a form of dimensional analysis). The remaining integral is

$$ T^n \int_0^\infty u^{n-1} e^{-u^C} du $$

and a change of variables to $v=u^C$ gives the gamma function:

$$ {T^n\over C} \int_0^\infty v^{{n\over C} -1} e^{-v} dv = \Gamma({n\over C}) {T^n\over C}$$

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+1 Thanks for the elaboration regarding the uniqueness of function expectations by their moments. I guess I know how to integrate via substitutation, so I'm not necessarily "bothered" by it (as the bounds are $0$ and $\infty$, that identity $\int E^{n-1}f(E/T)dE=c T^n$ is true by dimentional arguments already). I guessed collecting fourier transformation data will be sufficient by the fact that Fourier transform spectroscopy etc is possible. In fact, the question was also motivated by the fact that I'm intimidated (not bothered) by other integrals sucking out information, e.g. Radon transform. –  NikolajK Jul 12 '12 at 9:53

''is there e.g. a statistical significance to $⟨f⟩_f$?''

No but $\langle -\log f\rangle_f$ is the entropy.

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