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I am trying to compute the moments of the Boltzmann distribution using a moment generating function, by taking the Fourier transform of the distribution and then taking derivatives to find the appropriate moment. However I am unsure of the integration limits of the Fourier transform step:

$$\int_{a}^{b}dE e^{-ikE} e^{-\beta E}$$

Its clear that this integral doesn't converge on the limits negative infinity to positive infinity (the usual limits for FT's). So my first thought is alright, lets rig it and integrate from zero to infinity. But I'm not sure this is justified in the FT step. The "rigged" limits give me an answer, but I think its wrong.

I want to ultimately relate the specific heat, to the moments of the Boltzmann distribution. Perhaps there is a better way to go with this. Thanks ahead.

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  • $\begingroup$ What's the physical range on $E$? $\endgroup$
    – Kyle Kanos
    Aug 8, 2014 at 1:13
  • $\begingroup$ Well the assumption is that we have a discrete set of energy levels. Presumably, that set is infinitely large. I guess it would be OK to redefine the lowest energy state to be zero. I suppose that would justify the integration limits. Any further comments, or references will be appreciated. $\endgroup$
    – wgwz
    Aug 8, 2014 at 1:46
  • $\begingroup$ Correct, the physical limits should be $0$ and $\infty$, corresponding to the lowest and highest allowed energies. I imagine that most every Plasma Physics textbook would cover the kinetic theory of the Boltzmann distribution (though not necessarily via the Fourier transform), I've always like F Chen's text. $\endgroup$
    – Kyle Kanos
    Aug 8, 2014 at 1:52

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