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I have just finished a course on statistical mechanics. It centers on deriving useful information from various partition functions. In the case of grand canonical partition function, the log of it can be seen as a sum of the log of Bose/Fermi function at various energies (momentum). The sum can be approximated by an integral in momentum with the following conversion factor: $$ dN = V\frac{d^{3}p}{(2\pi\hbar)^{3}} $$ Which is number of states allowed by quantum mechanics for momentum interval dp. We can than change the variable from momentum to energy by introducing density of states. Equation for ideal gas is an approximation in the limit of high temperature.

All of what I've done involves quantum mechanics, without which none of it would make sense. Is it true that Boltzmann invented statistical mechanics? If so, how did he come up with this theory without quantum mechanics as quantum mechanics would not be invented until after Boltzmann's death? I mean, without quantum mechanics, Boltzmann cannot apply his theory to a classically continuous system, can he? Since momentum is classically continuous, number of states within any non-zero momentum interval should be infinite instead of finite as predicted by quantum mechanics, then the above integral would not make sense.

I mean, his formulation and interpretation of microstate and entropy can still be applied to classically discrete system, but how could his theory be widely accepted if it cannot be applied to system as simple as ideal gas, let alone other continuous systems?

Am I wrong in assuming quantum mechanics must be used in calculation involving ideal gas? Thank you for all your assistance, I have an interest in historical development of physics and statistical mechanics seems way ahead of its time.

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  • $\begingroup$ I'd suggest that you consult the treatise "Lectures on Gas Theory" written by Boltzmann himself. $\endgroup$ – Maxis Jaisi Dec 17 '16 at 9:02
  • $\begingroup$ There's practice called regularization and very much used in various areas of physics - introduce cutoff to the source of infinities to make calculations sensible and then consider limit to the original model when it's possible. Boltzmann used it but as far as I know there's some evidence that he actually considered it as not just mathematical trick and there is some speculation that he could in some sense give that idea to Planck. I think there are more knowledgeable people who can give proper answer with more details. $\endgroup$ – OON Dec 17 '16 at 9:29
  • $\begingroup$ Like in his 1872 paper on H-theorem he first assumes that kinetic energies of molecules can only have finite number of possible energies with constant step and later considers the continuous limit. $\endgroup$ – OON Dec 17 '16 at 9:35
  • $\begingroup$ I suppose more likely is that he did not actually give that idea to Planck but Planck tried to use his computational technique and discovered that he gets the desired answer already with finite step. $\endgroup$ – OON Dec 17 '16 at 9:41

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