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In Statistical thermodynamics Maxwell-Boltzmann statistics is considered a pre-quantum statistics. However in the mathematical treatment in all textbooks, and also in Wikipedia article, there is the concept of 'energy level' ($E_i$) involved in it. As far as I understand, energy level implies a quantization, which obviously is a quantum idea. How is this possible? What is it that I missed to understand?

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  • $\begingroup$ Can you mention the source where it is written that maxwell-Boltzmann statistics is pre-quantum statistics? $\endgroup$ Commented Oct 18, 2020 at 13:46

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Yes, Maxwell and Boltzmann produced their theories well before quantum ideas were dreamed of, and they were developed by Gibbs into the form we know today using purely classical ideas. This involves some quite deep assumptions about what macrostates are equally probably in an ensemble. Introducing quantum microstates makes it a lot easier to understand. So today it is taught that way round, and a good thing too.

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The energy levels used in Boltzmann distribution are normally ranges of energy, when we approximate discrete functions by continuous ones. They are not discrete energy levels in the quantum meaning.

For example: each gas molecule has an energy $E_k = \frac{1}{2}mv_k^2 = \frac{p_k^2}{2m}$.

The sum for all particles: $\sum_0^N{e^{-\beta E_k}} = \sum_0^N{e^{-\beta (\frac{p_k^2}{2m})}}$ can be approximated by: $\int_0^{\infty}{e^{-\beta (\frac{p^2}{2m})}dp}$, when N is very large, as is the case of the number of gas molecules in a container.

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