Both Fermi-Dirac and Bose-Einstein statistics reduce to the Maxwell-Boltzmann statistics in the classical limit of very high temperatures or very low concentration. In the case of very high temperatures, I suppose that the particles should be distributed uniformly among it's accessible states, so is Maxwell-Boltzmann distribution a uniform distribution?

  • $\begingroup$ I'm supposing you don't mean "uniform" in the sense of $P(x,a,b)=1/(b-a)$ sense of the PDF? $\endgroup$ – Kyle Kanos Apr 30 '19 at 15:34
  • $\begingroup$ That was my confusion at start: uniform in the sense of having equal number of particles in each state. $\endgroup$ – RicardoP Apr 30 '19 at 15:46

The Maxwell-Boltzmann distribution at high temperatures is a uniform distribution. Each energy level with energy $E_i$ comes with a weight (neglecting normalization) $e^{-E_i/k_B T}$. If $T$ is very large, we can expand this in orders of $1/T$. The leading term, as with any exponential, is just $1$. Of course, you can do this expansion for any temperature, but it only makes sense to focus on the first term at large $T$.

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