# Maxwell-Boltzmann Statistics is equivalent to a uniform distribution among acessible states?

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?

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