The Classical Limit of Maxwell-Boltzmann Distribution

[JohnnyGui]

I have been reading about the quantum effects that limit the Maxwell-Boltzmann Distribution under certain conditions which leads to the Bose-Einstein or Fermi-Dirac Distribution.
I have difficulty grasping the reasons why these quantum-effects occur only at certain conditions and why exactly they make the MB distibution invalid. I have some questions regarding this.

Several quantum-effects and conditions are pointed out that limit the MB distribution. These are:

1. Low temperature: From what I read, low temperature leads to "overpopulation" of energy levels by particles which makes these particles compete for states. What I don’t understand is why particles having the same state, or quantum state for that matter, should be a problem. Doesn’t each particle have its own quantum numbers that can independently be changed regardless of what quantum numbers other particles have? As long as the total energy is constant?

2. High density: What quantum-effect(s) occur at high density that makes the MB distribution inaccurate?

3. Indistinguishable particles: I’m not sure if this quantum-effect belongs either low temperature or high density. If it does belong to either of these conditions, what is the mechanism of low temperature or high density that makes particles all of a sudden indistinguishable?

4. Heisenberg Uncertainty Principle: Again, not sure if this quantum-effect belongs to one of the above reasons. Is this effect namely present in the case of particles with very low velocities which is the case at a low temperature? Why is it so important for the MB distribution to have an accurate measurement of the particles’ positions? Isn’t statistically calculating the population of the energylevels sufficient?

5. Degenerate energy levels: I don’t know why this limits the MB distribution since I have seen derivations of the MB statistics formula that takes degeneracy into account.

6. Spacing between energy levels at low temperture: I reckon the increase in spacing between the energy levels should not be a problem for the MB stastistics formula because MB stastics already considers energy levels as discrete?

Hope someone could clear (some of these) up for me.

 

[PeterDonis]

I have been reading about the quantum effects that limit the Maxwell-Boltzmann Distribution under certain conditions which leads to the Bose-Einstein or Fermi-Dirac Distribution.

You have it backwards. The quantum effects don't "limit the Maxwell-Boltzmann distribution" and make it into something else. The Bose-Einstein and Fermi-Dirac distributions are the fundamental ones; the Maxwell-Boltzmann distribution is an approximation that works reasonably well under certain conditions.

Several quantum-effects and conditions are pointed out

Where? Please give a specific reference.

 

[PeterDonis]

Note, btw, that even your thread title is incorrect. The Maxwell-Boltzmann distribution doesn't have a classical limit; it is a classical limit (i.e., it's what you get in the limit where quantum effects can be ignored).