A good conceptual starting point is to like the example that is mentioned at The Harvest of a Century by Siegmund Brandt (2008).
Consider a system with 2 particles and 3 states. Remember that:
Therefore, all the possible way to put those two particles in three states are for:
  • Maxwell-Boltzmann distribution: both A and B can go anywhere:
    State 1State 2State 3
    AB
    AB
    AB
    AB
    BA
    AB
    BA
    AB
    BA
  • Bose-Einstein statistics: because A and B are indistinguishable, there is now only 1 possibility for the states where A and B would be in different states.
    State 1State 2State 3
    AA
    AA
    AA
    AA
    AA
    AA
  • Fermi-Dirac statistics: now states with two particles in the same state are not possible anymore:
    State 1State 2State 3
    AA
    AA
    AA
Both Bose-Einstein statistics and Fermi-Dirac statistics tend to the Maxwell-Boltzmann distribution in the limit of either:
TODO: show on forumulas. TODO experimental data showing this. Please.....
Most applications of the Maxwell-Boltzmann distribution confirm the theory, but don't give a very direct proof of its curve.
Here we will try to gather some that do.

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