I was searching for the reason that why at higher energies, the FD and BE distributions behave as MB distribution. In Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles by Eisberg and Resnick, 2nd edition, page 386 it is stated that:

At high energies ($\epsilon>>kT$) where the probable number of particles per quantum state for the classical distribution is much less than one, the quantum distributions merge with the classical distribution. That is $n_{Fermi} \approx n_{Boltz} \approx n_{Bose}$, if $n_{Boltz}<<1$.

My question is why in the first place, at higher energies, the number of particles in the MB distribution for per state will be less than $1$? As per my understanding, the three distributions should merge to the classical one only when one state could occupy more than one particle (violating the exclusion principle) and they become distinguishable. The math is alright, but I want to know the physics behind this phenomenon.

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    $\begingroup$ Consider what happens when the probability of occupation of a given state is so low that two particles will never try to occupy it at the same time.. So they no longer care if they should be arguing (Fermi-Dirac) or not (Bose-Einstein) since it just doesn't matter. $\endgroup$
    – Jon Custer
    Nov 8, 2021 at 19:55

1 Answer 1


Intuitively, the difference between the statistics of identical particles (bosons or fermions) and non-identical particles (what you think of classically as being a particle) should become visible when there are more particles than available states, or only slightly more particles than available states. Then, by the pigeonhole principle, particles are going to tend to try to overlap if they can, and it will become obvious if they "prefer" to overlap, if they "can't" overlap, or if it does not matter. Conversely, when there are many more states than particles to fill them, it will not matter very much whether particles are identical or not. It will be very rare for two particles to stumble upon the same state by chance, and so there's never a need to use the information about whether particles can or can't fill the same state.

There tends to be a small number of available states when the system is cold. This is because the thermal energy available to the particles is small, so they will tend to "fall to the bottom of the potential" (if they are bosons) or "stack up to the Fermi energy" (if they are fermions). There tends to be a large number of available states when the system is hot. There is a huge thermal energy, so the particles can "explore the space of states" without getting stuck in one place.


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