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What is the distribution statistics for the mean occupancy of a many-body state?

How can I show that this reduces to the single-particle Bose-Einstein or Fermi-Dirac ones when interactions tend to 0?

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It really depends on the system and the nature of interactions amongst the particles.

For example, in a free fermionic system if the interactions are weak enough and repulsive, then the only effect of interactions is a change in the relation between the energy and momentum of free particles. Usually the mass of particles change and the dispersion relation becomes something like $E(p)=\frac{p^2}{2m^*}$ instead of the original relation $E(p)=\frac{p^2}{2m}$. $m^*$ is called effective mass or renormalized mass and usually $m^*>m$ since it is harder for particles to move in a crowd than when they can move through each other (putting Pauli's exclusion principle aside). Therefore in this case the distribution function is Fermi-Dirac but with the new energy formula.

The assumption of repulsion among particles was crucial in the previous paragraph. It is well known that even a small net attractive interaction in a fermionic system makes the system a superconductor at sufficiently low energies which is a completely different state of matter from a free gas or metal.

In summary, interactions can have huge effects on the behvaior of many-body systems, therefore it is not possible to give a general statement about their effects on any of the properties of these systems.

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