Source of the Chemical Potential

Question

I know that the chemical potential is the energy required to add another particle to the system but what is it's origin on the microscopic level? Take for example the case of the ideal Fermi gas, the Hamiltonian:
$$H=sum_i frac{p_i^2}{2m}-mu hat N$$
clearly the first term is the kinetic energy but what interaction is causing the energy associated with the second term?

Quantum spaghettification · Answer

There is a misconception in this question about the nature of the chemical potential.  Although the chemical potential is often described as 'the energy associated with adding a single particle' this is has to be interpreted with extreme care. In this saying 'energy' refers to the 'mean energy' and 'adding a single particle' refers to 'changing the mean particle number by one'.

The change in 'mean energy' is not due to any interaction but rather the statistical effect that changing the mean particle number changes which microscopic states are more likely then others.

That said the presence of the chemical potential in the Grand partition function is equivalent to that of an interaction which acts on all particles equally. But this interaction does not have any physical meaning.

Source of the Chemical Potential

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