I understand that both Fermions and bosons have the chemical potential $\nu <0$ when it is T>0, but still behave classically, the fermions would increase its chemical potential at T=0, whereas the boson chemical potential decreases to zero. According to my understanding, chemical potential negative indicates the system is happy to gain the particles as the free energy change is negative, i.e. energetically favourable, whereas positive indicates energy is required to put a particle in.

My physical picture is as follows,

imagine a typical fermionic system, like delocalised electrons around Cu metals, since more and more particles would overlap to lower its total energy, (tind binding model), at high temperature, the system is happy to take in particle, whereas at low temperature, due to Pauli exclusion, due to the "repulsion" between same spin particles, one has to put in energy, hence $\nu$>0. Hence, one has to shine laser to excite atoms from other atoms, if one wants to remove the energy previously put in. Whereas, in the case of a boson, the particles initially at roughly high temperature to below critical temperature, happy to merge together to form a Bose einstein condensate, the particle would keep getting into the system, up to a certain point, the number saturates, i.e. $\nu$ = 0. i.e. adding or removing particles doesn't change energy of the condensate. I hope anyone can clarify my understanding. I think my picture is very vague and not really explaining the role of chemical potential.

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