Does the entropy of ideal fermi gas go to zero , in accordance with third law of thermodynamics?
Consider a system of three fermions in a 3D box. The first fermion goes to the ground state of the box. Second fermion and the third will go to the first excited state as the first excited state of the box in triply degenerate.
This will be the ground state of the system, but the entropy of this ground state is not zero because two fermions can be distributed among three states in three ways.From the Gibbs definition of entropy, we have three micro states for the ground state of this system and hence entropy $=k_B\ln3$.
If we had $N$ fermions and we are adding a new fermion to the system this fermion can occupy any state on the surface of the fermi sphere. In this case also the ground state is degenerate and hence the entropy is non-zero.
The chemical potential is defined as the change in internal energy w.r.t change in particle number while keeping the volume and entropy constant.
In the fermi gas case we cannot add a particle to the system while keeping entropy constant. So how can we conclude that the fermi energy is the chemical potential at $T=0$?