# Help on precise definition of chemical potential T->0 FD Distribution

1. Aug 9, 2011

### pam d

Hello everyone, first post here.

In trying to decipher the meaning of chemical potential, I feel as if at least in the context of the Fermi-Dirac Distribution I have almost nailed it down. As I understand it, the chemical potential is the change in energy associated with the addition of one particle to an otherwise closed system of N particles. Looking at a system of particles obeying Fermi-Dirac statistics as the temperature becomes arbitrarily close to zero, I have read several times that the chemical potential is given by the Fermi energy.

Now I think I understand the Fermi energy, it is just the highest energy occupied by a fermion as T -> 0. Assuming no degeneracy, I am confused as to why the addition of one particle would add energy equal to the Fermi energy. If this state is already occupied, wouldn't the particle be forced to occupy the one state with energy above the Fermi energy? Would this not force the chemical potential for N particles to then be the Fermi energy of N+1 particles rather than that of N particles?

I often find in physics problems that when I become stuck on an issue like this one, I am making a fundamental error that requires outside guidance.

Thanks!

2. Aug 12, 2011

### PRB147

It is just a statistical problem, not just a 1+1=2 problem.

3. Aug 13, 2011

### pam d

Understood perb, which means I guess I'm not looking for a 1+1=2 response! Seems like you might have some kind of knowledge on the subject though, so I'll bite. I'll put away my arithmetic tables, but what next?

4. Aug 19, 2011

### PRB147

I mention 1+1=2 doesn't mean that your question is a simple question.
I think that the chemical potential mentioned here is referred to metal, not insulator.