What is the relation between chemical potential and the number of particles?

Question

Chemical potential is defined as the change in energy due to change in the number of particles in a system. Let we have a system which is defined by the following Hamiltonian: $$H = -t \sum_i^L c_i^\dagger c_{i+1} + V\sum_i^L n_i n_{i+1} -\mu \sum_i^L n_i$$ where $c^\dagger (c)$ are creation (annihilation) operators, $n$ is number operator, $t$ is hopping parameter, $V$ is nearest-neighbor interaction, $L$ is the total number of sites and $\mu$ is chemical potential.

What I understand by chemical potential is, if we set $μ=$some constant, then no matter how many sites ($L$) we add to the system, the number of particles will always be conserved. (Please correct me if I am wrong)

QUESTION:

What is the relation between chemical potential and the number of particles? i.e. if I set $μ = 10$ then how many particles are allowed in the system?

Answer 1

At zero temperature, to find the relation between $\mu$ and the particle number you have to know the ground-state energy $E_N$ of the system with $N$ particles, and then $\mu= E_{N+1}-E_N$ Consequently you have to solve your Hubbard model exactly before anything else.

Once you have done this, you can approximate the definition of $\mu$ as $$ \mu = \frac{\partial E}{\partial N} $$ (where $E=E(N)=E_N$) and from this obtain $N$ as a function of $\mu$ by means of Legendre transformation. Set $$ \Phi= E-\mu N $$ Then $$ \frac{\partial \Phi}{\partial \mu} = \frac{\partial E}{\partial\mu} -N-\mu \frac{\partial N}{\partial \mu} $$ $$ =\frac{\partial E}{\partial N}\frac{\partial N}{\partial\mu } - N- \frac{\partial E}{\partial N}\frac {\partial N}{\partial \mu} $$ $$ = -N $$

At finite temperature a thermodynamic system with a fixed chemical potential must be in a grand canonical ensemble and therefore free to exchange particles with a reservoir. Consequently the particle number is not fixed but instead its average $<N>$ is determined by $$ <N> = - \frac{\partial \Phi}{\partial \mu} $$ where now $$ \Phi \to E-TS-\mu N $$ is the thermodynamic grand potential.