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