The Hubbard model is often expressed as $$H=-J\sum\limits_{<i,j>} \sum_\sigma c_{i,\sigma}^{\dagger}c_{j,\sigma} +h.c.+U\sum\limits_{i} c_{i,\uparrow}^{\dagger} c_{i,\downarrow}^{\dagger} c_{i,\downarrow} c_{i,\uparrow} -\mu\sum\limits_i n_i~~~~~~~~~(1)$$ My question is now about the signs. As it appears, the factors $J$, $U$ and $\mu$ are supposed to be positive. Therefore it appears like the hopping term decreases the energy of the system and also more particles (larger $\sum_i n_i$) seem to decrease the energy of the system. The $U$ term increases the energy, if two particles with different spin occupy the same site.
Why should more particles decrease the energy ($\mu>0$)? As I understand it, the chemical potential is the energy needed, to increase the particle number by 1. Here it seems like the chemical potential acts like it lowers the energy, if particles are added.