# Is Boltzmann distribution contradicting with the fundamental assumption of statistical thermodynamics?

In equilibrium statistical physics the fundamental assumption of statistical thermodynamics states that the occupation of any microstate is equally probable (i.e. $p_i=1/\Omega, S=-k_B\sum p_i\,{\rm ln}\,p_i=k_B{\rm ln}\,\Omega$). But for isolated system in equilibrium we also have Boltzmann distribution which states $p_i=e^{-\beta E_i}/Z$, where $E_i$ are the allowed energy levels. So the two $p_i$ matches if and only if there is one single allowed energy level. How can we resolve this conflict?

With the Boltzmann distribution (AKA canonical ensemble) this assumption doesn't apply since we have knowledge about the system. In particular we know that if the system is put into contact with a thermodynamic heat bath of temperature $T$ (the same $T$ as in the $e^{-E/(kT)}$ distribution), the system will remain in statistical equilibrium (the distribution will not change). This property, of being in equilibrium with other systems of the same temperature, is special to the Boltzmann distribution and is what makes it so useful.