In the canonical ensemble, the total energy is not conserved. ( ). What are the fluctuations in the energy? The energy fluctuations are given by the root mean square deviation of the Hamiltonian from its average :
Therefore, the relative energy fluctuation is given by
Now consider what happens when the system is taken to be very large. In fact, we will define a formal limit called the thermodynamic limit, in which and such that N/V remains constant.
Since and E are both extensive variables, and ,
But would be exactly 0 in the microcanonical ensemble. Thus, in the thermodynamic limit, the canonical and microcanonical ensembles are equivalent, since the energy fluctuations become vanishingly small.