where N is the number of particles in the gas. This result is a specific manifestation of the ETH, in that it results in a prediction for the value of an observable in one energy eigenstate which is in agreement with the prediction derived from a microcanonical (or canonical) ensemble. Note that no averaging over initial states whatsoever has been performed, nor has anything resembling the H-theorem been invoked. Additionally, one can also derive the appropriate Bose–Einstein or Fermi–Dirac distributions, if one imposes the appropriate commutation relations for the particles comprising the gas.