  The canonical ensemble describes an open system in equilibrium with a heat
bath of fixed temperature. The probability distribution of such a system, the
Boltzmann distribution, is derived from the uniform probability distribution of
the closed universe consisting of the open system and the heat bath, by taking
the limit where the heat bath is much larger than the system of interest.
Alternatively, the Boltzmann distribution can be derived from the Maximum
Entropy Principle, where the Gibbs-Shannon entropy is maximized under the
constraint that the mean energy of the open system is fixed. To make the
connection between these two apparently distinct methods for deriving the
Boltzmann distribution, it is first shown that the uniform distribution for a
microcanonical distribution is obtained from the Maximum Entropy Principle
applied to a closed system. Then I show that the target function in the Maximum
Entropy Principle for the open system, is obtained by partial maximization of
Gibbs-Shannon entropy of the closed universe over the microstate probability
distributions of the heat bath. Thus, microcanonical origin of the Entropy
Maximization procedure for an open system, is established in a rigorous manner,
showing the equivalence between apparently two distinct approaches for deriving
the Boltzmann distribution. By extending the mathematical formalism to
dynamical paths, the result may also provide an alternative justification for
the principle of path entropy maximization as well.
