Microcanonical Origin of the Maximum Entropy Principle for Open Systems

TL;DR

This paper rigorously establishes that the maximum entropy principle for open systems derives from the microcanonical ensemble of the closed universe, linking two fundamental approaches to the Boltzmann distribution.

Contribution

It demonstrates the microcanonical origin of the maximum entropy principle for open systems, unifying the derivations of the Boltzmann distribution.

Findings

Shows the uniform distribution arises from maximum entropy in a closed system.

Derives the open system distribution by partial maximization over the heat bath.

Extends formalism to dynamical paths, suggesting a basis for path entropy maximization.

Abstract

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.