Quantum Microcanonical Entropy, Boltzmann's Equation, and the Second Law.

A recent proposal for a quantum entropy S univ Q for a pure state of a system-environment "universe" is developed to encompass a much more realistic temperature bath. Microcanonical entropy is formulated in the context of the idea of a quantum microcanonical shell. The fundamental relation that holds for the classical microcanonical ensemble - TΔ S univ = Δ F sys is tested for the quantum entropy Δ S univ Q in numerical simulations. It is found that there is "excess entropy production" Δ S x due to quantum time-energy uncertainty and spreading of states in the zero-order basis. The excess entropy production is found numerically to become small as the magnitude of the system-environment coupling nears zero, as one would hope for in the limit of the classical microcanonical ensemble. The quantum microcanonical ensemble and the new "universe entropy" thereby appear as well-founded concepts poised to serve as a point of departure for time-dependent processes in which excess entropy production is physically significant.