Quantum Thermodynamics, Entropy of the Universe, Free Energy, and the Second Law

TL;DR

This paper proposes an alternative definition of the universe's entropy consistent with the second law, and validates it through quantum simulations showing agreement with thermodynamic relations involving free energy.

Contribution

It introduces a new entropy concept for the universe that aligns with the second law and demonstrates its validity via quantum dynamics simulations.

Findings

New entropy definition aligns with second law in quantum systems.

Simulation results agree with thermodynamic relation involving free energy.

Observed deviations from microcanonical behavior in large quantum systems.

Abstract

We take the view that the standard von Neumann definition, in which the entropy $S^{vN}$ of a pure state is zero, is in evident conflict with the statement of the second law that the entropy of the universe $S_{univ}$ increases in spontaneous processes, $ \Delta S_{univ} > 0$. Here we seek an alternative entropy of the universe $S_{univ}$ that is in accord with the second law, in a spirit not dissimilar to von Neumann himself in lesser-known work. We perform simulations of time dependent dynamics for a previously developed \cite{polyadbath} model quantum system becoming entangled with a quantum environment. We test the new definition of the entropy of the system-environment "universe" against the standard thermodynamic relation $ \Delta F_{sys} = - T \Delta S_{univ} $, calculating system properties using the reduced density matrix and standard von Neumann entropy. Good agreement is obtained, showing the compatibility of an entropy for a pure state of a universe with the statement of the second law and the concept of free energy. Interesting deviation from microcanonical behavior within the zero order energy shell is observed in a context of effectively microcanonical behavior within the much larger total basis of the time dependent universe.