Emergence of equilibrium thermodynamic properties in quantum pure states. II. Analysis of a spin model system

TL;DR

This paper investigates how thermodynamic properties emerge in large quantum spin systems in pure states, showing that canonical behavior and temperature equivalence arise naturally in the random pure state ensemble without microcanonical assumptions.

Contribution

It demonstrates that thermodynamics and canonical typicality emerge in large quantum spin systems from the random pure state ensemble, without microcanonical assumptions, and proves temperature and entropy equivalences.

Findings

Thermodynamics is recovered in the large system limit from the RPSE.

Canonical statistics emerge without microcanonical assumptions.

Global and local temperatures are shown to be equivalent.

Abstract

A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the "global temperature," derived from the entropy equation of state, with the "local temperature" determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.