Derivation and Improvements of the Quantum Canonical Ensemble from a Regularized Microcanonical Ensemble

TL;DR

This paper introduces a Gaussian regularization of the quantum microcanonical ensemble, providing a rigorous derivation of the canonical ensemble and explaining its effectiveness for small, isolated systems.

Contribution

It offers a novel derivation method linking microcanonical and canonical ensembles with rigorous bounds and asymptotic expansions, applicable to systems near the thermodynamic limit.

Findings

Derived a Gaussian microcanonical ensemble that explains the canonical ensemble's effectiveness.

Established direct parameter correspondence between microcanonical and canonical ensembles.

Provided asymptotic expansions for microcanonical corrections in near-thermodynamic systems.

Abstract

We develop a regularization of the quantum microcanonical ensemble, called a Gaussian ensemble, which can be used for derivation of the canonical ensemble from microcanonical principles. The derivation differs from the usual methods by giving an explanation for the, at the first sight unreasonable, effectiveness of the canonical ensemble when applied to certain small, isolated, systems. This method also allows a direct identification between the parameters of the microcanonical and the canonical ensemble and it yields simple indicators and rigorous bounds for the effectiveness of the approximation. Finally, we derive an asymptotic expansion of the microcanonical corrections to the canonical ensemble for those systems, which are near, but not quite, at the thermodynamical limit and show how and why the canonical ensemble can be applied also for systems with exponentially increasing density of states. The aim throughout the paper is to keep mathematical rigour intact while attempting to produce results both physically and practically interesting.