An Introduction to Lieb's Simplified approach to the Bose gas

TL;DR

This book introduces Lieb's Simplified approach to the Bose gas, a set of effective equations that accurately predict ground state properties and correlations across densities, offering insights different from traditional theories.

Contribution

It provides a comprehensive introduction to Lieb's Simplified approach, including derivations, analytical and numerical results, and open problems, with minimal background required.

Findings

Simple equation predicts ground state energy accurately in all regimes

Predictions for condensate fraction and correlations match Bogolyubov theory at low density

Big equation yields numerically accurate results across densities

Abstract

This is a book about Lieb's Simplified approach to the Bose gas, which is a family of effective single-particle equations to study the ground state of many-body systems of interacting Bosons. It was introduced by Lieb in 1963, and recently found to have some rather intriguing properties. One of the equations of the approach, called the Simple equation, has been proved to make a prediction for the ground state energy that is asymptotically accurate both in the low- and the high-density regimes. Its predictions for the condensate fraction, two-point correlation function, and momentum distribution also agree with those of Bogolyubov theory at low density, despite the fact that it is based on ideas that are very different from those of Bogolyubov theory. In addition, another equation of the approach called the Big equation has been found to yield numerically accurate results for these observables over the entire range of densities for certain interaction potentials. This book is an introduction to Lieb's Simplified approach, and little background knowledge is assumed. We begin with a discussion of Bose gases and quantum statistical mechanics, and the notion of Bose-Einstein condensation, which is one of the main motivations for the approach. We then move on to an abridged bibliographical overview on known theorems and conjectures about Bose gases in the thermodynamic limit. Next, we introduce Lieb's Simplified approach, and its derivation from the many-body problem. We then give an overview of results, both analytical and numerical, on the predictions of the approach. We then conclude with a list of open problems.