Ergodic Optimization and Ground States: a brief Introduction

TL;DR

This paper provides a concise overview of Ergodic Optimization, highlighting its basic concepts, connections to Statistical Mechanics, and potential procedures for explicit solutions, aimed at newcomers.

Contribution

It offers a non-technical, schematic introduction to Ergodic Optimization, emphasizing its relevance and relationship with Statistical Mechanics without presenting new results.

Findings

Provides a global overview of Ergodic Optimization concepts.

Highlights connections between Ergodic Optimization and Statistical Mechanics.

Suggests procedures for obtaining explicit solutions.

Abstract

Our goal in this short note is to briefly and succinctly describe some basic concepts and properties of Ergodic Optimization for readers unfamiliar with the subject. We avoid technical issues in order to provide a global overview of this topic. We will not attempt to cover all of the many contributions of various authors, who have greatly enriched the theory with invaluable results. The author has made a personal selection of the topics to be addressed, keeping in mind two main objectives: to motivate the reasons for studying the subject, and to describe schematically and pictorially its relationship with relevant concepts and properties of Statistical Mechanics, which is one of the sources of inspiration for the theory. We will not present new results or detailed proofs. Some examples will be provided. We describe some procedures that may help in obtaining explicit solutions. We present some references that by no means aim to exhaust the bibliography on the subject, where possible, minimizing the number of references.