The entropy, from Clausius to functional inequalities

TL;DR

This paper explores the concept of entropy from its origins in Clausius's thermodynamics to its applications in gradient flows and functional inequalities, highlighting its fundamental role across various mathematical fields.

Contribution

It provides a comprehensive overview of how entropy links thermodynamics, gradient flows, and functional inequalities, emphasizing its broad mathematical significance.

Findings

Entropy underpins key functional inequalities like Sobolev inequality.

Gradient flows are fundamental in understanding entropy's role in analysis.

The paper connects classical thermodynamics to modern mathematical theories.

Abstract

In this document we are interested in entropy. Entropy is multiple, the idea is to describe the definition proposed by the physicist Clausius. Indeed, Clausius exposes in 1865 the second principle of thermodynamics and also proposes the concept of entropy. Instead of simply defining a functional, central point for the development of the second principle, it will in fact define a concept sufficiently general to be used in many fields of mathematics.In these few pages, I wish to show the role played by entropy in the field of gradient flows and functional inequalities. Starting from the definition of Clausius in 1865, I will try to explain how fundamental functional inequalities like Sobolev inequality, key point in analysis, are natural inked with a particular entropy. This path allows me to give an overview of the use of gradient flows in finite or infinite dimension, of Bakry-Emery theory and more recently of Otto's calculus.