  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.
