TITLE: Mathematical difference between entropy and energy
QUESTION [13 upvotes]: I have a rather soft question. Let's assume that we consider the heat equation posed in $S^1$:
$$
\partial_t u=\partial_x^2u.
$$
It is well known that if we define the functionals
$$
H(t)=\int_{-\pi}^\pi u\log(u)-u+1dx,
$$
$$
E(t)=\int_{-\pi}^\pi u^2dx,
$$
they decay. As far as I know, in the literature, a functional similar to $H$ is called entropy while a functional like $E$ is called energy. I understand the physical reasons behind that. 
The question is: are there some subtle mathematical differences between these two Lyapunov functionals that we emphasize by using different names entropy/energy? Or, on the other hand, it's just a reminder from the physical origin of both terms?

REPLY [5 votes]: It is probably worth mentioning that the the heat equation is inherently linked to energy and entropy in two ways:

The heat equation is the gradient flow of the energy in the Hilbert space $L^2$ (classical).
The heat equation is the "gradient flow of the entropy in Wasserstein space $W_2$" (Jordan-Kinderlehrer-Otto, also Ambrosio-Gigli-Savare).