Is Dirichlet energy related with entropy?

Question

Intuitively, I feel that Dirichlet energy is related with entropy. And entropy seems to be equivalent with some discrete form of Dirichlet energy.

Is this a nice intuition?

Is there something worth discussing here?

Answer 1

There is a loose relationship I would say. Consider the thermal diffusion you mentioned in your comment, i.e. the heat equation on $\mathbb{T}^d$ (for simplicity), \begin{align} \partial_t \rho = \Delta \rho\\ \rho(0)=\rho_0 \, , \end{align} where $\rho_0$ is a Borel probability measure on $\mathbb{T}^d$. There are two ways of interpreting this equation: it is gradient flow of the Dirichlet energy with respect to the $L^2$ metric or it is gradient flow of the entropy with respect to the $2$-Wasserstein metric on the space of probability measures (see this paper by Jordan, Kinderlehrer, and Otto: https://francahoffmann.files.wordpress.com/2018/07/302ca7465ae824f3d2d629bfeaacfb56b4b8.pdf ). Both quantities measure, in some sense, how far you are from the constant and they are both zero when $\rho$ is constant.