Consider some Markov process $X$ on a finite state space $E$ admitting a reversible measure $\mu$. The following two quantities are extremely useful in its study:
The relative entropy $$ H(\nu_t|\mu) = H(f_t) = -\mathbb{E}_\mu[f_t\log f_t]\geq 0, $$ where we assume for simplicity that the distribution $\nu_t$ of $X_t$ has density $f_t$ w.r.t. $\mu$.
The Dirichlet Form $$ D(f) := \langle f, (-L)f\rangle_\mu \geq 0, $$ where $L$ denotes the generator of the Markov process and $\langle\cdot,\cdot\rangle_\mu$ denotes the $L^2(\mu)$-inner product.
It is relatively easy to show (see e.g. Appendix 1, Theorem 9.2, of Scaling Limits of Interacting Particle Systems by Kipnis and Landim) that $$ \partial_t H(f_t) \leq -2D(\sqrt{f_t}) $$ which constitutes a very important statement as it allows us to estimate the rate of convergence of $X_t$ to its stationary state.
The problem that I have is that I cannot find any intuitive reason for this relationship to hold. The proof itself does not help as it uses a simple but uninformative inequality to prove it. I am wondering if there is a heuristic why this should hold true or if I am too optimistic...