Adopting the philosophy that the first variation of a functional (as defined e.g. here) is like the gradient of a function, is there a way to test whether a given function is the first variation of some functional? I am specifically interested in finding a functional $E$ such that $$\frac{\delta E}{\delta \rho} = \log(\rho \ast \phi_\varepsilon)$$ where $\phi_\varepsilon$ is the probability density function of the Gaussian distribution with mean $0$ and variance $\varepsilon$, and $\rho \ast \phi_\varepsilon$ denotes convolution of two real-valued functions on $\mathbb{R}$.
If it helps, I know that $$\frac{\delta E}{\delta \rho} = \phi_\varepsilon \ast \log(\rho \ast \phi_\varepsilon)$$ for $$E(\rho) = \int_\mathbb{R} \phi_\varepsilon \ast \rho \log(\rho\ast\phi_\varepsilon).$$
The motivation for this question is extending the entropy functional to a discrete measure $\rho$. I came across a very similar question, and from the second answer it seems that no such functional exists. But I am not sure if I understood that answer well enough to trust my calculation.
