Functional Derivative Calculation

Question

Given the functional: $$ F[\phi] = \int_V \frac{k_B T}{a^3}\phi\ln(\phi) \ ds = \int_V I(\phi) ds $$ I want to find the functional derivative. I believe this would result in: $$ \frac{\delta F}{\delta \phi} = \frac{\partial I}{\partial \phi}=\frac{k_B T}{a^3}[\ln(\phi)+1]$$ However, the paper I am following along has only the first term. Is my calculation correct? Note, in this case I set the functional derivative equal to the partial derivative because the functional doesn't contain any higher derivatives - hence those partials vanish.

Answer 1

Your attempt points in the right direction, but note that the functional derivative is not the partial derivative as you're deriving with respect to a function and not a variable. Nonetheless, they are connected for certain functionals.

Take a compactly supported smooth function $\psi$, then by the definition of the functional derivative: \begin{align*} \int_V\frac{\delta F[\phi]}{\delta\phi}\psi\; ds \stackrel{!}{=}\left[\frac{\mathrm d}{\mathrm d\varepsilon}F[\phi+\varepsilon \psi]\right]_{\varepsilon=0} =\ldots =\int_V\frac{k_\mathrm{B}T}{a^3}(\ln(\phi)+1)\psi\mathrm ds, \end{align*} from which the result follows with the fundamental theorem of the calculus of variations. I think you can fill in the two missing steps, where you just have to put in your expression of the functional, for yourself.