The potential energy of a system of charged particles, characterized by particle charge \(e\) and number density \(n(\boldsymbol{r})\), is given by

\[
U=\frac{e^{2}}{2} \iint \frac{n(\boldsymbol{r}) n\left(\boldsymbol{r}^{\prime}ight)}{\left|\boldsymbol{r}-\boldsymbol{r}^{\prime}ight|} d \boldsymbol{r} d \boldsymbol{r}^{\prime}+e \int n(\boldsymbol{r}) \phi_{\mathrm{ext}}(\boldsymbol{r}) d \boldsymbol{r}
\]

where \(\phi_{\mathrm{ext}}(\boldsymbol{r})\) is the potential of an external electric field. Assume that the entropy of the system, apart from an additive constant, is given by the formula

\[
S=-k \int n(\boldsymbol{r}) \ln n(\boldsymbol{r}) d \boldsymbol{r}
\]

compare to formula (3.3.13). Using these expressions, derive the equilibrium equations satisfied by the number density \(n(\boldsymbol{r})\) and the total potential \(\phi(\boldsymbol{r})\), the latter being

\[
\phi_{\mathrm{ext}}(\boldsymbol{r})+e \int \frac{n\left(\boldsymbol{r}^{\prime}ight)}{\left|\boldsymbol{r}-\boldsymbol{r}^{\prime}ight|} d \boldsymbol{r}^{\prime}
\]