On the controversy over the stochastic density functional equations

This letter aims at justifying the stochastic equations in terms of the number density variable, which are still controversial, via complementing Dean’s approach [Dean D S 1996 {\itshape J. Phys. A} {\bf 29} L613]. Our course is twofold: First, we demonstrate that standard manipulations straightforwardly transform the stochastic equation of density {\itshape operator}, derived by Dean, to the Fokker-Planck equation for the (c-number) density distribution functional $P(\{\rho \},t)$. Moreover, we verify the associated static solution of $P(\{\rho \},t)$ with the help of the conditional grand canonical partition function.