In this talk, we extend a previously introduced one-dimensional diffusion model on probability measures, defined via the rearranged stochastic heat equation, by penalizing the dynamics with an additional entropy-driven gradient-descent term. By means of a splitting argument, we prove that despite the opposite effects of rearrangement and entropy minimization, the resulting penalized stochastic heat equation is well defined. We study several properties of the associated dynamics and show, in particular, that solutions admit a density satisfying a corrected version of the Dean--Kawasaki equation. Moreover, smoothing properties established for the stochastic heat equation are also shown to persist.