In this paper we consider an entirely new-previously unstudied to the best of our knowledge-type of density fluctuations stochastic partial differential equation with a singular coefficient involving the inverse of a probability density. The equation was recently introduced by Schieber [36] while working on a new polymer molecular dynamics approach that pertains to the generally called polymer reptation (aka tube) theory. The corresponding probability density is the solution of an evolution equation (a stochastic transport) defined on a dynamical one-dimensional subspace. A peculiarity of the here studied equation is its very singular pattern, even though it exhibits a well-posed structure. As a first step towards furthering the understanding of this new class of equations we prove the existence and uniqueness of solutions under suitable conditions. Moreover, capitalizing on the assumed gradient structure, we prove that the existing solutions converge to an equilibrium density.