On a particle approximation to the Dean-Kawasaki type equation with logarithmic interactions

We consider a class of Dean-Kawasaki type equations on $\mathbb{T}$ with logarithmic repulsive interactions depending on the inverse temperature $\beta$ and a new spectral approximation to the noise part, which approximately features Otto's metric in $\mathbb{P}(\mathbb{T})$. Following the idea of intrinsic constructions of Brownian motions on the Wasserstein space, we construct a class of particle models whose fluctuating hydrodynamic limits, denoted as $p_t^\beta$, are solutions to the martingale problems of this class of equations. Specifically, we give a quantitative convergence rate of the particle approximation, which allows us to identify a unique limit distribution depending on $\beta$. As the inverse temperature rises, the regularizing effect of repulsive interactions becomes stronger. We prove that there exists three thresholds $0<\lambda_0\leq\lambda_1<\lambda_2$ depending on the noise such that, when $\beta>\lambda_0$, $p_t^\beta$ is a non-atomic measure process in $\mathbb{P}(\mathbb{T})$; when $\beta>\lambda_1$, $p_t^\beta$ is absolutely continuous with respect to Lebesgue measure almost surely; when $\beta>\lambda_2$, the expectation of the R\'enyi entropy of $p_t^\beta$ satisfies an exponential decay estimate.