Hyperuniformity of expected equilibrium density distributions of Brownian particles via designer external potentials

Disordered hyperuniformity (DHU) is a recently discovered novel state of many-body systems that is characterized by vanishing normalized infinite-wavelength density fluctuations similar to a perfect crystal, yet possesses an amorphous structure like a liquid or glass. Due to their unique structural characteristics, DHU materials are typically endowed with unusual physical properties, such as large isotropic photonic band gaps, optimal transport properties and superior mechanical properties, enabling a wide spectrum of novel applications. Here we investigate hyperuniformity of expected equilibrium distributions of Brownian particles induced by external potentials. In particular, we analytically derive sufficient conditions on the external potentials in order to achieve distinct classes of DHU density distributions of Brownian particles in thermal equilibrium, based on the stationary-state solutions of the corresponding Smoluchowski equation. We show for a wide spectrum of tight-binding potentials, the desirable DHU distributions of Brownian particles can be controlled and achieved by imposing proper hyperuniformity conditions on the potentials. We also analyze the evolution dynamics of an initial density distribution (hyperuniform or non-hyperuniform) to the desirable equilibrium DHU distribution determined by the prescribed external potentials, which is shown to be coupled with the full spectra of the force fields associated with the imposed potentials. We find that although the transient density distribution can rapidly develop local patterns reminiscent of those in the equilibrium distribution, which is governed by the fast dynamics induced by the external potential, the overall distribution is still modulated by the initial density fluctuations which are relaxed through slow diffusive dynamics. Our study has implications for the fabrication of designer DHU materials.