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Disordered hyperuniformity (DHU) is a recently discovered novel state ofmany-body systems that is characterized by vanishing normalizedinfinite-wavelength density fluctuations similar to a perfect crystal, yetpossesses an amorphous structure like a liquid or glass. Here we investigateequilibrium DHU states of Brownian particles induced by external potentials. Inparticular, we analytically derive sufficient conditions on the externalpotentials in order to achieve distinct classes of DHU density distributions ofBrownian particles in thermal equilibrium, based on the stationary-statesolutions of the corresponding Smoluchowski equation. We show for a widespectrum of tight-binding potentials, the desirable DHU states of Brownianparticles can be controlled and achieved by imposing proper hyperuniformityconditions on the potentials. Moreover, we find that thermal motions in thesesystems tend to enhance hyperuniformity. We also analyze the evolution dynamicsof an initial density distribution (hyperuniform or non-hyperuniform) to thedesirable equilibrium DHU state determined by the prescribed externalpotentials, which is shown to be coupled with the full spectra of the forcefields associated with the imposed potentials. We find that although thetransient density distribution can rapidly develop local patterns reminiscentof those in the equilibrium distribution, which is governed by the fastdynamics induced by the external potential, the overall distribution is stillmodulated by the initial density fluctuations which are relaxed through slowdiffusive dynamics. Our study has implications for the fabrication of designerDHU materials.

Hyperuniform Density Distributions of Brownian Particles via Designer External Potentials