  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. Here we investigate
equilibrium DHU states 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 states of Brownian
particles can be controlled and achieved by imposing proper hyperuniformity
conditions on the potentials. Moreover, we find that thermal motions in these
systems tend to enhance hyperuniformity. We also analyze the evolution dynamics
of an initial density distribution (hyperuniform or non-hyperuniform) to the
desirable equilibrium DHU state 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.
