Enhanced hyperuniformity from random reorganization

Diffusion relaxes density fluctuations toward a uniform random state whose variance in regions of volume v = ℓ^d scales as σ²_ρ ≡ 〈ρ²(ℓ)〉 - 〈ρ〉² ∼ ℓ^-d. Systems whose fluctuations decay faster, σ²_ρ ∼ ℓ^-λ with d < λ ≤ d + 1, are called hyperuniform. The larger λ, the more uniform, with systems like crystals achieving the maximum value: λ = d + 1. Although finite temperature equilibrium dynamics will not yield hyperuniform states, driven, nonequilibrium dynamics may. Such is the case, for example, in a simple model where overlapping particles are each given a small random displacement. Above a critical particle density ρ_c, the system evolves forever, never finding a configuration where no particles overlap. Below ρ_c, however, it eventually finds such a state, and stops evolving. This "absorbing state" is hyperuniform up to a length scale ξ, which diverges at ρ_c. An important question is whether hyperuniformity survives noise and thermal fluctuations. We find that hyperuniformity of the absorbing state is not only robust against noise, diffusion, or activity, but that such perturbations reduce fluctuations toward their limiting behavior, λ → d + 1, a uniformity similar to random close packing and early universe fluctuations, but with arbitrary controllable density.