Hyperuniformity in phase ordering: the roles of activity, noise, and non-constant mobility

Abstract

Hyperuniformity emerges generically in the coarsening regime of phase-separating fluids. Numerical studies of active and passive systems have shown that the structure factor S(q) behaves as q ς for q → 0, with hyperuniformity exponent ς = 4. For passive systems, this result was explained in 1991 by a qualitative scaling analysis of Tomita, exploiting isotropy at scales much larger than the coarsening length. Here we reconsider and extend Tomita’s argument to address cases of active phase separation and of non-constant mobility, again finding ς = 4. We further show that dynamical noise of variance D creates a transient ς = 2 regime for q ^ ≪ q ^ ∗ ∼ D t [ 1 − ( d + 2 ) ν ] / 2 , crossing over to ς = 4 at larger q ^ . Here, ν is the coarsening exponent for the domain size ℓ , such that ℓ ( t ) ∼ t ν , and q ^ ∝ q ℓ is the rescaled wavenumber. In diffusive coarsening ν = 1 / 3 , so the rescaled crossover wavevector q ^ ∗ vanishes at large times when d ⩾ 2 . The slowness of this decay suggests a natural explanation for experiments that observe a long-lived ς = 2 scaling in phase-separating active fluids (where noise is typically large). Conversely, in d = 1, we demonstrate that with noise the ς = 2 regime survives as t → ∞ , with q ^ ∗ ∼ D 5 / 6 . (The structure factor is not then determined by the zero-temperature fixed point.) We confirm our analytical predictions by numerical simulations of continuum theories for active and passive phase separation in the deterministic case and of Model B for the stochastic case. We also compare them with related findings for a system near an absorbing-state transition rather than undergoing phase separation. A central role is played throughout by the presence or absence of a conservation law for the centre of mass position R of the order parameter field.