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 $q^{\varsigma }$ for $q\to 0$, with hyperuniformity exponent $\varsigma =4$. 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 $\ell$. Here we reconsider and extend Tomita's argument to address cases of active phase separation and of non-constant mobility, again finding $\varsigma =4$. We further show that dynamical noise of variance $D$ creates a transient $\varsigma =2$ regime for $\hat{q}\ll {\hat{q}}_{\ast }\sim \sqrt{D}{t}^{[1-(d+2)\nu ]/2}$, crossing over to $\varsigma =4$ at larger $\hat{q}$. Here, $\nu$ is the coarsening exponent, with $\ell \sim t^{\nu }$, and $\hat{q}\propto q\ell$ is the rescaled wavenumber. In diffusive coarsening, $\nu =1/3$, so the rescaled crossover wavevector ${\hat{q}}_{\ast }$ vanishes at large times when $d\ge 2$. The slowness of this decay suggests a natural explanation for experiments that observe a long-lived $\varsigma =2$ scaling in phase-separating active fluids (where noise is typically large). Conversely, in $d=1$, we demonstrate that with noise the $\varsigma =2$ regime survives as $t\to \infty$, with ${\hat{q}}_{\ast }\sim D^{5/6}$. (The structure factor is not then determined by the zero-temperature fixed point.) We confirm our analytical predictions by numerical simulations of active and passive continuum theories 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 of the order parameter field.

34 pages, 4 figures

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