Noise, Diffusion, and Hyperuniformity

We consider driven many-particle models which have a phase transition between an active and an absorbing phase. Like previously studied models, we have particle conservation, but here we introduce an additional symmetry—when two particles interact, we give them stochastic kicks which conserve the center of mass. We find that the density fluctuations in the active phase decay in the fastest manner possible for a disordered isotropic system, and we present arguments that the large scale fluctuations are determined by a competition between a noise term which generates fluctuations, and a deterministic term which reduces them. Our results may be relevant to shear experiments and may further the understanding of hyperuniformity which occurs at the critical point.

Figure 1

(a) An illustration of the random organization model with center of mass conserving dynamics. (b) The one-dimensional COMCon model. In both cases blue particles are inactive and red are active.

Figure 2

2D RandOrg with c.m. conservation. The structure factor for $\rho >{\rho }_c$ scales as $k^2$ as $k\to 0$. Here, $L=400$, and the number of realizations is 50. At the critical point, ${\rho }_c\approx 0.487$, the usual scaling $k^{0.45}$ is seen.

Figure 3

2D Manna model with c.m. conservation for $\rho >{\rho }_c$. Here, $L=400$, and the number of realizations is 50. Here, ${\rho }_c\approx 1.7591$.

Figure 4

Real space fluctuations of the 2D c.m. Manna model and the COMCon model. In both cases the density fluctuation at high density is maximally hyperuniform, ${\sigma }^2(\ell )\propto {\ell }^{-d-1}$. Near the critical point of the Manna model (${\rho }_c\approx 1.7591$), ${\sigma }^2(\ell )\propto {\ell }^{-2.45}$, similar to the case where c.m. symmetry is absent.

Figure 5

The structure factor, $S(k)\propto k^2$, of COMCon for density $\rho >1$, corresponding to the active phase. The system size here is $L=1000$, and the number of realizations is 20 for each density.