Large compact clusters and fast dynamics in coupled nonequilibrium systems

We demonstrate particle clustering on macroscopic scales in a coupled nonequilibrium system where two species of particles are advected by a fluctuating landscape and modify the landscape in the process. The phase diagram generated by varying the particle-landscape coupling, valid for all particle densities and in both one and two dimensions, shows novel nonequilibrium phases. While particle species are completely phase separated, the landscape develops macroscopically ordered regions coexisting with a disordered region, resulting in coarsening and steady state dynamics on time scales which grow algebraically with size, not seen earlier in systems with pure domains.

Figure 1

Schematic presentation of different phases in one and two dimensions. (a) shows the coarsening mechanism in one dimension for $b^{\prime }\le 0$. $H$ ($L$) particles are shown by solid (open) circles. (b) and (c) show typical configurations in two dimensions. In (b) the $H$ ($L$) cluster is shown in red (blue) and (c) shows the equal height contours.

Figure 2

Phase diagram in the $b-b^{\prime }$ plane. For $b>0$ and $b^{\prime }>0$, the system shows SPS. The dotted horizontal and vertical lines are related to each other via interchange of the two particle species. On these lines the system is in IPS phase. The striped region ($-b<b^{\prime }<0$) represents the FPS phase. The $b=-b^{\prime }$ line corresponds to the FDPO phase. The dotted region ($b^{\prime }<-b$) corresponds to the disordered phase.

Figure 3

Occurrence of three regimes in the steady state dynamics of the deepest point of the valley. Main plot: Mean squared displacement of the deepest point of the valley as a function of time. The displacement shows an initial diffusive growth, followed by a plateau, and finally another diffusive regime at large time. Bottom inset: Small time diffusivity $D_1\sim 1/N$. Top inset: For large time, the diffusivity $D_2\sim e^{-\alpha N}$, with $\alpha \simeq 0.26$. These data correspond to $b^{\prime }=0,b=E,a=D$ and have been averaged over 5000 steady state configurations.

Figure 4

Scaling of particle density correlation in the coarsening phase in both one and two dimensions. The equal time density correlation for the particles $C(r,t)$ shows a collapse when $r$ is scaled by $\mathcal{L}\left(t\right)\sim t^{1/z}$. (a) and (c) show scaled data for $b=0.5,b^{\prime }=0$ in one and two dimensions, respectively. Here we find $z\simeq 2$. (b) and (d) show similar plots for $b=0.3,b^{\prime }=-0.2$. Here, in one dimension, $z\simeq 2$ and in two dimensions $z\simeq 2.6$. We have used $N=1024$ (a), $N=16384$ (b), and $N=256\times 256$ [(c) and (d)] here.