Ordered phases in coupled nonequilibrium systems: Static properties

We study a coupled driven system in which two species of particles are advected by a fluctuating potential energy landscape. While the particles follow the potential gradient, each species affects the local shape of the landscape in different ways. As a result of this two-way coupling between the landscape and the particles, the system shows interesting new phases, characterized by different sorts of long-ranged order in the particles and in the landscape. In all these ordered phases, the two particle species phase separate completely from each other, but the underlying landscape may either show complete ordering, with macroscopic regions with distinct average slopes, or may show coexistence of ordered and disordered regions, depending on the differential nature of effect produced by the particle species on the landscape. We discuss several aspects of static properties of these phases in this paper, and we discuss the dynamics of these phases in the sequel.

Figure 1

We show typical configurations of our system in each of the phases. $h\left(x\right)$ denotes height of the landscape at a position $x$. The thick (red) lines represent the region occupied by the $H$ particles, while remaining parts (blue) are occupied by $L$ particles. (a) A typical configuration in the SPS phase. The clean V-shape of the valley indicates complete phase separation between the up slopes and down slopes in the landscape. (b) A representative configuration in the IPS phase. The landscape occupied by the $H$ particles shows pure domains, while the rest has a linear profile with a gradient $1/N$. (c) A typical configuration in the FPS phase. Unlike SPS or IPS phases, the landscape does not have a compact domain of up-slope and down-slope bonds. Although there is a macroscopic valley, the finite fraction of down slopes (up slopes) present in majority up slope (down slope) domain makes it have a smaller slope. Also, the valley bottom is often rough and more than one minimum can be present. (d) In the FDPO phase, the order in the surface is completely lost. However, the particles congregate into several macroscopic clusters. (e) In the disordered phase, both the particles and the landscape are devoid of any order.

Figure 2

Phase diagram in the $R-R^{\prime }$ plane, where $R=\frac{E-b}{E+b}$ and $R^{\prime }=\frac{E+b^{\prime }}{E-b^{\prime }}$ with model parameters $E$, $b$, and $b^{\prime }$ defined in Eq. (2) as part of model description. Here, $R>1$ ($R^{\prime }<1$) indicates a downward bias imparted by $H$ ($L$) particles on the landscape, and $R^{\prime }>1$ indicates that $L$ particles push the landscape upward. For $1<R^{\prime }<\infty$, one has the SPS phase. In this regime, detailed balance in satisfied in the system on the straight line $R=q^2{R}^{\prime }$, where $q=(D-a)/(D+a)$. The LR model is shown by a solid circle on this line. The dashed lines shown in the diagram correspond to the IPS phase. The dotted region corresponds to the FPS phase ($R^{\prime }<R<1$), while the white region corresponds to the disordered phase ($R<R^{\prime }<1$). Disordered phase is also seen when $R^{\prime }<1$ and $R=1$. For $R=R^{\prime }<1$ FDPO phase is observed.

Figure 3

Schematic representation of different transitions that are allowed to occur in the system. $\mathrm{\Delta }{E}^{\prime }s$ denote the energy costs involved in the transitions, as per the Hamiltonian in Eq. (8). Solid (empty) circles represent $H$ ($L$) particles.

Figure 4

Temperature variation of $s_{\rho }$ and $s_m$ for three different values of $N$ [plots (a) and (c)]. (b) and (d) show the cumulants $f_{\rho }$ and $f_m$ for three $N$ values. We have used $\lambda =0.2$ here. We obtain the best linear fits to the data points for each value of $N$. From the point of intersection of the straight lines, ${\beta }_c$ is estimated to be $15.65\pm 0.001$ for $f_{\rho }$ and $15.76\pm 0.005$ for $f_m$, which is close to theoretical prediction $5\pi$. The data shown here have been averaged over at least ${10}^8$ histories.

Figure 5

Breakdown of Kolmogorov loop condition in the IPS phase. Starting from the first configuration, the system passes through a sequence of configurations and comes back to the starting configuration again, but the ratio $Q=\frac{W(1\to 2)W(2\to 3)W(3\to 4)W(4\to 5)}{W(2\to 1)W(3\to 2)W(4\to 3)W(5\to 4)}\ne 1$. This shows that the system does not obey detailed balance.

Figure 6

We measure the density of up slopes $S^{+}(k,N)$ as a function of the scaled distance $k/N$, where $k$ is the distance measured from the position of the single $H$ particle in a system of size $N$. The profile decreases linearly with a gradient $\sim 1/N$ as predicted by our mean field analysis. The data shown here have been averaged over at least ${10}^4$ initial histories.

Figure 7

(a) Away from the valley bottom, $S^{+}(r,N)$ saturates at a value $m$, which depends on $b$ and $b^{\prime }$. For $b^{\prime }=-0.3$, we vary $b$ and plot $m$, which matches well with the mean-field result. Discrete points are from simulation data and the continuous line shows mean-field solution of Eq. (30). We have used $N=1024$ here. (b) $S^{+}(r,N)$ changes from the value $(1-m)$ to $m$ across the domain boundary of width $\sqrt{N}$.

Figure 8

We plot $S^{+}(r,N)$ after subtracting the disordered phase value $1/2$ against $(r-N/4)$ for two different system sizes with $b=0.3,b^{\prime }=-0.2$. The decay to a disordered phase occurs algebraically with an exponent $1/2$ (inset). The data shown here have been averaged over at least ${10}^6$ initial configurations.

Figure 9

Representative plots for diamond- and wedge-type configurations in the IPS phase in two dimensions on a $64\times 64$ square lattice. In (a) and (c), $H\left(L\right)$ clusters are shown in red (blue) while (b) and (d) show the equal-height contour plots.