Ordered phases in coupled nonequilibrium systems: Dynamic properties

We study the dynamical properties of the ordered phases obtained in a coupled nonequilibrium system describing advection of two species of particles by a stochastically evolving landscape. The local dynamics of the landscape also gets affected by the particles. In a companion paper we have presented static properties of different phases that arise as the two-way coupling parameters are varied. In this paper we discuss the dynamics. We show that in the ordered phases macroscopic particle clusters move over an ergodic time scale growing exponentially with system size but the ordered landscape shows dynamics over a faster time scale growing as a power of system size. We present a scaling ansatz that describes several dynamical correlation functions of the landscape measured in steady state.

Figure 1

Schematic description of various ordered phases. We show the average height profile of the landscape; $H$ ($L$) particles are shown by solid (empty) circles. In the SPS phase, due to complete phase separation between up-slope and down-slope bonds, average height grows linearly from the deepest point, with a slope 1. In the IPS phase, the height profile remains similar to SPS till the edge of the $H$ cluster and then it gradually flattens out. In the FPS phase, owing to partial phase separation between up-slope and down-slope bonds in the $H$ region, the height grows linearly with a slope smaller than unity, before flattening out in the $L$ region.

Figure 2

The different transitions in the model and the rates for each of them. (a) and (b) show the rates of particle movements. An $H$($L)$ particle slides down (up) with a rate $D+a$, while the reverse moves occur with a rate $D-a$, where $D>a>0$. (b) and (c) show the rates at which the flipping of local hills and valleys occupied by, respectively, $H$ and $L$ particles takes place. A local hill (valley) occupied by an $H$ flips with a rate $E+b(E-b)$, where $E>b>0$. A local hill (valley), occupied with an $L$ on the other hand, can flip with rate $E-b^{\prime }(E+b^{\prime })$ where the parameter $b^{\prime }$ can be either zero, or positive or negative such that $E>\mid b^{\prime }\mid$.

Figure 3

The deepest point $V$ of the valley moves a distance $x$ from the center of mass $C$ of the $H$ cluster. This causes a height asymmetry $\mathrm{\Delta }h$.

Figure 4

Scaling collapse of dynamical correlation functions in IPS phase. The scaling argument is $t/N^z$ with dynamical exponent $z=2$. For small argument, the scaling function grows as a power law with an exponent $\alpha$. (a) For ${\sigma }_0^2\left(t\right)$ we find $\gamma =1,\alpha =1$. (b) Scaling collapse of ${\mathrm{\Delta }}^2\left(t\right)$ shows $\gamma =1,\alpha =1$. (c) For ${\delta }_{\mathrm{CM}}^2\left(t\right)$ best collapse is obtained for $\gamma =0.94$ and $\alpha =0.82$. The maximum error bar in these exponent values are $\pm 0.02$. All data are averaged over at least ${10}^4$ initial histories.

Figure 5

Motion of $V$ in IPS phase as an Ornstein-Uhlenbech process. (a) Velocity of $V$, when at a distance $x$ away from $C$, increases linearly with $x$. (b) $C$ experiences a restoring bias, when at a height $\delta h$ above $V$ and the bias increases linearly with $\delta h$. The data shown for three different system sizes have been averaged over at least ${10}^4$ initial histories.

Figure 6

Scaling collapse of dynamical correlation functions in FPS phase. In this case the scaling argument is $t/N^{3/2}$. For small argument the scaling function grows with a power $\alpha$ and for large argument saturates to a constant value. (a) Data for ${\sigma }_0^2\left(t\right)$ give $\gamma =0.92$ and $\alpha =0.5$. (b) For ${\mathrm{\Delta }}^2\left(t\right)$ we find $\gamma =1$ and $\alpha =0.67$. (c) For ${\delta }_{\mathrm{CM}}^2\left(t\right)$ best collapse is obtained for $\gamma =0.83$ and $\alpha =0.31$. The maximum error bar is $\pm 0.02$. All these data have been averaged over at least ${10}^4$ steady-state configurations.

Figure 7

We explicitly illustrate that it is possible for $V$ in an FPS phase to move without causing any change in the height difference between the edges of the $H$ cluster. Panels (a), (b), (c), and (d) show time-evolution of a local configuration in the FPS phase (also see the average structure of the phase in Fig. 1). We find that the finite fraction of impurities present within the majority up(down)-slope domains causes the deepest point to shift without causing any change in the heights of the right ($i_R$) and left ($i_L$) edges of the $H$ cluster.

Figure 8

Velocity of $V$ when at a height $\delta h$ below $C$ has a nonlinear dependence on $\delta h$ in the FPS phase. This motion cannot be described as an Ornstein-Uhlenbeck process, unlike IPS phase. The data shown for three different system sizes have been averaged over at least ${10}^4$ histories.

Figure 9

The correspondence between the $L$ region in the IPS phase and an open-chain SEP. (a) Scaled auto-correlation function of an up-slope bond at a distance $r+L/4$ from $C$, where the average density of up slopes is denoted by $S^{+\left(r\right)}$. We plot $a_r\left(t\right)=\frac{\sqrt{2\pi }}{S^{+\left(r\right)}[1-S^{+\left(r\right)}]}A(r,t)$ for three different $r$ and show that it decays as $t^{-1/2}$. The system size used is 2048. The data shown here have been averaged over at least ${10}^6$ initial histories. (b) Tagged mean-squared displacement of an up-slope bond at a distance $r+L/4$ from $C$. We plot ${\sigma }_r^2\left(t\right)=\sqrt{\frac{\pi }{2}}\frac{S^{+\left(r\right)}}{1-S^{\left(r\right)}}{\sigma }^2(r,t)$ for three different values of $r$ and show that it grows as $t^{1/2}$. We have used $N=512$ here. These data have been averaged over at least ${10}^4$ initial histories.

Figure 10

Main plot shows the sliding tag correlation function ${\mathrm{\Lambda }}^2\left(t\right)$ for an up-slope bond in the $L$ region in FPS phase. It grows as $t^{0.67}$. Inset shows the variance ${\sigma }^2\left(t\right)$ of a tagged up-slope bond displacement in the same phase. When the relative shift between the tagged bond and the density patches is not taken into account, measurement of the tagged mean-squared displacement does not capture the correct dissipation and shows a diffusive behavior. We have used $N=1024,2048$ here. We have taken the density of $H$ particles to be $1/2$, and hence the length of the open chain ASEP in maximal current phase is of length $N/2$. These data have been averaged over at least ${10}^3$ initial histories.

Figure 11

Tagged mean-squared displacement (${\sigma }_{CM}^2$) of the center of mass of the $H$ cluster for IPS and FPS phases. ${\sigma }_{\mathrm{CM}}^2$ grows diffusively in time with a diffusivity $D_{\mathrm{CM}}\sim \exp (-\alpha N)$. Plots (a) and (b) show the behavior in the IPS phase, while (c) and (d) correspond to the FPS phase. The values of $\alpha$ in IPS and FPS phases are $\sim 0.23$ and $\sim 0.1$, respectively. These data have been averaged over at least ${10}^5$ initial histories.

Figure 12

Landscape dynamics in two dimensions. (a) Mean-squared displacement ${\sigma }^2\left(t\right)$ of the deepest point of the valley along the $x$ direction as a function of time for three different system sizes, $24\times 24$ (solid line), $32\times 32$ (dotted line), $40\times 40$ (dashed line) in the IPS phase. Top and bottom insets show the short- and large-time diffusion constants $D_1\sim 1/N$ and $D_2\sim e^{-\alpha N}$, where $\alpha =0.25$. (b) ${\sigma }^2\left(t\right)$ in the FPS phase for four different system sizes. While the initial growth and saturation does not show any dependence on the system size, the large-time diffusivity falls off exponentially as the system size (data not shown here). (c) The height fluctuation ${\mathrm{\Sigma }}^2\left(t\right)$ as a function of time in the part of the surface holding the $L$ cluster in IPS phase. The fluctuations grow logarithmically in time as is characteristic of an Edwards Wilkinson surface in two dimensions. (d) ${\mathrm{\Sigma }}^2\left(t\right)$ in the FPS phase shows a subdiffusive regime that remains valid up to $t\sim 100$. This is followed by a diffusive growth at large times with the diffusion constant falling off as a power law $\sim N^{-1.5}$.

Figure 13

In FPS phase, the tagged mean-squared displacement ${\sigma }^2(r,t)$ of an up-slope bond inside the $H$ cluster initially at a distance $r$ from the deepest point shows a subdiffusive behavior and grows as $t^{1/2}$. Here, we tag an up-slope bond at a distance $N/16$ from the deepest point and show the plots for three different $N$ values. We observe an initial nonscaling behavior for $t\lesssim 100$ where the growth is steeper. These data have been averaged over at least ${10}^5$ initial configurations.