Phase growth in bistable systems with impurities

A system of coupled chaotic bistable maps on a lattice with randomly distributed impurities is investigated as a model for studying the phenomenon of phase growth in nonuniform media. The statistical properties of the system are characterized by means of the average size of spatial domains of equivalent spin variables that define the phases. It is found that the rate at which phase domains grow becomes smaller when impurities are present and that the average size of the resulting domains in the inhomogeneous state of the system decreases when the density of impurities is increased. The phase diagram showing regions where homogeneous, heterogeneous, and chessboard patterns occur on the space of parameters of the system is obtained. A critical boundary that separates the regime of slow growth of domains from the regime of fast growth in the heterogeneous region of the phase diagram is calculated. The transition between these two growth regimes is explained in terms of the stability properties of the local phase configurations. Our results show that the inclusion of spatial inhomogeneities can be used as a control mechanism for the size and growth velocity of phase domains forming in spatiotemporal systems.

Figure 1

(Left-hand side) Spatial support showing active sites $(○)$ and impurities $(●)$. Impurities are randomly placed keeping a minimum distance $d$ between them. (Right-hand side) The density of impurities scales as $\rho =0.625d^{-2}$.

Figure 2

Stationary patterns on a lattice of size $100\times 100$ with impurities (represented by black dots), for different values of parameters. Top left-hand side: $\rho =0.0097$ $\left(d=8\right)$, $ϵ=0.73$. Top right-hand side: $\rho =0.039$ $\left(d=4\right)$, $ϵ=0.73$. Bottom left-hand side: $\rho =0.0097$ $\left(d=8\right)$, $ϵ=0.95$. Bottom right-hand side: $\rho =0.0097$ $\left(d=8\right)$, $ϵ=0.98$.

Figure 3

Asymptotic $⟨R⟩$ as a function of $ϵ$ for $d=\infty$ $\left(\rho =0\right)$, averaged over 40 realizations of initial conditions. Error bars correspond to the standard deviations.

Figure 4

Logarithm-logarithm plot of the average domain size $⟨R_t⟩$ vs $t$ for $d=\infty$ $\left(\rho =0\right)$ (empty circles), and $d=6$ $\left(\rho =0.017\right)$ (solid circles), with fixed $ϵ=0.73$, averaged over 40 realizations. The values of the scaling exponents ${\alpha }_1$ and ${\alpha }_2$, as well as a typical error bar are indicated on each curve.

Figure 5

Local configurations taken into account by the fractions $G\left(k\right)$, for $k=0$, 1, 2, 3, and 4.

Figure 6

Fractions $G_t\left(k\right)$ vs $t$ for different values of $d$, with fixed $ϵ=0.73$: $G_t\left(0\right)$ (continuous line), $G\left(1\right)$ (long-dashed line), $G_t\left(2\right)$ (short-dashed line), and $G_t\left(3\right)$ (dotted line). (a) $d=\infty$, (b) $d=6$.

Figure 7

Top: Mechanism of fast phase growth when two domains of the same phase join together to form a larger domain. The arrow indicates the direction of time. Bottom: The local configuration changes occurring in the boxes marked in the top panel are illustrated.

Figure 8

Phase growth exponents ${\alpha }_1$ (dashed line) and ${\alpha }_2$ (continuous line) vs $ϵ$ for different values of $d$. The values shown are averages obtained over 40 realizations of initial conditions for each value of $ϵ$. Empty circles, ${\alpha }_1$ for $d=\infty$ $\left(\rho =0\right)$; empty squares, ${\alpha }_1$ for $d=6$ $\left(\rho =0.017\right)$; solid squares, ${\alpha }_2$ for $d=\infty$; solid circles, ${\alpha }_2$ for $d=6$. Typical error bars are shown on each curve.

Figure 9

The fast phase growth exponent ${\alpha }_2$ as a function of $ϵ$ and $d$. The critical boundary ${ϵ}_c\left(d\right)$ that separates the fast growth regime from the slow growth regime is indicated by a thick continuous line on the plane $\left(ϵ,d\right)$. The values are obtained as in Fig. 8.

Figure 10

The critical exponent $\gamma$ as a function of the distance between impurities $d$, obtained as averages over 40 realizations of initial conditions for each parameter value. Error bars are shown.

Figure 11

Fractions $G_{\infty }\left(k\right)$ versus $ϵ$ for $\rho =0$: $G_{\infty }\left(0\right)$ (continuous line), $G_{\infty }\left(1\right)$ (long-dashed line), $G_{\infty }\left(2\right)$ (short-dashed line), and $G_{\infty }\left(3\right)$ (dotted line). Values shown correspond to averages over 40 realizations of initial conditions.

Figure 12

Phase diagram of the system equation (1) on the space of parameters $\left(ϵ,\rho \right)$. The regions where homogeneous, heterogeneous, and chessboard phases exist are indicated by labels. The dashed line indicates the critical boundary for the onset of the regime for fast growth of domains characterized by ${\alpha }_2>0$. The critical value ${ϵ}_o$ for the occurrence of domain growth in the absence of impurities is marked on the vertical axis.