Directed percolation with long-range interactions: Modeling nonequilibrium wetting

It is argued that some phase transitions observed in models of nonequilibrium wetting phenomena are related to contact processes with long-range interactions. This is investigated by introducing a model where the activation rate of a site at the edge of an inactive island of length $\ell$ is $1+a{\ell }^{-\sigma }$. Mean-field analysis and numerical simulations indicate that for $\sigma >1$ the transition is continuous and belongs to the universality class of directed percolation, while for $0<\sigma <1$, the transition becomes first order. This criterion is then applied to discuss critical properties of various models of nonequilibrium wetting.

Figure 1

Schematic configuration of an interface bound to a substrate. Within the DP framework bound sites may be considered as active while islands of depinned sites are inactive.

Figure 2

Power law critical behavior of the generalized contact process (1) for $\sigma =1.1$ and $\sigma =1.5$, compared to the expected DP behavior (dashed lines): (a) decay in time of the density of active sites $\rho \left(t\right)$; (b) stationary density of active sites as a function of the distance from criticality. Circles correspond to the case $\sigma =1.5$, while squares to $\sigma =1.1$. Both graphs are plotted in a doubly logarithmic scale.

Figure 3

Doubly logarithmic graph of the (un-normalized) size distributions of inactive islands $P\left(\ell \right)$ as a function of size $\ell$. From top to bottom the solid lines correspond to $\sigma =1.5$, $\sigma =1.1$, $\sigma =0.9$, and $\sigma =0.5$. The dot-dashed line marks the power-law decay expected in the case of a DP phase transition. The long dashed line decays as $1∕{\ell }^2$, discriminating in $1+1$ dimensions between first order and continuous phase transitions.

Figure 4

Typical cluster grown from a single seed for $\sigma =0.5$ at criticality, simulated up to 5000 Monte Carlo sweeps. Notice that large inactive islands are suppressed and the growing cluster can be regarded as effectively compact.

Figure 5

Time evolution of the average number of active sites close to the phase transition. In the left panel the average number of active sites $N\left(t\right)$ is plotted as a function of time for $a=2$, $\sigma =0.5,0.8,0.9$ and for different values of $\lambda$, close to ${\lambda }_c$. All curves indicate that $N\left(t\right)$ converges to a constant after a transient time which increases with $\sigma$. In the right panel the time-rescaled number of active sites $N\left(t\right)t^{-\eta }$ (with $\eta =0.313686$) is shown for $a=2$, $\sigma =1.1,1.2,1.5$, and for different values of $\lambda$, close to ${\lambda }_c$. In order to avoid a messy overlap, the curves have been shifted vertically by an arbitrary value. Their asymptotic behavior indicates good agreement with the expected DP critical scaling. The estimates of ${\lambda }_c$ are reported in Table . Both graphs are plotted in a doubly logarithmic scale.

Figure 6

The survival probability $P_s\left(t\right)$ (left panel) and the average number of active sites $N\left(t\right)$ (right panel) at criticality for $\sigma =0.5,0.8,0.9,1.1,1.2,1.5$ (shown from bottom to top) in a doubly logarithmic representation. The predicted asymptotic slopes of directed percolation (DP) and Glauber-Ising (GI) are indicated as bold lines.

Figure 7

Phase diagram of the RSOS model for $q_0>q_0^{*}\left(p\right)$ (left panel) and $q_0<q_0^{*}\left(p\right)$ (right panel). The phase transition takes place along the full line, while the dashed line in the right panel marks the lower border of the phase-coexistence region (see text).

Figure 8

RSOS model—decay of the activation rate (see text) as a function of size of depinned islands. Numerical simulation have been performed close to the special point $p=0$. Circles refer to $T_0=(p=0,q_0=0.2,q=0.755)$, while squares to $T_1=(p=0.01,q_0=0.2,q=0.76)$. The graph is plotted in a doubly logarithmic scale. For the sake of clarity the data are shifted vertically by an arbitrary value.

Figure 9

RSOS model—decay of the activation rate (see text) as a function of size of depinned islands. Numerical simulation have been performed at the critical points $T_2=(p=0,q_0=0.4,q=0.70)$ (squares) and $T_3=(p=0.2,q_0=0.3,q=0.745)$ (circles). The graph is plotted in a doubly logarithmic scale. Data at $p=0.3$ show a power law decay with an exponent $\sigma =0.69\left(2\right)$, while $q_0=0.4$ data decays with an exponent $\sigma =0.89\left(2\right)$.

Figure 10

Updating rule of the SSW model. The full line represents the interface, while the shaded area represents the wall. Dashed segments indicate interface growth occurring in randomly chosen local minima (see A and B in upper panel) and in all sites located below the wall after it has been shifted upwards by one.

Figure 11

SSW model—Double logarithmic plot of the activation rate $\overline{\lambda }\left(\ell \right)$ as a function of the size $\ell$ of depinned islands. Numerical simulations have been performed at criticality: circles correspond to $q=0.7$, $v_w=0.4297$ while squares to $q=0.8$, $v_w=0.3489$. Dashed lines mark our best fits with Eq. (1), rendering respectively the estimates $\sigma =1.20\left(5\right)$ and $\sigma =1.30\left(5\right)$.

Figure 12

Qualitative “finite-size” phase diagram of the RSOS model in the case $q_0<q_0^{*}\left(p\right)$. We expect the entire transition line for $p<1$ to belong to the DP universality class. The full line (line A) on the left of the equilibrium point (EQ) marks a region in which the transient first-order behavior is the only numerically accessible one, with a crossover length $L_c$ to DP behavior which diverges as $p\to 1$. On the other hand, also systems of moderate size can exhibit a DP critical behavior over the dashed line (B). The dot dashed transition line (C) on the right of the equilibrium point is genuinely first order (see text). Finally, the light dashed line where the free interface velocity changes sign marks the lower boundary of the phase-coexistence region (PC).