Dynamics of fluctuation-dominated phase ordering: Hard-core passive sliders on a fluctuating surface

We study the dynamics of a system of hard-core particles sliding downwards on a one-dimensional fluctuating interface, which in a special case can be mapped to the problem of a passive scalar advected by a Burgers fluid. Driven by the surface fluctuations, the particles show a tendency to cluster, but the hard-core interaction prevents collapse. We use numerical simulations to measure the autocorrelation function in steady state and in the aging regime, and space-time correlation functions in steady state. We have also calculated these quantities analytically in a related surface model. The steady-state autocorrelation is a scaling function of $t∕L^z$, where $L$ is the system size and $z$ is the dynamic exponent. Starting from a finite intercept, the scaling function decays with a cusp, in the small argument limit. The finite value of the intercept indicates the existence of long-range order in the system. The space-time correlation, which is a function of $r∕L$ and $t∕L^z$, is nonmonotonic in $t$ for fixed $r$. The aging autocorrelation is a scaling function of $t_1$ and $t_2$ where $t_1$ is the waiting time and $t_2$ is the time difference. This scaling function decays as a power law for $t_2⪢t_1$; for $t_1⪢t_2$, it decays with a cusp as in steady state. To reconcile the occurrence of strong fluctuations in the steady state with the fact of an ordered state, we measured the distribution function of the length of the largest cluster. This shows that fluctuations never destroy ordering, but rather the system meanders from one ordered configuration to another on a relatively rapid time scale.

Figure 1

Illustrating the linear drop of $A(t,L)$ for short times $t\lesssim 1$ in the SP model for system size $L=128,256,512$.

Figure 2

Scaled autocorrelation function in steady state for SP and CD models for (a) EW and (b) KPZ interfaces. In both cases, we used $L=512,1024,2048$. The cusp exponents were estimated using the plots shown in the inset.

Figure 3

Aging autocorrelation for CD and SP models with (a) EW and (b) KPZ interfaces. The cusp exponent ${\beta }^{\prime }$ was determined for $t_1⪢t_2$, after subtraction from $m^2$. The CD model data in (a) have been multiplied by $1.5$ to distinguish them from the SP model data points. The inset shows the power law behavior in the regime $t_1⪡t_2$. We used $L=2048$ in (a) and $L=8192$ in (b). The inset shows the data with $t_1=500,2000,8000$ in both (a) and (b). For extraction of ${\beta }^{\prime }$, we used $t_1=2000,8000,32000$.

Figure 4

The time dependence of $G(r,t,L)$ is shown for particles on an (a) EW and (b) KPZ surface for $r∕L=0.016$. The values of $L$ are $256,512,1024$ for (a) and $512,1024,2048$ for (b). The scaled autocorrelation is also shown, for comparison. The insets show the same quantity calculated for the corresponding CD model with $r∕L=0.125$ for both cases.

Figure 5

The distribution of the length of the largest cluster $P(l_{max},L)$ for particles advected by the KPZ surface is shown for $L=256,512,1024$, with the scaling collapse in the inset. The curves to the left show the same distribution in disordered state, after rescaling the $y$ axis by $0.2$.