Scaling of ballistic deposition from a Langevin equation

An exact lattice Langevin equation is derived for the ballistic deposition model of surface growth. The continuum limit of this equation is dominated by the Kardar-Parisi-Zhang (KPZ) equation at all length and time scales. For a one-dimensional substrate the solution of the exact lattice Langevin equation yields the KPZ scaling exponents without any extrapolation. For a two-dimensional substrate the scaling exponents are different from those found from computer simulations. This discrepancy is discussed in relation to analytic approaches to the KPZ equation in higher dimensions.

Figure 1

Local growth rates at a randomly chosen site $i$ for several representative configurations of the BD model using $a=0.1$ in Eq. (14). Height differences between this site and nearest-neighbor sites are shown at the sides of each configuration.

Figure 2

The saturated interface width $w_{\mathrm{sat}}$ obtained from Eq. (17) for BD versus the linear system size $L$ for (a) $d=1$ and (b) $d=2$. The error bars associated with the data points are smaller than the symbol size. The lines with slope $\alpha =0.495$ in (a) and $\alpha =0.362$ in (b) are optimal fits.