Exact solutions of a restricted ballistic deposition model on a one-dimensional staircase

Abstract

Surface structure of a restricted ballistic deposition(RBD) model is examined on a one-dimensional staircase with free boundary conditions. In this model, particles can be deposited only at the steps of the staircase. We set up recurrence relations for the surface fluctuation width $W$ using generating function method. Steady-state solutions are obtained exactly given system size $L$. In the infinite-size limit, $W$ diverges as $L^α$ with the scaling exponent $α=\frac{1}{2}$. The dynamic exponent $β$ $(W\sim t^β)$ is also found to be $\frac{1}{2}$ by solving the recurrence relations numerically. This model can be viewed as a simple variant of the model which belongs to the Kardar-Parisi-Zhang (KPZ) universality class $(α_{KPZ}= \frac{1}{2} , β_{KPZ}=\frac{1}{3})$. Comparing its deposition time scale with that of the single-step model, we argue that $β$ must be the same as $β_{KPZ}/(1-β_{KPZ})$, which is consistent with our finding.