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

TL;DR

This paper derives exact solutions for a one-dimensional restricted ballistic deposition model, revealing its surface fluctuation scaling behavior and its relation to the KPZ universality class.

Contribution

It provides the first exact steady-state solutions for the surface fluctuation width in a restricted ballistic deposition model on a 1D staircase.

Findings

Surface fluctuation width diverges as L^{1/2} in the infinite limit.

Dynamic exponent β is 1/2, matching numerical solutions.

Model belongs to the KPZ universality class with specific scaling exponents.

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 $L^\alpha$ with the scaling exponent $\alpha=\frac{1}{2}$. The dynamic exponent $\beta$ $(W\sim t^\beta)$ 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 $(\alpha_{KPZ}= \frac{1}{2} , \beta_{KPZ}=\frac{1}{3})$. Comparing its deposition time scale with that of the single-step model, we argue that $\beta$ must be the same as $\beta_{KPZ}/(1-\beta_{KPZ})$, which is consistent with our finding.