General self-flattening surfaces

TL;DR

This paper provides an analytical framework for understanding the scaling properties of self-flattening surfaces in multiple dimensions, extending previous numerical results and deriving explicit formulas for roughness and window exponents.

Contribution

It introduces a general analytic approach to self-flattening surfaces in D dimensions, deriving explicit formulas for roughness and window exponents based on the equilibrium roughness exponent.

Findings

Derived formulas for roughness and window exponents in D dimensions.

Confirmed known exact value of roughness exponent in 1D as 3/2.

Estimated exponents for 1D self-flattening surfaces as 3/5 and 2/5.

Abstract

Recently Jeong and Kim [Phys. Rev. E {\bf 66}, 051605 (2002)] investigated the scaling properties of equilibrium self-flattening surfaces subject to a restricted curvature constraint. In one dimension (1D), they found numerically that the stationary roughness exponent $\alpha\approx 0.561$ and the window exponent $\delta\approx 0.423$. We present an analytic argument for general self-flattening surfaces in $D$ dimensions, leading to $\alpha=D\alpha_0 /(D+\alpha_0)$ and $\delta=D/(D+\alpha_0)$ where $\alpha_0$ is the roughness exponent for equilibrium surfaces without the self-flattening mechanism. In case of surfaces subject to a restricted curvature constraint, it is known exactly that $\alpha_0=3/2$ in 1D, which leads to $\alpha=3/5$ and $\delta=2/5$. Small discrepancies between our analytic values and their numerical values may be attributed to finite size effects.