Related Experiment Video
Updated: Jun 5, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids
Published on: December 4, 2017
One-dimensional model of interacting-step fluctuations on vicinal surfaces: analytical formulas and kinetic Monte
Paul N Patrone1, T L Einstein, Dionisios Margetis
1Department of Physics, University of Maryland, College Park, Maryland 20742, USA. ppatrone@umd.edu
Abstract:
We study analytically and numerically a one-dimensional model of interacting line defects (steps) fluctuating on a vicinal crystal. Our goal is to formulate and validate analytical techniques for approximately solving systems of coupled nonlinear stochastic differential equations (SDEs) governing fluctuations in surface motion. In our analytical approach, the starting point is the Burton-Cabrera-Frank (BCF) model by which step motion is driven by diffusion of adsorbed atoms on terraces and atom attachment-detachment at steps. The step energy accounts for entropic and nearest-neighbor elastic-dipole interactions. By including Gaussian white noise to the equations of motion for terrace widths, we formulate large systems of SDEs under different choices of diffusion coefficients for the noise. We simplify this description via (i) perturbation theory and linearization of the step interactions and, alternatively, (ii) a mean-field (MF) approximation whereby widths of adjacent terraces are replaced by a self-consistent field but nonlinearities in step interactions are retained. We derive simplified formulas for the time-dependent terrace-width distribution (TWD) and its steady-state limit. Our MF analytical predictions for the TWD compare favorably with kinetic Monte Carlo simulations under the addition of a suitably conservative white noise in the BCF equations.
Related Concept Videos
Reaction Mechanisms: The Steady-State Approximation
First Law: Particles in One-dimensional Equilibrium
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
First Law: Particles in Two-dimensional Equilibrium
Newton's first law tells us about the...
Theories of Dissolution: The Danckwerts' Model and Interfacial Barrier Model
Basic Postulates of Kinetic Molecular Theory: Particle Size, Energy, and Collision
