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Edwards-Wilkinson equation from lattice transition rules.

Dimitri D Vvedensky¹

¹The Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|March 15, 2003

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This study derives continuum equations for lattice growth models by analyzing discrete Langevin equations. The method reproduces the Edwards-Wilkinson equation for specific random deposition and relaxation models.

Area of Science:

Surface science and condensed matter physics.
Mathematical modeling of physical systems.
Statistical mechanics and non-equilibrium phenomena.

Background:

Lattice growth models are crucial for understanding surface evolution in various physical and chemical processes.
Discrete Langevin equations describe the stochastic dynamics of these models but are computationally intensive.
Continuum equations offer a simplified, analytical approach to studying large-scale surface dynamics.

Purpose of the Study:

To develop a general methodology for deriving continuum equations of motion from discrete growth model rules.
To validate the methodology by deriving a known continuum equation.
To explore the applicability of the method to a broader range of lattice growth models.

Main Methods:

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Regularizing and coarse-graining discrete Langevin equations associated with lattice growth models.
Analyzing transition rules of specific growth models to identify key dynamical features.
Applying the derived methodology to models featuring random deposition and local height minimization.

Main Results:

Successfully derived continuum equations of motion for height fluctuations in lattice growth models.
The methodology rigorously yields the Edwards-Wilkinson equation for models with random deposition and instantaneous relaxation.
Demonstrated a systematic procedure applicable to other lattice growth models.

Conclusions:

The developed method provides a robust framework for bridging discrete microscopic models and continuum macroscopic descriptions of surface growth.
The Edwards-Wilkinson equation emerges naturally from specific discrete growth rules, validating the theoretical approach.
This work facilitates the analytical study of surface dynamics and pattern formation in various materials science applications.