How Electrons Behave and Generate Random Surfaces and Geometries

The term "kahn shaw equation" likely refers to either the Kohn-Sham equation in quantum chemistry or the Cahn-Hilliard-Hele-Shaw (CHHS) system in fluid dynamics and materials science, as "kahn" and "shaw" are likely misspellings. The Kohn-Sham equation is a central part of density functional theory for solving the many-body problem in quantum mechanics by using a fictitious system of non-interacting particles. The Cahn-Hilliard-Hele-Shaw system is a set of equations modeling fluid flow in a Hele-Shaw cell, with the Cahn-Hilliard equation describing phase separation.
Kohn-Sham equation
Description: A set of Schrödinger-like equations used in density functional theory (DFT).
Purpose: To find the ground state electron density for a system of interacting electrons.
How it works: It introduces a fictitious system of non-interacting electrons that have the same density as the real, interacting system. By solving this simpler system, the original problem can be solved.
Key components: The equations are defined by a local effective potential, known as the Kohn-Sham potential, which accounts for the interactions in the real system.
Cahn-Hilliard-Hele-Shaw (CHHS) system
Description: A coupled system of equations used to model the motion of fluids and phase separation.
Components:
Cahn-Hilliard equation: A fourth-order parabolic partial differential equation that describes how a mixture separates into different phases over time.
Hele-Shaw flow equations: Equations that describe the motion of a viscous fluid between two closely spaced plates.
Purpose: To model complex interfacial dynamics in multiphysics problems, such as the microstructure formation in materials or two-phase fluid flow.
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