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theory.html
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theory.html
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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
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<title>Theory</title>
</head>
<body>
<iframe src="header.html" onload="this.before((this.contentDocument.body||this.contentDocument).children[0]);this.remove()"></iframe>
<main id="main">
<h1>Theory</h1>
<p>The phase-field model is based on a free energy functional written as function of the system's order parameter \(\phi\) (that distinguishes the different phases present in the system).
In this formalism, the interface has a width \(\varepsilon\) and the order parameters vary continuously along its width.
As a consequence of that, the system evolution as a whole is described by a unique set of equations applied to every phase of the system, without taking into account the boundary conditions at the interfaces.</p>
<h2>Allen-Cahn Equation</h2>
<p>
\[
\left( \frac{\partial } { \partial t} + \mathbf{v}_m \cdot \nabla \right) \phi_m= -\frac{\delta F}{\delta \phi_m} \, ,
\]</p>
<h2>Cahn-Hilliard Equation</h2>
<p>\[
\left( \frac{\partial }{\partial t} + \textbf{v}_m \cdot \nabla \right) \phi_m = M(l_m) \nabla^2 \frac{\delta F}{\delta \phi_m} \textrm{ .}
\]</p>
<h2>Actomyosin model</h2>
<p>Try it yourself our Python implementation in a <a href="actomyosin.html">live terminal.</a></p>
</main>
</body>
</html>