Published February 17, 2021 | Version 1
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Lecture "Allen-Cahn-type phase-transition dynamics"

  • 1. WWU Münster, Institut für Theoretische Physik

Description

The data deposit contains an introductory  video lecture by Uwe Thiele, WWU Münster, Institut für Theoretische Physik on "Allen-Cahn-type phase-transition dynamics" together with the slides in pdf format and this info file.

The lecture gives a basic introduction to the archetypical case of a nonconserved gradient dynamics, i.e., Allen-Cahn-type models as obtained in phenomenological nonequilibrium thermodynamics with the help of Onsager’s variational principle.

The lecture should be suitable for advanced Bachelor students, Master students and beginning PhD students of the natural sciences and other interested people. It is a stand-alone lecture on the behaviour described by the Allen-Cahn equation for a single order parameter field, but was given in the context of a lecture course "Introduction to the Theory of Phase Transitions" (ITPT). Background information on gradient dynamics may be obtained in the lecture "Introduction to Nonequilibrium Thermodynamics - Onsager’s variational principle"  (see https://dx.doi.org/10.5281/zenodo.4545320).

We start with a brief recapitulation of the concept of gradient dynamics on an underlying energy functional, discuss Allen-Cahn-type dynamics of uniform states before introducing the proper  Allen-Cahn equation by specifying a particular energy functional (square-gradient terms and local double-well potential). Then homogeneous steady states are discussed, their linear stability is analysed (dispersion relations are calculated and discussed), and the fully nonlinear behaviour is illustrated by time simulations. Then, heterogeneous steady states are discussed (where reference is made to the continuation tutorial ACCH, see https://dx.doi.org/10.5281/zenodo.4545409). Subsequently, steady and moving fronts are discussed distinguishing pulled and pushed fronts and touching the question of front speed selection. Bifurcation diagrams for these fronts are given including a glimpse into the more intricate behaviour of a cubic-quintic Allen-Cahn equation. Finally, the growth of droplets and domain coarsening are briefly discussed before concluding the lecture.

The second archetypical case of conserved gradient dynamics, i.e., Cahn-Hilliard-type dynamics, is considered in the accompanying lecture "Cahn-Hilliard-type phase-transition dynamics" (see https://dx.doi.org/10.5281/zenodo.4545361).

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