The von Neumann stability analysis is a great tool to assess stabilty of discretizing schemes for PDEs. But too often, imho, the discussion gets blurred with trigonometric expressions. Here, I try to provide a shortcut.
BTW: it’s “fon no ee man”
OK, I have encountered this theory again, after many years. The idea is to describe separation between two phases as the minimum of a free energy with respect to an order parameter :
When a becomes negative, the minimum of g changes from 0 to other values, . The function then has the celebrated double minimum feature, which features prominently in many symmetry-breaking theories, including of course the appearance of particle mass mass and, you know, the Big Bang and the Universe.
But here we are just considering phase separation in materials. The interface between two coexisting phases must have some associated cost, and the simplest (lowest order) way to include it is by introducing a total free energy functional
This is also called a London-Linzburg-Landau free energy, also appearing in their theory of superconductors.
Now, parameters a, b, and c are not easy to measure (or, at least, estimate) experimentally, but they are related to: the surface tension, the width of the interface, and the magnitude of the bulk equilibrium order parameter (i.e. ). Here I show how to obtain it in a slightly more general setting, since I was not able to find it on the internet (it can be found e.g, in the book by Rowlinson & Widom).
Just a list of The Onion articles about science that I find funny:
The LaTeX package SIunits (which seems to be a descendant from the units package that I once used) addresses one of the most nagging issues in physics and engineering: getting the units well typeset. While the source is a bit cumbersome, it is quite readable:
There are prefixes, so that “\mega\ohm” is ok. Many ready-to-use units are predefined, such as \kilogrammetrepersecond for your momentum, or\kilogrammetrepersecondnp if you prefer negative powers instead of division.
As a bonus, \celsius adds the little “degree” symbol that’s so easy to get wrong, and \kelvin does not, as it should be:
Also, units maybe inside formulae or outside:
$9.81 \newton$ or$9.81$\newton
The space in the first equation is ignored, btw, so both produce the same output. This space, is moreover, one of the few things left to the author:
$9.81$ \newton or$9.81$\newton ???
It seems the first one is correct, but one may use this to get things always right:
This later choice is a bit more cumbersome, but convenient: math mode is invoked, no need to use $$; also, no need to remember the extra \ at the end of the command: “a force of $1$ \newton\ is required \ldots”.
This is a very interesting website: TED. A collection of interesting talks, all of them downloadable and in the public domain (CC license). There isn’t much on physics yet, and certainly not in my area. There is a very nice talk, though, by Steven Strogatz on synchronization. Strogatz is a remarkable speaker, I had the pleasure of attending a set of lectures on “living polymers” that were very interesting, and I remember he used some of this examples on his introduction. Some of his demonstrations are quite ingenuous, and not so difficult to reproduce.
Er, I forgot to embed the video in the first version of this entry, which is funny since I came across this site from a wordpress announce. There it goes:
Here is an interesting application of Voronoi tesselations / Delaunay triangulations (see previous post The alpha shapes for another one.) Suppose you have a set of points carrying some information, let’s call them particles; the simplest case is just a scalar number representing some field whose value is only known for the particles. Then, you want to compute the value of the field in some other, arbitrary point. (In the simplest instance, you may want to plot the field, and you need to know the values at nodes of a grid.) Continue reading
Done! It seems unbelievable, but yesterday I managed to finish The Road to Reality, by Sir Roger Penrose. I must say I liked it a lot. It has its failures, of course, and it is sometimes contentious (willingly so), specially at the end. I have made a list of things I have found outstanding in the book.