Quick von Neumann stability analysis

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”

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Van der Waals’ square gradient theory

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 \phi:

g(\phi) = \frac a2 \phi^2 + \frac b4 \phi^2

When becomes negative, the minimum of g changes from 0 to other values, \pm \phi_0. 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

F = \int dr \,  f(\phi, \nabla\phi)

f(\phi) = g(\phi) + \frac c2 (\nabla\phi)^2 .

This is also called a London-Linzburg-Landau free energy, also appearing in their theory of superconductors.

Now, parameters ab, 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. \phi_0 ). 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).

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The Onion on Science

Just a list of The Onion articles about science that I find funny:

  • Christian Right Lobbies To Overturn Second Law Of Thermodynamics. The second law of thermodynamics, a fundamental scientific principle stating that entropy increases over time as organized forms decay into greater states of randomness, has come under fire from conservative Christian groups, who are demanding that the law be repealed.
  • Bush Finds Error In Fermilab Calculations. President Bush met with members of the Fermi National Accelerator Laboratory research team Monday to discuss a mathematical error he recently discovered in the famed laboratory’s “Improved Determination Of Tau Lepton Paths From Inclusive Semileptonic B-Meson Decays” report.
  • National Science Foundation: Science Hard. The National Science Foundation’s annual symposium concluded Monday, with the 1,500 scientists in attendance reaching the consensus that science is hard.
  • World’s Top Scientists Ponder: What If The Whole Universe Is, Like, One Huge Atom? Gathering for what members of the international science community are calling “potentially the most totally out-to-lunch freaky head trip since Einstein postulated that space and time were, like, curved and shit,” a consortium of the world’s top physicists descended upon Stanford University Monday to discuss some of the difficult questions facing the cutting edge of theoretical thinking.
  • Raving Lunatic Obviously Took Some Advanced Physics… “Where’s my cheese? Don’t take my rowboat! Got no room!” the lunatic screamed from his regular spot near the Campus Drive bus stop. “I need space! Gimme space! Infinite dimensional separable Hilbert space!”.
  • Evangelical Scientists Refute Gravity With New ‘Intelligent Falling’ Theory. As the debate over the teaching of evolution in public schools continues, a new controversy over the science curriculum arose Monday in this embattled Midwestern state. Scientists from the Evangelical Center For Faith-Based Reasoning are now asserting that the long-held “theory of gravity” is flawed, and they have responded to it with a new theory of Intelligent Falling.
  • High-School Science Teacher Takes Fun And Excitement Out Of Science. Verona High School ninth-grade science teacher Mark Randalls has a unique talent for taking the fun and magic out of science, students of his comprehensive survey class reported Tuesday.


Britney Spears .ac, including Britney Spears’ Guide to Semiconductor Physics, with articles such as Semiconductor Transport: The Einstein Relations.

Get your units right with LaTeX

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:

$R=8.314$ \joule\per(\mole\usk\kelvin)

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:

$0$\celsius=$273.15$ \kelvin.

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”.

TED, and Steven Strogatz

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:

Natural coordinates


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