## Wednesday, July 18, 2012

### Making gnuplot plots look like Mathematica plots

For an article I am writing I wanted to plot the following function:  $$\theta \left(x > \frac{1}{2} \right) \left(\sin(4 \pi x) + \sin(4 \pi y) +2 \right)$$
Wolfram Alpha gives the following pretty plot:
 Plot from Wolfram Alpha
However, since you need to pay now to export this picture in a more usable format (like svg, eps or pdf) I decided to recreate the picture in gnuplot. Friendly advice for you others writing articles: always use vector formats where possible, otherwise your graphics will look horrible on higher resolutions!

To plot the function in gnuplot we first have to do a little trick, because the step function is not included in gnuplot. It does have the sign function, however. So we can do the following: $$\theta(x > \frac{1}{2}) = \frac{1 + \text{sgn}(x-\frac{1}{2})}{2}$$
A first attempt, with default settings is:

set terminal postscript eps
set pm3d at s hidden3d
unset hidden3d
unset surface
set xrange [0:1]
set yrange [0:1]
set samples 20,20

splot (1+sgn(x-0.5))*(sin(4*pi*x) + sin(4*pi*y)+2)/2 title ''


This gives

This doesn't really look too good. First thing that has to be changed is the color palette and the color of the grid. The color palette can be changed like this:

set terminal postscript eps
set palette defined ( 0 "#bd2c29", 2 "#ffd35a", 6 "white")


The numbers in front of the color codes are the values of the function that the color should correspond to. Thankfully, gnuplot automatically interpolates the colors.

Next up is the grid color. The color can be set to a given line style by adding the style number at the end of the pm3d options:
set pm3d implicit at s hidden3d 100
set style line 100 lc rgb '#000000' lt 1 lw 0.6

Here I have defined the grid style to be a thin black line. I have chosen index 100 to make sure it doesn't clash with other settings.

After removing the color bar and shifting the image down with ticslevel, we get the final image:
This looks a lot better! The only thing missing is the shadowing of surfaces angled away from the camera. I am not aware of a way to produce these in gnuplot. If anyone has additional advice to improve the plot, please let me know!

The final version of the code:

set terminal postscript eps
set output '| epstopdf --filter --outfile=plot.pdf'
set palette defined ( 0 "#bd2c29", 2 "#ffd35a", 6 "white")
set pm3d implicit at s hidden3d 100
set style line 100 lc rgb '#000000' lt 1 lw 0.6
unset hidden3d
unset surface
set border 4095 front linetype -1 linewidth 0.8
set ticslevel 0
set xrange [0:1]
set yrange [0:1]
set ztics 1
set isosamples 40,40
set samples 20,20
unset colorbox

splot (1+sgn(x-0.5))*(sin(4*pi*x) + sin(4*pi*y)+2)/2 title ''


## Wednesday, July 4, 2012

### What is the Higgs boson?

Today the Higgs boson has been found at CERN, a great accomplishment in the field of theoretical particle physics. This discovery means that the model we are currently using is completed: the final ingredient has been found. This model, called the Standard Model, yields very good predictions of what is happening in nature.

But what is this Higgs boson, what does it do and why didn't we see it before?

## The mystery of the massive particles

When physicists were deriving the physics of elementary particles, they used certain assumptions about the universe, most of them are related to symmetries. What's a symmetry? Take for example rotational symmetry. Imagine you are standing in the middle of a round football stadium. It does not matter if you rotate, you will still see the same image. There is a very fancy theorem that says that every symmetry has an accompanied conserved quantity. The rotational symmetry, for example, yields conservation of angular momentum, whereas translational symmetry in time (meaning physics looks the same now and 10 minutes from now) yield conservation of energy.

Using some more advaced symmetry groups all of our beloved standard particles can be found:

The standard model (without Higgs)

The group in red are called gauge bosons. They are resultants of a very special symmetry and they can be interpreted as forces working on all the other particles. In fact, every elementary force has a matching gague boson. The photon (symbolised with a gamma), or "light", transfers electromagnetic forces, the gluon the strong force and Z and W the weak force.

But there is a problem. The theory requires the W and Z boson to be massless in order for the theory to work. This is obviously not the case, because these particles have been found to have mass. In fact, they are pretty heavy!

## The Higgs mechanism

When physicist discovered that the theory was in trouble, they started searching for solutions. The symmetry of nature that can be used to find the gauge bosons worked far too good to just abandon it. Then, a new idea rose: we keep the symmetry, but we say nature broke it spontaneously. Wait, what?

How does this work? Imagine you have a piece of plastic between your fingers and you apply a little pressure to it. Initially, the plastic stick is straight, even when you apply a little pressure to it. Now the system (nature) has a symmetry: there is for example rotational symmetry if you are standing at the position of the rod (if you ignore the hand).

But remember the hand was applying a force? This means that the stick could spontaneously jump to either the left or the right, as can be seen in the picture (try this at home ;) ). Now the symmetry that nature had is broken, spontaneously and without external help. Note that in the analogy, the hand should be seen as an initial energy that the universe has.

Spontaneous symmetry breaking

A similar scenario this of a toy car that is sitting on a small mountain, but it has some energy which makes the car shift a little. Given enough time, the car will roll off the mountain into a valley, where it will remain.

 Imagine a toy car in this figure

Now, it turns out that breaking this symmetry can be done by introducing a particle: the Higgs particle. When analysing the properties of this particle, it turns out that this particle interacts with all the other particles, including itself! Due to these interactions, the particles gain mass.

## Why did this take so long to find?

Well, we didn't know where to look. Basically, in the LHC particles are shot at eachother with incredible speeds, and thus incredibly high energies. Since E=mc^2, it takes very high energies to create particles with high mass. The problem is, we don't know the mass of the Higgs, it is a free parameter of the theory. This means the mass is not predicted by the theory. And because of that, the whole LHC range has to be explored (with no guarantee that the Higgs boson is actually in this range).

Complications arise due to the fact the Higgs particle decays intoa lot of other particles very, very quickly. So we can actually only see the remnants of the Higgs particle. When there is enough energy to create a Higgs, we'd expect to see more of these remnants: a peak in a graph.

Today, the people at CERN announced that the peaks they found in many expirements, are trustworthy enough to say that the Higgs particle has been found at about 126 GeV. This is great news, because now we can confidently continue the search for theories that expand the standard model, like supersymmetry and other theories that may introduce gravity into the picture.

We aren't quite there yet, but a theory of everything is coming!