Wednesday, 18 December 2013

Spring Time


Picture a spring. You know that the harder you squash or stretch it the harder it tries to get back to its "natural length". So you know that the force must be related to the difference between the natural length and the length it's been forced it to. \[F = f(dx)\] where $dx$ is the difference between the natural length and the stretched or squashed length. The equation above summarizes this in mathematical form, the force is given by some function of $dx$. $17^{\text{th}}$ century weirdo Robert Hooke gave the first guess for what this function was in the form of a Latin anagram ceiiinosssttuv which is solved to give Ut tensio, sic vis, "As the extension, so the force" or "Force is proportional to the extension." This gives us \[F = k dx \] where $k$ is some constant that tells us how stiff the spring is. Of course as advanced $21^{\text{st}}$ century scientists we can put in a more complicated function, \[F = k dx + \alpha dx^2 + \beta dx^3 \] $\alpha$ and $\beta$ are just some numbers. Any force law, other than Hooke's one is called "non-linear".

Now you could take a bunch of springs and join them together, making something like a mattress, pull on it and let it oscillate. For springs obeying Hooke's law it will oscillate in a pretty boring way. For non-linear springs the motion will be crazier. Joining a bunch of non-linear springs together is called the Fermi-Pasta-Ulam problem. Fermi and Ulam are famous guys. Pasta was a American computer scientist with an unfortunate name. Also involved in the first paper was a lady called Mary Tsingou, but for some reason she doesn't get to be included with the first three guys in the name of the problem. The idea was that if you had a non-linear force the system would oscillate in such a crazy way that it would go through every possible arrangement. When they actually simulated it however they found it didn't, it oscillated in a surprisingly regular way. The conclusion in fancy physics jargon: non-linearity does not imply ergodicity.

From the force we can get a potential energy, using the non-linear force above gives us the following four basic shapes of the potential.

You can think of the system as a little ball rolling on the potential hill. The top left is Hooke's law, small oscillations about the mean value, imagine a little ball rolling back and forth inside the "well". If you go to the simulation below you can see what that looks like by setting $\alpha = \beta = 0$ and $k = 1$ (and clicking Submit) for example. Much more interesting is the top right potential. Imagine carefully placing a little ball on top of the hump in the middle. Unless you were extremely careful it would roll down to the left or right. This is in fact exactly the sort of thing that happens in the Higgs mechanism which explains the mass of the electron. In our model, if we start almost at the top of the hump (press the "random" button and try $k=-1$, $\beta = 80$, $\alpha = 0$) we find that the springs want to compress together making clumps. These clumps are a little like our Higgs bosons. The symmetry, where all the points were more or less equal, is broken and we rolled down the hill.

The third one is also interesting, again imagine a little ball rolling on the hill on the bottom left. If it starts near the middle it will roll happily inside the little well, but if it gets above the hill it will roll down forever getting faster and faster. You can see this happen, and break the simulation, by choosing say $k = 1$ and $\beta = -250$. After a little while some of the springs should, by chance, accumulate enough energy to get over the hump and fall to their doom (infinity). The bottom right potential is what it looks like if you turn $\beta = 0$ and turn on $\alpha$. You have one stable direction and one unstable one.

Try clicking on the dots, you can move them around if you want a different starting position. You can set up some interesting looking oscillations by choosing a good starting point for all the points. If you really want to, you can break some springs. Easiest is to make a potential with positive parameters. The forces can become huge and the algorithm that evolves the positions forward in time will break as the numbers overflow the maximum allowed size. These masses will break off. By pulling really hard on a spring you might be able to get it to snap off also, depending on the parameters. If you do break some springs clicking "random" will regenerate them.

No comments:

Post a comment