Solution to Buffon’s Experiment.

Thanks to Vic and Xian Han for their nice suggestion. This has been done on Thursday.

The basic problem in the previous question is that we can’t produce some random number within the range 0-Pi or, say, 0-2Pi without any knowledge of Pi. But this can be done by extending the circle to a disc.

 

It is pretty easy to produce random number between 0-1. So it is straightforward to produce a uniform distribution with in a square area (0,1)x(0,1). Then we just put on some constraint: x^2+y^2<=1 to choose the proper number (x, y). The ratio of the area of the disc to the square is not low so the efficiency of this program will not be suppressed by this method. What’s more important is that the uniform distribution in the disc can induce a uniform distribution of the angle from 0-2Pi. And it is not hard to get other quantities through those coordinates.

 

Haha~~

Near, Far, Wherever you are

Today, we talked about the neutrino oscillation in the elementary particle class. Usually, we set up two detectors, one is near and the other is far. This really reminds me of the song in Titanic: Near, far, wherever you are. Especially the last sentence is so vivid. Yes! We are searching it but hard to know where it is and don’t care where it is.Another thing I am thinking about is the computer simulation. I just want to repeat Buffon’s needle experiment with Monte Carlo simulation on computer and solve for Pi. I met a problem in this process: It happens when I tried to transfer the random angle (in the unit of degree) in to a sin function, which will be used to determine whether the needle touches the lines or not. This seems to be impossible without the number of Pi. If we just pay attention in computer simulation, we know that the “sin” is just a ratio and can be measured directly in the physical process (in fact, whether the needle touches the edge can be observed directly.). So we can just use the internal math function of a certain language and get the value for sin(angle) without caring about how it comes (with Pi or not). However, logically, we don’t know Pi at this point and Pi is our goal. So my question is principally, can we simulate the Buffon’s needle experiment on computer? There should be some way to solve it but I need to find it out.

How to test the Extra Dimension?

Generally speaking, there are three to test the extra dimensional theories:

1. Test the gravity in short distance. This is the most directly idea since if we access to the distance comparable to the extra dimension, the gravity will be modified and we should sence that.

2. Test the energy missing in the collider experiments. K-K Graviton is not visible in present experiments. So energy loss in Collider experiments will be signals of extra dimension. This kind of experiment will be possible will the new accelerator, such as LHC.

3. Test the energy missing in astrophysics. Present testing is based on the cooling rates of supernova. A faster cooling rate indicate the graviton leaking to extra dimensions.

However, I am wondering, Is there other possible signals?…

Supernovae constraint on ADD model

There is  a brief paper by V.H. Satheesh Kumar et.al[1], which discuss the bound from the supernova(specified as SN 1987A) on the large extra dimension. The constraint came from the cooling process of the supernova, in which 99% energy will be released in the form of neutrino burst, about according to the “standard model” of supernova. And the observation has a good agreement with this theoretical prediction. So if there is another channal for the supernova to release energy, such as an extra dimension for the graviton emission, the emissivity shouldn’t be too high to destroy this agreement. Raffelt has proposed a simple analytic criterion based on detailed supernova simulations: if any energy los mechanism has an emisivity greater than then it will remove sufficient energy from the explosion to invalidate the current understanding of Type-II supernovae’s neutrino signal. At present, those astrophysical signals put the strongest constraint on the ADD model, say, for n=2, the dimension of the extra dimension should be in the order of mm. I will move on to some reading about the detailed calculation of this boundary to see how it works.

Dark matter

“Don’t be cheated by the brightness, the dark side dominates the Universe”

There is a replaced article on arxiv by Michael E. Peskin(here), nice illustration on dark matter and some beautiful pictures.

The discovery of dark matter can be traced back to early 20th century, by Fritz Zwicky. The in the 70s astronomers took the more precise measurement of the rotation curve of clusters. As well known, if masses are concentrated in the visible stars, the rotation velocity should be propotional to by virtue of Kepler’s law. However, the result is we obtained a flat or even slowly increasing curve, which indicates that there are some matters that we didn’t see, named dark matter.

As suggested in this article, the candidate for dark matter is the WIMPs(Weak Interaction Massive Particles), which are supposed to have mass around 100GeV and is stable. They are supposed to be produced and annihilated in pairs. And there is some predicted signal in the future LHC and ILC. Using certain SUSY model, in this article, Peskin showed the possible signal of WIMPs on LHC and ILC, which can be produced by the decay process of gluinos and squarks. Tevatron gave the bound to the later two as 300 GeV.

It is suggested that once we get the mass of the WIMPs from high energy collider, we need to compare that with the cosmological observation. There are two ways to take the measurement from the cosmological side. One is by using ultra sensitive detector to test the single event of the WIMP. However, the cross section is rather small in this process(about zb()). The other way is to obseve the annihilation of WIMP pairs. Even though the rate is very low, it is possible in some region where WIMPs have a relatively high density. Then we can observe the spectrum of the gamma ray produced in this process. The sharp cutoff of the spectrum should give out the mass of the decay origin, i.e. the WIMP.

There are something more that we can do, such as calculation of the cross section of the annihilation process of two WIMPs. There are two possible processes as suggested in this article. One decay mode is through the interaction with the bosonic partners of leptons and the other is by way of the fermionic partners of weak interaction gauge bosons, as W, Z.

 Anyway, the conclusion is, with LHC and ILC, this dark matter problem may be solved in few years.

A start in the formal thoughts

Hi, this is a new place for me to post my thoughts in sciences, especially in Physics, since many of my friends complain that I post so many physics in my other two blogs.

This blog will be constructed in English and hopefully we can have a good communication through it.