Dark matter

Today I attended this colloquium on the self-interacting dark matter, and I find the topic very interesting. I will try my best to explain it here in simple terms.

1. Basic concepts

First of all, what is dark matter? As the name suggests, dark matter is some strange matter that’s “dark”, meaning that we know they are there but we just can’t see it. Sound mysterious isn’t it? Well, you may ask, if we can’t see them, how do we know they exist? The idea is very simple. Imagine that you are holding one end of a long rope that has the other end hidden behind a closed door. In general, we don’t know if there is anything at all behind the door, but if there is a force pulling on the rope, we know that something must be there, better if we know roughly how strong the pulling force is, then we will have some ideas of what might be pulling it, just like if the pulling force is weak, it’s probably not an elephant that’s pulling the rope. The same concept applies to gravity. If there is a gravitational pull, we know that something must be there, no matter if we can see it or not. With a good measurement of how strong the pulling force is, we can figure out how massive the object is that pulls us. In the case of our galaxy, if we look at some outermost stars in our galaxy, we can measure the gravitational pull that they experience by observing their motions, and to people’s surprise, the pulling force that we estimate based on the motion of these outer most stars is much stronger than we expect from looking at the visible matter! In our rope example, if we feel a tremendous pulling force but only see a kitten in the room, something is wrong! An invisible monster is hiding in the room! This is how surprising the discovery of dark matter is to scientists.

2. Rotation curves

We briefly mentioned that by studying the motion of a star, one can deduce the gravitational force that the star experiences and hence the mass of the matter that exerts the force. How exactly is it done? This brings us to the concept of rotation curves. The idea is very simple, imagine that you are trying to swing an object in a circular motion in the air through a string, daily experiences tell us that to make the object rotates faster, we have to pull the string harder. Similarly, stars rotate faster with stronger gravitational pull. In other words, how fast a star rotates about the center of our galaxy tells us how much matter there is inside the galaxy that attracts it. What’s more interesting is that if we look at how fast stars rotate at various distances away from the center of our galaxy, we get a good understanding of how matter is distributed in our galaxy. This profile of how fast stars rotates across a variety of distances away from the galaxy is a good characteristic of the distribution of matter in the galaxy and is given the name “rotation curve” of the galaxy.

3. Self-interacting dark matter

The self-interacting dark matter model is one of the many dark matter models that can explain some of the observed features in the rotation curve. In simple terms, the strength of the self-interaction between dark matter particles determines many observable features in the rotation curves. Reversely, by matching the rotation curves of different galaxy systems, we can estimate how strong the self-interaction is between the dark matter particles within the galaxy system. This is extremely useful as in one can now study the properties of dark matter at various interaction strengths by looking at different types of galaxies. What do I mean by that? Take atoms as an example, when atoms interact through low energy bombardment, an atom behaves like a solid sphere, but when atoms interact through high energy collisions, the inner structure of atoms, such as nucleus and electrons, emerge and affect the observable physics. The same may be true for dark matter particles, at different strengths of interaction, the dark matter particles may behave differently, showing their underlying structures. Therefore, to be able to study dark matter particles at different strengths of self-interaction through looking at different types of galaxies is a great way to figure out what dark matter really is.

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1. Brain magic

This is a fascinating talk that showed us the remarkable capability of our brain to support learning. Contrary to the common belief that brain doesn’t change during adulthood, and it only degrades during aging, the speaker showed that brain is constantly changing, both chemically and structurally based on our daily behaviors. There are a few points that she mentioned that I found very interesting –

  1. The brain structure changes more significantly when we are faced with difficult learning tasks, and these changes support long-term memories. Interestingly, our brain behaves just like a character in a role-playing game where the character gains the most experiences from difficult fights and unlock new skills. On the other hand, it implies that despite however much we like to do simple mundane tasks, it’s the challenging ones that make us grow. I find this perspective very valuable.
  2. After studying the brain of numerous individuals, one remarkable fact that she discovered is the huge variability in how each individual’s brain changes to support learning. In other words, each individual learns differently. There is no universal pattern of learning that works for everyone. Some may find traditional classroom teaching effective, some may find self-study effective, while some may prefer peer study. This lets me think about our education system. In fact, our education system is biased towards only one type of learning, classroom learning. The attendances for lectures, recitations, are factors in a student’s final evaluation that may affect his or her future in some way. In other words, we are selecting only students who learn well in classroom setting despite the fact that a large portion of students does not learn effectively in this manner. This is a waste of intellectual resources, the most valuable resources of mankind.

Plan to write

It only occurs to me recently that writing is something that I want for my life. For I like science, and I like to talk about science, yet the best way to communicate science is through writing. To be able to consolidate understanding of the universe into a few lines of words that everyone can understand and appreciate sounds like magic to me. I hope that, by picking up my stashed blog, I may have some slightest hope of becoming a better writer and a future science communicator.

 

 

Robotics Thought Experiment

With the ever growing field of artificial intelligence, it’s not only necessary but also crucial to evaluate what lies in front of us, the tangible future when robotics become so abundantly available that all low level labours are replaced by robotics. What’s the implications of such scenario and how will it impact our daily life? Let’s have a fun thought experiment.

Imagine that the world has been running happily as the way it is for an indefinite amount of time, until one seemingly unspectacular morning, when everyone wakes up and sees a massive amount advertisements from a company called RoboticX featuring their $99 robotics that are demonstrated to accomplish almost all low level human labour such as manufacturing daily products, undertaking construction works, etc. The Owners of manufacturing companies will think, “Jeez! It’s so cheap! Why am I still hiring workers! I don’t want to pay for their freaking health insurance!”. Owners of construction companies will think, “Jeez! It’s so cheap! Robots don’t get accident! Robot works 24-7!”. Next day, all manufacturing and construction workers lost their jobs. How many are they? Not too many, just about one third of the world population.

Next day, everywhere on the streets you will find people crying and protesting against RoboticX which is claimed to be the sole trigger of the disaster. RoboticX doesn’t even bother, and the protesters don’t seem to get much government support. Soon the protestors realize that “Wait a sec! RoboticX is getting so rich and they are paying a lot of tax! The government is also getting rich!”. Soon the protestors redirect their fire to the government, “Government is conspiring with RoboticX! They want us to lose our jobs so they can all get rich!” Stress from one third of the population is certainly strong enough to shake any government. New politicians start to emerge and get strong support by advocating “No Robot Movement”. To compete with them in president election, the party in charge is also forced to restrict the use of robots and place heavy constraints on RoboticX. The story ends with Robotic getting a bankruptcy due to heavy political constraints and lack of people support. The world is back to the old and happy way as it was for an indefinite amount of time.

Although being an imaginary thought experiment, it occurs to me clearly that our present political structure is no longer fit for a technology dominated future. The fact that all politics are local but all economics are global is one of the biggest paradoxes in the present political environment. Singapore is a good example of such paradox. As a small island with limited human resources, Singapore has to rely on importing foreign talents from nearby countries to boost its economics. At the same time it also means that many Singaporeans’ jobs are taken up by foreigners. As a local nation deemed to serve the benefits of its local people, Singapore government faces heavy stress from Singaporeans and are forced to place constraints on importing foreigner talents which may likely to slow down Singapore’s economic growth. Such paradox exists everywhere in the political world. The problem may lie in the very notion of country itself which has its origin in the prehistorical times when forming a tribe is crucial to survival, but is such an old notion still adequate to support human being’s continuous development after we have witnessed such immense advancement in the technology world? Shouldn’t we rethink about our political structure in an ever-growing global economy?

Having read so many unfortunate stories in wars, I truly look forward to a global political reform.

Interstellar

I watched this movie last week. I guess it’s now an appropriate time to write a little about it without spoiling the fun of some. Walking out the theater, I have nothing in mind but a deepest respect for the director Christoper Nolan. How remarkable is his imagination! Everything springs out of fundamental principles in physics, being creative but not absurd. How dazzling is the ring of light surrounding the gigantic black-hole! How magnificent is the journey through the wormhole, full of lavishly blossoming optical illusion! How marvelous is his world beyond the event horizon, scientifically known to permit traveling through time! The ordinary concept of distance and time vanishes in a cosmic scale, and it is exploited exquisitely in the movie with no lack of an emotional touch. What I found the most remarkable is the effort that the production team puts in to elevate ordinary people from their daily necessities to ponder upon the alluring wonders of our universe. Such contemplation, followed by a careful reflection, will surely give people a better perspective into many important aspects of our ordinary life.

An Interesting Maths Problem

There was one day when i walked into SPS room I saw this question on the whiteboard, Anyone can prove that

\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{...}}}}=3

I knew it’s a waste of time to work on this and it doesn’t seem to carry any physical significance, but I just couldn’t help thinking about it and trying to figure it out! Is it a disease?

Math1 Math2

Two Commutation Relations

In previous post I mentioned that the First Commutation Relation and Second Commutation Relation are equivalent. Here is a proof from Classical Perspective.

We assume one of them to be true. For example, we assume First Commutation Relation to be true. i.e.

[\hat{x}, \hat{p}] = i\hbar\rightarrow \hat{p} = -i\hbar\frac{\partial{}}{\partial{x}}

Classical Hamiltonian is given by

\hat{H} = \frac{\hat{p}^2}{2m} + \hat{V} = -\frac{\hbar^2}{2m}\frac{\partial{}^2}{\partial{x^2}} + V

We next compute the commutation between Hamiltonian and Position

[\hat{H},\hat{x}] = [\frac{\hat{p}^2}{2m} + \hat{V},\hat{x}]=\frac{1}{2m}[\hat{p}^2, x] =i\hbar\frac{\hat{p}}{m}

We see the operator in the last term is precisely the velocity operator \hat{v}

\hat{v}\psi = v\psi

Therefore from definition of velocity

\hat{v}\psi = \frac{\partial{x}}{\partial{t}}\psi = [\frac{\partial{}}{\partial{t}}, \hat{x}]\psi

So we conclude that

[\hat{H},\hat{x}]\psi = i\hbar[\frac{\partial{}}{\partial{t}}, \hat{x}]\psi

\hat{H} = i\hbar\frac{\partial{}}{\partial{t}}

It then follows

[\hat{H},\hat{t}] = [i\hbar\frac{\partial{}}{\partial{t}}, \hat{t}] = i\hbar

Equivalence proved.

(Note it’s only proved for non-relativistic hamiltonian, i still haven’t worked out the most general case at the moment)