White Dwarfs

The existence of White Dwarfs is explained by quantum statistics – EDP or Electronic Degeneracy Pressure. Fermi-Dirac Statistics and Pauli’s exclusion principle help us understand how these old tiny stars sustain themselves.

But first some interesting facts about White Dwarfs.

1. As the name suggests, they are small in size and mass : about the mass of the Sun squeezed into the volume of the Earth. This translates to an immense density. But still not immense enough to make them a Neutron Star or a Blackhole. A White Dwarf can’t be too heavy. If it exceeds the Chandrashka limit of 1.46 times the mass of the Sun, the EDP force can no longer support the collapsing mass and it will explode and eventually become a Type Ia Supernova.
2. They are no longer fusing Hydrogen to make Helium : the fuel had been used up. When Hydrogen fusing stopped, they still could fuse Helium to make Carbon and Oxygen or even Neon and Magnesium, depending on their mass. The heat generated at this stage made Red Giants, the precursors to White Dwarfs. So they were big before becoming small. When the outer layer of a Red Giant left as Planetary Nebula, the leftover core is finally a White Dwarf.
3. White Dwarfs are old. Several billions years old. Since it takes many billions years more for them to cool down to lose their shine and turn into Black Dwarfs, we are still able to observe many of them at this time.

Now back to EDP.

When fusion stops, gravity causes the collapsing mass to fall inwards, bringing on a snowball effect which is only slowed down by Pauli’s exclusion principle : no two electrons spinning the same way can occupy the same space. The matter, although being squeezed by gravitational collapse, creates an immense pressure of several hundred billion Pascals to resist the inward falling masses.

The total energy of a White Dwarf comes from two sources : Internal energy and gravitational potential energy and they are expressed as follows :

To minimise this total energy, take the derivative with respect to R and set it to be zero. The calculated R is about 7000 km, which corresponds to the observed radius of Sirius B, one of the more famous White Dwarfs in our Milkyway galaxy.

 

Leonard

My good friend Leonard passed away in mid-July 2016.

I last saw Leonard on 28 March 2016. We met at the TeaRoom in Chinatown to listen to some AI expert talk about neural network, Monte-Carlo simulation, crazy stones and whatnot. The Go craze brought on by the Lee Sedol vs Alphago match had made us wanting to learn more about the game and the machine. We had dinner afterward and Leonard ordered a few large-portion dishes. I was happy to see him enjoy his food. We talked about how some strange moves by Alphago caught everyone off-guard, speculated whether KeJie was up there enough to take on Alpha’s next challenge. We laughed at how Lee Sedol was speechless after losing the second match and how relieved one felt after seeing him won the one-and-only precious game for the honour of humanity. We also talked about Lionel Messi. Leonard felt sorry for him that he couldn’t add the World Cup to his medal list and that made Messi a slightly under-achiever when compared to Pele and Maradona. After dinner we walked to his car and when we passed by some wine bar it reminded me of our time in Paris. I asked him if he still recalled the guy he ‘played well’ with (a codeword we understood between us as playing chess for money) in quartier latin many years back. Of course he did. Those were our most care-free days and we always had a good laugh talking about it. As always, when I didn’t drive, Leonard was kind enough to drive me home. In his moyen-de-transport (old car) We talked about the carnage in Paris, the radical islamists and how LKY had foreseen the problems many years back, and had suggested the only solution is through the effort of their own moderates. We talked about the delisting of NOL and how indecisions to buy big ships or small ships were already ringing warning bells about the company. Leonard invited me to join in some jam sessions with his NUSS band, but I felt I didn’t know enough people there it’d be a little intrusive. Then we reached my home. I bid him goodbye and that was the last time I saw him.

A week before our last meetup, we were at the Singapore Weiqi Clubhouse in Bishan Community Centre. Again it was for listening to analysis by Go experts on the Alphago vs Lee match. I witnessed for myself how popular Leonard was in this tight community of Go enthusiasts. So many of them knew Leonard by name (Peng Yeow in Mandarin was how they addressed him), eager to talk to him and show warm camaraderie. After the talk, Leonard stayed on for a while talking to a young chap about the enormous advantage  in memory capacity machines had over humans. We then proceeded for supper at a nearby food court. We were both happy and sentimental about Alphago’s victory. After Deep Blue’s win over Gary Kasparov in 1997, if one continued with using the same brute-force approach in chess-programming, we thought it’d take at least another 50 years before machine could finally  beat human in Go, given its many order of complexity over chess. So when Alphago came along it was so exciting to both of us. During those 5 games, Leonard and I followed the moves live on internet and commented via sms. I still kept some of these chat messages in my phone.

Those were our last two precious time together before he left us.

Last year it was music that brought several occasions for us to meet. The French Alumni annual dinner was an especially memorable one. Leonard had messaged me beforehand that he would ‘Sing Well’ on that day. Boy he didn’t disappoint. “Mes deux amours” was a beautiful rendition with his typical humorous jeux de mots. “My Way” told the story of us taupins only we could comprehend. He made the evening a huge success, for I couldn’t remember an Alumni dinner as enjoyable as that last one.

I also watched him and his band “The Silverstone” played music at the Guild House a few months back. He was the Bass Guitarist keeping the tempo with mathematical precision. As usual in a group he was that jovial one keeping everyone’s spirit up with his friendly and playful sense of humour. I’d always enjoyed the company of Leonard, a most unpretentious friend. We talked about fun facts on Mathematicians and Physicists, Technology, Go, Computer Chess, Gödel’s undecidability theorem, Turing’s machine, our old days in France and Singapore, human stupidity and a whole lot of stuffs. The conversation never gets into arguments or gossips or bitterness. It’s always for a good laugh. Leonard is simple in spirit, deep in thoughts and kind in person.

I missed him very much and that’s why I wanted to record some memories of him here.

The TdS equations

The TdS equations are fun to play with. Using partial derivatives and the Maxwell relations, many nice identities are discovered.

First, let’s work with S(T, V) :

Next S(T, p) :

 

 

 

 

 

Lastly S(p, V) :

If we equate the first 2 TdS equations, we can find an expression for the difference between the two heat capacities Cp and Cv (hint : write dT in terms of dV and dp).

The Gamma function

The Gamma function is defined as :

We can find an approximation for n! by the Stirling formula, which is derived this way :

One application of this approximation is in the expression for the Free Energy of a dilute gas :

(Now we know where that 1 in F come from !)

Operator equations for Superconductors

It’s easy to write down the Schrödinger equation for a superconductor in a magnetic field :

But to work on it we must be very careful with the order of operation : Always begin from the right and do not multiply operators unless it is unmistakably obvious.

In the above equation, the first round of operation should lead to :

Similarly, for the equation of continuity, carry out the operation starting from the right, one step at a time :

If we do everything right and by suggesting the wavefunction to be written as  , we can put it into the Schrödinger equation above, then after equating the real and imaginary parts on both sides to obtain two equations, and some further hard work on the operators (always be careful on the order of operation), and also recognising  we should be able to arrive at these two equations :

By making a nabla operation on each term of the second equation, and by recalling Maxwell’s equations, we will eventually get to this equation that resembles the one in hydrodynamics :

Wave Mechanics Equations (Big 5)

Here are the Big 5s of wave mechanics :

In the beginning, there was Schrödinger’s equation (Non-relativistic, spin-less) :

Then came Pauli’s equation (Non-relativistic, spin-½) :

After that, Klein, Gordon and Fock found this (Relativistic, spin-0) :

Getting nearer to the ultimate, Weyl wrote this equation (Relativistic, spin-½, mass-less) :

At last, Dirac discovered his equation, which he claimed to be cleverer than him 🙂 (Relativistic, spin-½, massive) :

The Gamma function and the Poission distribution

There is an interesting link between the gamma function (which defines the factorial function) and the Poisson distribution formula.

Gamma function is defined as follows :

The Poisson probability density formula is :

The two are therefore linked by :

Nice.

Reflection from dielectric surfaces

Here we have light going through 3 regions of different refractive indices. For example, from air, through a thin layer of anti-refractive coating, to glass. For normal incident light, what is the ratio of the transmitted intensity to that of the incident intensity ?

What is the thickness of the coating so that this ratio equals 1, i.e., 100% transmittance ?

Now we know how high-index coatings help to reduce reflection from lenses and glasses and to allow as much light to pass through them as possible.

 

Simultaneity is relative

An observer positioned at the centre of a moving spaceship, length 10m measured in his reference frame, watches 2 flashes of light reaching him simultaneously from both ends of the spaceship.

Another observer from outside the spaceship, seeing the spaceship fly past her, found that the two flashes of light were emitted with a time delay of 13 ns between them.

What’s the speed of the spaceship ?

3 ways to show that time runs faster upstair than downstair

According to Einstein, the physics a person experiences in an accelerated frame is identical to that experienced by another person in a gravitational field.

If a person has two clocks in a rocket that accelerates forward with acceleration g, the clock at the head of the rocket will run faster than the clock at the tail. Since accelerated frame is equivalent to gravity, a person on earth should see the difference in clock rates for clocks positioned at different heights.

We can see how this works by applying special relativity to light clocks, i.e., with photons from higher position emitted towards lower position  (the relative speed between the emitter and the receiver is v)  :

A second way to prove this is by conservation of energy :

The third way is by Quantum Physics :

The difference in the frequencies is very small, but still measurable. The experiment performed by Rebka and Pound confirmed this unusual prediction.