A Candle in the Dark

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Archive for the ‘videos’ Category

Storm – Tim Minchin

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Worth watching, if you haven’t seen it before.

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Written by parseval

October 14, 2010 at 8:01 am

Posted in rant, videos

Terror in Mumbai by Dan Reed

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You should watch this revealing documentary by Dan Reed on the recent Mumbai terror attacks. From the Mirror,

Dan Reed’s exhaustively researched film builds up a chilling, detailed picture, using CCTV footage, survivors’ testimonies and from police on the scene – who, with their outdated rifles, were powerless to stop the gunmen.

As compelling as these accounts are, it’s the phone conversations between the gunmen and their controllers back in Pakistan – recorded by the Indian intelligence services and aired here for the first time – that are the most revealing.

I have a lot of emotions right now, but I’ll try to keep this short. The actual recorded telephone conversations and CCTV images are gruesome to watch, but highlight the complete lack of preparation our security forces had in tackling such an attack. There’s footage of atleast a dozen police officers running away in the train station together, instead of trying to fire at the terrorists.

It’s also almost surreal listening to the recorded conversations between the terrorists and their handlers, as they receive *live* instructions on where to attack and what to set on fire *after* getting into the hotel. Was this information even available during the counter-terrorism response? If this was known, why were live feeds of the security activities allowed to be broadcast by the TV networks? I can only hope that our security forces have learned a lot from this incident.

There’s also a conversation with Kasab which is shown, where the police questioner asks him about his motivations. It’s easy to label the terrorists as inhuman, and devoid of human emotion, but the religious conviction which they’ve been brainwashed with in order to justify their acts is incredibly sad to watch. The recorded telephone conversations show how they are repeatedly reminded by their handlers that they should not get arrested and should die, and that God will reward them. It’s going to be education, rationalism and literacy which can prevent such people from turning into terrorists.

Written by parseval

July 24, 2009 at 6:12 am

Posted in events, terrorism, videos

Homeopathy is dangerous

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This is an example of the harm that can happen when alternative ‘medicine’ is tolerated as an acceptable treatment by society. In fact, I think that the whole label of ‘alternative medicine’ is silly. There’s medicine which is shown to work in controlled studies, and there’s garbage like homeopathy.

P.S Have I already told you that India has a national department which ‘studies’ homeopathy?

Written by parseval

May 5, 2009 at 2:19 am

Posted in pseudoscience, videos

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Crikey!

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Written by parseval

December 4, 2008 at 2:44 am

Posted in people, videos

When solitons collide

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Adapted from notes I copied during a course I attended

Consider a 1+1 dimensional scalar field \phi(x,t) where x corresponds to the space dimension, and t corresponds to the time dimension. The Lagrangian and the Hamiltonian energy density for the field are given by,

\mathcal{L}=\frac{1}{2} \partial_{\mu}\phi \partial^{\mu}\phi - U(\phi)

\mathcal{H}=\frac{1}{2}(\partial_0 \phi)^2 + \sum_i\frac{1}{2}(\partial_i \phi)^2 + U(\phi)

Here, the Einstein summation convention is used, where repeated indices are summed over. \partial_{\mu} is used to denote the four-vector (\frac{1}{c}\frac{\partial}{\partial t}, \nabla)

The evolution of this scalar field is governed by the Euler-Lagrange equation,

\frac{\partial \mathcal{L}}{\partial \phi} = \partial_{\mu}\left(\frac{\partial \mathcal{L}}{\partial \left( \partial_{\mu}\phi \right)}\right)

Plugging in Lagrangian for the scalar field I’ve written above, the E-L equation reduces to,

\Box \phi = -\frac{\partial U(\phi)}{\partial \phi}

where, \Box is the D’Alembertian. ie, \Box \phi = \phi_{tt}  - \phi_{xx}

A specific example, where the solutions of the E-L equation have interesting properties, is given by the potential,

U(\phi)=\frac{\lambda}{2} \left(\phi^2-a^2\right)^2

The lowest energy solutions which satisfy the E-L equations are known as the classical vaccum. In this specific example, they are given by

\phi(x)=a
\phi(x)=-a

You can see that these solutions correspond to the minimum of the potential. Additionally, the total energy is also minimized by these solutions, because the contribution of the space and time derivatives to the Kinetic Energy density term is zero.

Apart from these vacuum solutions, there exist finite energy solutions of the E-L equations for this potential. From the criteria that the energy must be finite when the Hamiltonian energy density is integrated over x, such a solution must tend to the classical vacuua solutions at the extremities.

Such a solution is known as the kink solition, and is given by,

\phi(x)=a \tanh{\mu x}

where, \mu^2=\lambda a^2

You can check that this solution satisfies the Euler-Lagrange equation by plugging it in. Note that the solution is time-independent. Furthermore, when x \rightarrow \pm \infty, \phi \rightarrow \pm a. Therefore, the solution interpolates between the classical vacuua solutions.

To understand why it’s called a kink soliton, let’s plot its functional form, and the corresponding Energy density. For the plots below, I’ve set a=1, \lambda=1

Plot of the kink soliton solution

Plot of the kink soliton solution

Hamiltonian density for the kink-soliton solution

Energy density for the kink-soliton solution

As you can see, the solution tends to the vaccum solution (here, a=1) at either end and the energy density appears as a stationary wave packet which is like a kink in a rope.

Now, there are certain global symmetries that Lagrangian has, which are absent in the kink soliton solution. For instance, if we set x \rightarrow -x, the Lagrangian in Eq. (1), and consequently the corresponding action, is invariant. However, the solution \phi(-x) \rightarrow -\phi(x), is a new solution and interpolates the other way (ie, \phi \rightarrow \mp a as x \rightarrow \pm \infty. This is called the anti-kink solition, and has the same energy as the kink soliton.

There are other symmetries where the Lagrangian is invariant which give rise to new solutions. For example, a spatial translation will give a solution \phi_{new}=\phi(x-x_0), which shifts the location of the center of the kink. A lorentz transformation x \rightarrow \gamma(x-vt) will give a new time dependent solution, \phi_{new}=\phi\left(\gamma(x-vt)\right), where \gamma=1/\sqrt{1-v^2/c^2}. This corresponds to a kink which starts out at the origin and moves to the right with a constant velocity.

Therefore, the most general form of the kink soliton can be written as,

\phi_K(x,t) = a \tanh{\mu(x-x_0-vt)}

This corresponds to a soliton which is initially centered at x_0 and moves with a velocity v to the right. The anti-kink soliton corresponding to this is simply -\phi_K(x,t)

Now, consider a box which has both a kink and an anti-kink solition placed away from each other and with velocities in opposite directions. What happens to them as they approach each other? They can’t pass each other, otherwise they’d violate the boundary condition (remember that the kink and anti-kink solitons tend to different vacuum solutions at either end). So you’d expect them to bounce off each other.

In fact, we can even numerically test this, because we know that they are governed by the Euler-Lagrange equation. If \phi(x,t) represents both the kink and anti-kink soliton, then initially, one has

\phi(x,t) = \left\{\begin{array}{c} a\tanh{\mu(x+x_0-vt)} \quad x \leq 0 \\ -a\tanh{\mu(x-x_0+vt)} \quad x \geq 0 \end{array}\right\}

then, the E-L equation can be rewritten as two first order differential equations

\left(\begin{array}{c} \partial_t \phi(x,t) \\ \partial_t \pi(x,t) \end{array}\right) = \left(\begin{array}{c} \pi(x,t) \\ \partial_x^2\phi(x,t) - \frac{\partial U(\phi)}{\partial \phi}\end{array}\right)

where, \pi(x,t) = \partial_t \phi(x,t) is the momentum density

To solve the system of equations above, I discretized the spatial derivative using the central difference method, and used a fourth-order Runge Kutta solver with corresponding intial and boundary condition discussed above. You can see how the kink and anti-kink soliton bounce off each other in the video below.

Kinky, ain’t it?

Wikipedia has a gallery of images on solitons for the Sine-Gordon model, where a Sine potential is used instead of the quartic potential. There are also a lot of nice websites on the fantastic properties that such non-linear PDE’s exhibit.

See Also
Sine-Gordon Solitons and Soliton Collisions:
Pendulum Model

Soliton-Lab Art Gallery

Written by parseval

November 30, 2008 at 6:20 am

Posted in physics, science, videos

Sunday Video

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For more, check out this nice introductory post by Prof. Arunn Narasimhan on optimization and genetic algorithms.

Written by parseval

July 12, 2008 at 9:38 pm

Unfair, unbalanced and completely whacko

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Written by parseval

May 29, 2008 at 8:12 pm

Posted in politics, videos