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  • Akarsh Simha 12:21 am on June 13, 2009 Permalink  

    Weeks 3 and 4 at TIFR 

    Ahh, a month has passed without me achieving the progress that I wanted to achieve in a month. I hope I will use the other half of the month very efficiently and effectively. All it takes is a little discipline, that I don’t have, but this might be a good chance to cultivate that.

    Week 3 was when I learnt about the Langevin Equation, Markov Processes, the Fokker-Planck equation, the Master Equation and the Boltzmann Transport Equation. I referred Balescu’s Equilibrium and Non-Equilibrium Statistical Mechanics for the earlier three topics, and some random article I found on the internet for the Master Equation. As for the Boltzmann Transport Equation, I just looked at Wikipedia without going through much detail. So I learnt some non-equillibrium statistical mechanics, but this stuff is yet to sink in. I’m going to make some notes about it very soon (started this today) and that should get things in place.

    Loganayagam (Dr. Shiraz’s doctoral student) helped me solve the “stirring of a colloid” problem, as I’d like to call it – the problem of Brownian Motion of a colloidal particle with an external force that has a curl. The idea is that there’s no detailed balance in the non-equillibrium system. Of course, we could solve only toy problems in 2 and 3 dimensions, while we’d have ideally wanted to solve it in an uncountably infinite number of dimensions! In the 2-D toy problem of Brownian diffusion under a force with a constant curl, we encountered an equation whose solutions were Bessel functions and that’s when I learnt about the Hankel transform. I thought it was pretty amazing. Loganayagam converted it into a Fourier transform in no time. Incidentally, the very next day, my co-VSRP Kaushik encountered the same equation and the same situation in a problem related to a wave propogating in AdS space. (He’s working on AdS/CFT correspondance)

    I also read a bit of Katepalli Srinivasan’s Physics Today article on Turbulence, which gave me more physical insight. I still need to complete reading it.

    I went home last weekend because I was feeling rather home-sick. I spent a wonderful time at home, two complete days with my family for a change (it’s usually astronomy sessions, visiting the planetarium, BAS meetings, meetings with old friends or something of that sort on a conventional weekend, where I see family for ~3 hours!). On the flight down to Bangalore, I tried working out the Correlation-generating functional of a free Quantum particle after Prof. Wadia gave me the path integral representation. The path integral was hard to evaluate, but waiting at the airport being the time when I’m most productive, I came to a point where it looked like I hadn’t got anything. However, that exercise made me comfortable with evaluating path integrals by first principles.

    Week 4 started with me playing further with the toy problem. I became a little more comfortable with that form of the Fokker-Planck equation. I worked with the Burgers’ equation (which is the same form as the Navier-Stokes equation in 1-D). We had to find the determinant of an infinite-dimensional operator, in the context of which I proved that det A = exp( Tr ln A ). I also learnt about the WKB approximation in the context of path integrals from Prof. Spenta and made some notes on it, but am yet to use it. I also tried to evaluate the determinant of the operator we had at hand, and there are still some minor glitches. I also verified that the Cole-Hopf transformation, which is mentioned in the Wikipedia page on Burgers’ Equation linearizes the viscous Burgers’ Equation – and this had some sort of “guage symmetry” which kept us confused for a while. I’m glad I could find that out!

    On Thursday, I joined the string theory group for lunch. While string lunch doesn’t make much difference to me (except I don’t pay the already highly subsidised amount for lunch), it was nice to many of the string theorists together. Everyone in the department, except Prof. Sandeep Trivedi, were present. To “make up” for his absence, Prof. Gopakumar from HRI was present, and so was Prof. Atish Dhabolkar. Once in a way, when I heard things like “the spectrum of a closed string with the left moving and right moving stuff”, I used to think ‘Hey, I know what that means’. (Thanks to Dr. Suresh’s boltzmann talk). I’m now so used to hearing the word ‘Gauge Theory’ (I usually get the spelling wrong), that I feel as comfortable with it as I would if I knew what it actually meant. (Well, I know it’s a “Quantum Field Theory” with some sort of Gauge Invariant action, but it takes more than words to know something in Physics.)

    This week, I attended three talks – one by artist and TIFR employee Sukant Saran on art-science interaction and a popular talk on the “Ubiquity of Symmetry” by Prof. Ramanan, a former professor at the TIFR Math department with well known contributions to mathematics. I also “half-attended” a talk by Dr. Atish Dhabolkar on Mock Modular Forms – obviously, I didn’t understand much, but I learnt at least something.

    I felt that Week 4 was not upto my full working potential. I feel I should’ve worked more. Will try and make sure that Week 5 is going to change that. It’s just that there’s a point in-between reading something where you get “frustrated”, and I end up breaking for long hours at that point. I also found out I take a really long time to see and reply to mail – that needs to be optimized a bit, which means more procmail filters to cut out mail that I amn’t interested in.

    On a side note, I’m trying to read some Quantum Field Theory, so that I can at least get an idea of what the subject is like. Dr. Suresh recommended the book by Srednicki, in which I’m slowly finishing the 1st chapter.

    On other aspects of life, I made some contributions to KStars this week – particularly those pending from GSoC 2008 and there’s a little more stuff to do. Prakash (whom I’m mentoring under GSoC this summer) is pretty much taking care of himself and needs my support only once in a way – and it has been very enjoyable working with him. As I know from earlier experience with him, he learns things really fast – so I had to explain things to him initially, but I no longer need to. He seems to be enjoying his summer work and is rather passionate about it. I realised I needn’t push him to do stuff, which I thought will be the hardest part of the game when I told Lydia that I’d chosen Prakash; now he needs to push me to do stuff!

    While I’ve realised that I’m more productive at work when I’m back in my flat, I took a break from that yesterday and played the flute for a whole 75 minutes! I put on the shruti looping on my laptop, and it started off with Behag, followed by a bit of Revati, followed by a pseudo-“RTP” in Kalyani, ending with my favourite – kApi. Of course, my flautistry is still very n00bish, but I can at least manage to entertain myself with it! I’m pretty sure my neighbours will be cursing my existance, that too now that the flute is pitched at G#.

    Yet again, I’m going to Bangalore for the weekend – not to visit home – but to attend a workshop by (to be Dr.) Vishnu Reddy on “Asteroids, Comets and Meteorites”. Vishnu has promised to teacch us some professional astronomers’ techniques, so I thought I’d learn something by attending it. Pavan, Vivek and Naveen Nanjundappa are doing a great job at organizing it – and for a change, I’ve been at the “customers’ end” rather than at the organizers’ end. I’ll be returning to Bombay on Monday evening (considering that my weekend will be rather intense) and I hope to finish notemaking from Frisch’s turbulence and get to the place where he constrains correlation functions using symmetries by the time I’m back at TIFR.

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    • Radha 12:08 am on June 22, 2009 Permalink | Log in to Reply

      This comes from an exTIFR… experimental cond. matter also interested in music….
      So you are a flautist? And the RTP was by…? (That is very important… equally imp the accompanists ‘coz generally RTPs are followed by Thani!)
      How did I land up in your blog? Thanks to Boltzmann!

    • Akarsh 10:05 am on June 24, 2009 Permalink | Log in to Reply

      I’m only an amateur flautist. I am far from playing well, let alone professionally :).

      You mean you found me on the Boltzmann Yahoo! Group?

  • Akarsh Simha 4:03 am on June 5, 2009 Permalink  

    Metric Tensor 

    After a discussion with Henry de Valence (hdevalence) on IRC, who wanted to know something about the metric tensor, I thought I should put down some stuff that I explained to him. This is like a “Why a metric tensor” article.

    A quadratic form, is a homogeneous polynomial of degree 2, like

    x^2 + y^2 + 2xy

    We can always represent such a polynomial of degree 2 by a symmetric matrix:

    [x y][1 1][x]
         [1 1][y]

    [I’m sorry, I still don’t know how to type matrices in LaTeX, and I want to put-off learning that for a later time]

    So, let’s try to represent the distance between two points, which is a quadratic form, as a matrix. Let’s consider very small distances – between two closely separated points: (x, y, z) and (x + dx, y + dy, z + dz) where dx, dy and dz are “very small” distances.

    Ordinary 3-D “Euclidian” Space, Pythagoras theorem reads:

    ds^2 = dx^2 + dy^2 + dz^2


    [dx  dy  dz][1 0 0][ dx ]
                [0 1 0][ dy ]
                [0 0 1][ dz ]

    Special relativity tells us that time is just like a “coordinate” and we shouldn’t treat it separately. Space and time are now put together and called “space-time” and there are 4 coordinates. The “distance” (Space-Time interval) between two points (t, x, y, z) and (t + dt, x + dx, y + dy, z + dz) is defined like this:

    ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2


    [dt dx dy dz][ -c^2           ][ dt ]
                 [       1        ][ dx ]
                 [           1    ][ dy ]
                 [              1 ][ dz ]

    Just like distances don’t change under rotations, space-time intervals don’t change under “Lorentz Transformations” in Special Relativity, which is why it is useful to have the notion of such a weird distance. Ordinary 3-D distances do change under “Lorentz Transformations” – as is demonstrated by the effect of “Length Contraction”.

    The matrix that we’ve written in the above is (a representation of what is) called a “metric tensor”.

    We can write the metric tensor for the surface of a “two-sphere”. That refers to the surface of the ordinary sphere in three dimensions, that we are accustomed to. It’s called a two-sphere because we are talking about the surface of a sphere in 3-D – which is 2-dimensional. Note that this metric varies from point to point (it’s a function of r and \theta) and hence, is a tensor field:

    ds^2 = dr^2 + r^2 d\theta^2 + r^2 sin^2(\theta) d\phi^2


    [dr d \theta d \phi][ 1                          ][ dr       ]
                        [    r^2                     ][ d \theta ]
                        [          r^2 sin^2(\theta) ][ d \phi   ]

    Finally, any “Riemannian Manifold” (curved spacetime) can be described by the metric tensor and its derivatives. So the metric tensor plays a central role in General Relativity. Formally, space-time intervals are written in the following manner:

    ds^2 = dx^\mu dx^\nu g_{\mu \nu}

    The above follows a notation called “Einstein Summation Convention”, which my 17-year-old friend already knew, which can be read from other sources.

     
    • Muthusubramanian NV 1:33 pm on June 4, 2009 Permalink | Log in to Reply

      Machee, use

      \begin{bmatrix}
      1 & 0 \\
      0 & 1 \\
      \end{bmatrix}

      This is an identity matrix of order 2. very simple 🙂

    • Naveen 9:54 am on June 11, 2009 Permalink | Log in to Reply

      typo: (t + dt*, x + dx, y + dy, z + dz)… 🙂

    • Akarsh Simha 10:21 am on June 11, 2009 Permalink | Log in to Reply

      Thanks. Fixed.
      M-x weblogger-start-entry
      [Put password]
      M-x weblogger-prev-entry
      C-s t + dy
      C-? t
      C-x C-s
      C-X C-c

      😉

    • SG 9:14 am on June 14, 2009 Permalink | Log in to Reply

      These as some remarks with respect to the sphere that you mention.

      1. First, it is either a 2D sphere or a sphere in 3D.

      2. The line-element in the non-LaTeX form is not correct. The row to the left should be [ dr d \theta d\phi] and a similar correction needs to be done in the column entry as well. Is there any particular reason to choose this non-LaTeX method of writing matrices?

    • Akarsh 2:41 pm on June 16, 2009 Permalink | Log in to Reply

      @SG: Ouch! Yes.
      Thanks for the corrections.
      I chose the non-LaTeX method because I didn’t know how to format matrices side-by-side in LaTeX and decided to put-off learning for a later stage.

      • SG 4:18 pm on June 18, 2009 Permalink | Log in to Reply

        Here is how you do it in LaTeX:
        $$
        \begin{pmatrix} dr & d\theta & d\phi \end{pmatrix}
        \begin{pmatrix}
        1 & 0 & 0 \\
        0 & r^2 & 0 \\
        0 & 0 & r^2 \sin^2\theta
        \end{pmatrix}
        \begin{pmatrix} dr \\ d\theta \\ d\phi \end{pmatrix}
        $$

        The above example illustrates the use of pmatrix environment (there are other ones like matrix etc.).

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