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# A mathematician's take on "what is light?"

Attached is an intriguing commentary on the scientific method through an example, written by my good friend, Luc Tartar. The specific example is that of trying to understand what 'light' might be, especially from a mathematician's point of view. The mathematician in this case is an extremely talented one, who also happens to actually understand a whole lot of physics and mechanics.

I am posting it especially for our younger members on imechanica, since I think there are interesting things to learn here. If you are an engineer or a physicist, it will not necessarily be a comfortable read, both on matters technical and philosophical. But my personal point of view is that not everything worth learning has to be within one's comfort zone. Being open-minded about learning, and recognizing when there is something to be learnt, is one of the best habits we can develop. One does not have to agree with all that is said, but the greatest intellectual progress happens when a collection of sincere, talented people operate at the boundary of their individual comfort zones - not necessarily agreeing, but definitely learning from each other.

So, enjoy!

Attachment | Size |
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WhatIsLight-3.pdf | 305.51 KB |

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## Comments

## Plasma

Thanks Dr Acharya

I enjoyed it although I did not get much at the end of it. I remember learning as a kid that there are 3 states of matter and later on realized everyone ignored plasma. And now eqn's representing such state of matter seems like a living object unrevealing some hidden stories. ..fascinating!!

Regards

Amit

## Re: Tartar's light

Tartar mentions that discontinuities in the governing equations can be dealt with by invoking the integrals -> distribution concept. Is that true when each material point has a different property?

-- Biswajit

## spatially varying coefficents

Biswajit,

I presume you mean what one could say about the limiting forms of energy in the case the material properties oscillate a lot.

The one result I know of is the following (by Tartar) - for the scalar, linear, second order wave equation with L^\inf coefficients - so only essentially bounded but otherwise could vary wildly (would accommodate what you describe) - one can take a sequence of approximating weak solutions that converge to a limit (say zero) in the sense of weak convergence - thus in the limit you have a solution that is wildly oscillating with mean zero. Then the action - the kinetic energy *minus* the potential energy evaluated along this sequence tends to the limit 0. This proves a type of equipartition where the total energy is exactly equally divided into the potential energy and the kinetic energy in the limit (note this meaning of equipartition is different from that used in statistical mechanics for discrete systems).The main issue in the above question is that the kinetic energy and the potential energy are both nonlinear functions of the state.

To evaluate the common limit of the potential energy and the kinetic energy in this case, one needs to assume more about the smoothness of the coefficients - I believe it is C^1. Then Tartar characterizes (i.e. gives a formula for) the limit by the H-measure of the sequence.

So, for the explicit formula one does not have a rigorous result for the type of material property variation you mention - but note that even a C^1 function can be made to oscillate very, very wildly.

The interesting thing about the above result (for me) is that this is probably the only rigorous result I know of where you assume almost nothing and have internal energy emerge, i.e. the difference of (the limit of the kinetic energy + potential energy along the sequence) And (the kinetic energy + potential energy evaluated on the weak limit of the sequence) is non-zero in general and we have an explicit formula for it.

I don't think there are comparable results in the system (even linear) case.

May be the above helps. I had to interpret what you meant by 'dealt with.'

- Amit

## Re: randomly varying coefficients

Amit,

You've "dealt with" the question quite well :) Weak convergence has been used to justify why PDEs with smooth coefficients are so good at predicting the behavior of structures even though there is considerable (and non-smooth) variation in microscopic properties.

Tartar writes : " ... One may have discontinuities of the coefficients, so that the partial dierential equations should be

understood in the sense of distributions, ... in the case of piece-wise smooth coefficients showing a discontinuity along a smooth interface, it is equivalent to writing the partial differential equations in a classical way on each side, and adding adapted transmissions conditions at the interface."

If you look at the literature on stochastic finite elements, people assume that one can differentiate a random variable in the standard way (and I'm not talking about discrete calculus). That has been bothering me and I wonder if there is a way of showing that such an assumption does not make any practical difference.

-- Biswajit

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## Great theory , I've already

Great theory , I've already thought a lot about the quantum wave-particle duality , I click here only for a pdf like that , thanks a lot mister Acharya . Can you post or link to Luc Tartars works ?