## Very Drunk Photons: A Challenge for Probability Boffins

May 5, 2009 by lukebarnes

I’ve been beating my head against a random walk problem for a few days now. I’ve become so muddled that I can’t even place it in the right formalism to begin looking for a solution.

For those who don’t know about these things, the stereotypical random walk problem involves a drunk man trying to walk home. He bravely steps away from the lamppost that he was using to steady himself. However, he is completely disoriented and thus each step is in a new direction. How far, on average, will be away from the lamppost after N steps?

The answer is the length of each step, multiplied by the square root of the number of steps. This result is ably proven here. There are, I am discovering, numerous ways to complicate this scenario, in ways that are relevant to physical situations. The particular problem I am looking at is within radiative transfer. Photons emitted from, say, stars will bounce around in the surrounding gas on their way to our telescopes. This will affect the spectrum and appearance of the source. In this case, the length of each drunken photon step is not constant, but depends on the photon’s frequency, the surrounding gas density and velocity, and good old-fashioned chance.

I’ve given the details of the particular challenge I’m facing here, or as a pdf here. I’m hoping that someone with more statistical brains than me will help me out. Feel free to ask questions in the comments.

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on May 25, 2009 at 2:04 pm |Berian JamesLuke, have you made any progress with this? I am a bit confused by parts of the description (I am looking at the html version): variously you refer to the particle scattering through ‘space’ and ‘frequency’. Is it just doing the latter, or doing both until a condition on the latter is met? Is it necessary to introduce the approximation to the Voigt profile? Have you obtained numerical results for simple cases?

Here are other comments:

– It seems like b should be folded in to the opacity distribution as the scale parameter;

– Why not set r0 = 0?; should x0 be in units of xcrit or xc?; should xcrit or xc be in units of the other?;

– Can the integrand of RII be manipulated further using a trig identity, or an expansion under the assumption that dx = x_{i+1} – x_i << x_i? Does it definitely need to be a function of both variables?

The expression for RII is the pdf? I.e. to generate the next step, one needs to throw a uniform random variable into the inverse cdf of RII? Or does the expression you give include that computation already? I will think more. I was going to write a matlab script to do a simple computation, but will wait for your reply. Also, you may have a script of your own you wish to post in the comments.

on June 2, 2009 at 11:17 am |lukebarnesIt is scattering through both space and frequency, until the exit condition is met. The approximation to the Voigt function is not necessary – use it or not as you desire. I don’t have any numerical results to guide me at the moment.

b can be folded in as desired.

You can set r_0 = 0 if desired. xcrit, x_c and x_0 are all dimensionless. R_II is difficult to manipulate. x_{i+1} – x_i <> x_c. R_II is indeed a pdf: the probability, given x_i, that x_{i+1} will lie in the range x_{i+1} , x_{i+1} +dx_{i+1}.

I don’t have a script already as such – I’ve confused myself over this so much that I can’t even write down a formal solution to throw a numerical code at. I can only do it Monte Carlo style …