Jack - Feb 12

Jim says

We talked about a couple of things during our meeting.

1. I think that there may be a back-of-the-envelope way of connecting this single particle distribution with the signal that you want to get from the laser power spectrum, or correlation of that or whatever it is. I have in mind something like saying that when the distance traveled is similar to the laser wavelength, the interference phase goes through one cycle. That would suggest to me that maybe there's a characteristic time τ such that \( \lambda^2 = \frac{2k_bT \tau}{\alpha}\) which gives you a characteristic frequency 1/τ in the power spectrum. Or something like that.

2. If you do a visualization like the one on the brownian motion wikipedia page, then you may be able to use it to show how to simulate that single particle distribution, at least in principle. It may even let you get the right time dependence, namely \( \sigma^2 = C t \) for some constant C.

Jack says

Oops, I forgot to make this page. Sorry, Jim, if you were looking for this earlier and it wasn't there.

Anyway, I found that the distribution for a single particle in x is \( \sigma^2 = \frac{2k_bTt}{\alpha}\). Since there is no preferential direction, it is the same for both y and z as well. I got this info from the PDF attached in pages 26-28 (solving the Langevin equation). This number is actually in the PDF for the actual experiment, I just didn't understand it there.

So for the simulation, t should always be \(\delta t\), the arbitrary time we choose for a single step. So for every step we pull from this distribution and move the particle that distance relative to its current position. We also have to keep track of the particle's original position for purposes of the vector r, used to determine the phase.

One of my current concerns is the laser. If we have these particles moving in some cube volume, I suppose the laser would be a cylinder going through it. The light scattering would only occur from the particles in the light, but I am uncertain how particles leaving and entering this cylinder of light would effect the scattering. I suppose leaving is simple, we just have that particle no longer contribute to the intensity of the scattered light. But I don't know what to do for particles that start outside the light that would later enter it.

Finally, I had an idea while I was looking for this distribution that might be cool. I thought that instead of just finding a distribution and using it, I could construct it from a simulation of a single particle interacting with the molecules around it. I thought that was very clever until I realized that the beads I'm dealing with undergo 10^14 collisions per second. While I certainly can't simulate that, at least I don't believe I can, I still think it might be fun to simulate a large particle undergoing Brownian motion as a result of collisions with a bunch of smaller particles. I don't think I could use anything from that in the rest of my program, but it would be really cool.

Honestly, maybe I shouldn't even try and do a 1:1 replica of my experiment in my code, but instead I go and simulate a much smaller number of light-scattering particles which are actually interacting with each other and a bunch of smaller particles. The calculation for light intensity would still be the same (albeit simpler). The results wouldn't be worth comparing to my actual experiment, but if I could have an actual simulation of interacting particles, especially if I could visually show the particles bouncing off of each other and stuff, it might be a better plan component than just pulling a few thousand numbers from a distribution, plugging them into a function and having it spit out a handful of graphs.

attachments

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	12._Maxwell_Boltzmann__Brownian_motion.ppt.pdf	Sat May 04 2024 04:17 am	2.8M