Proving Einstein Wrong…ish: Measurement of the Instantaneous Velocity of a Brownian Particle

Last summer, there was a fair bit of hype about a paper from Mark Raizen’s group at Texas which was mostly reported with an “Einstein proven wrong” slant, probably due to this press release. While it is technically true that they measured something Einstein said would be impossible to measure, that framing is a little unfair to Einstein. It does draw media attention, though…

The experiment in question involves Brownian motion, and since I had to read up on that anyway for something else, I thought I might as well look up this paper, and write it up for the blog.

OK, so what did they do that Einstein said they couldn’t? The title pretty much give it away: they measured the instantaneous velocity of a particle undergoing Brownian motion. They made very careful measurements of the position and velocity of a tiny glass bead suspended in air, and showed that they don’t fit the prediction of Einstein’s model of Brownian motion.

Let’s pretend that I’m too lazy to click that link above and read about Brownian motion, and have you explain it quickly here. Brownian motion is a sort of jittering motion of small particles suspended in a fluid. The motion was observed by lots of people, but takes its name from the British botanist Robert Brown, who was the first to rule out the presumed explanation that had been believed to be the cause, namely that the jittering was the motion of tiny living creatures.

What’s this got to do with Einstein? Are these things jittering at the speed of light, or something? Einstein’s most famous for his work on relativity, but his background was in what we’d now call statistical mechanics. His Ph.D. thesis, one of his five great 1905 publications, was about the diffusion of molecules in a solution, and provided a clever way to estimate Avogadro’s number. Following closely on that work was a paper explaining Brownian motion– Abraham Pais says in Subtle Is the Lord… that it was finished just 11 days after the thesis. Einstein’s model for Brownian motion made a quantitative prediction that could be directly measured, and put together with the work of a few other people at around the same time, this helped conclusively settle the question of the existence of atoms.

Wait, what? I thought people knew about atoms in, like, ancient Greece, and stuff. The name comes from the atomist philosophies of the ancient world, but up until the early 20th century, there was active debate about whether the notional atoms used in physics and chemistry were real microscopic particles or just a convenient mathematical fiction. Einstein’s work on Brownian motion helped conclusively prove that atoms are real physical entities.

OK, how? Einstein showed that the characteristic jittering of Brownian motion could be explained as collisions between the atoms making up the fluid colliding with the larger particle and causing it to move. Any object in a fluid is constantly bombarded from all sides by the background atoms, and each collision causes a corresponding change in the motion of the observed particle. When you carefully work through the implications of this, you find that on average, the displacement of the object from it starting position should increase in a very particular way– as the square root of the time since the measurement started.

And Einstein said this could never be measured? No, what Einstein said was that this average displacement was the only thing that could be measured. That is, he said that the only thing you could hope to measure was the aggregate effect of vast numbers of atomic collisions, through the displacement, and not the instantaneous velocity changes caused by the collisions.

And that’s wrong? Well, it was perfectly true in 1905. The technology for doing this sort of thing has advanced quite a bit over the intervening 105 years. Hence this paper.

All right, so how did they measure the instantaneous velocity that Einstein said couldn’t be measured? The key issue is that Einstein’s model only holds for relatively long times. If the time over which you measure the displacement is much longer than the time between collisions, Einstein’s model is the only way to look at Brownian motion. If you can look at times comparable to or shorter than the time between collisions, though, you should see a different sort of behavior, which was described mathematically by a bunch of different people, starting with Langevin in 1908.

This is essentially impossible for the traditional Brownian motion system of small bits of stuff floating in water, because water is so dense that the time between collisions is extremely short. In this paper, they say that direct measurement of this “non-diffusive” motion would require position measurements with a resolution of a couple of picometers, every ten nanoseconds, which is an awfully tall order.

So how did they do it? They didn’t. They moved to a different system, where the technical requirements aren’t nearly so extreme. Rather than looking at particles in a dense fluid like water, they looked at small glass beads floating in air, which has much lower density, and thus a much longer time between collisions.

Yeah, but how did they get glass beads to float in air? Last I checked, glass is heavier than air. They didn’t look at free-floating glass beads, but at bead that were suspended in a particular region of space, trapped by optical tweezers. The tweezers are a pair of tightly focused laser beams, that trap small particles in the focus of the beams.

But if it’s trapped, it’s not really moving, is it? It’s trapped, but not that tightly. The bead can move around by small amounts, and this micromotion will be Brownian in nature, caused by collisions with the air inside their apparatus. The trap keeps the bead from falling under gravity, or otherwise wandering off by a large amount, but allows Brownian motion. It also makes it possible to measure the position of the bead very precisely and very quickly.

How does that work? They split the beam in half, and send each half to its own detector. If the glass bead is right in the center of the focus, the two beams should have equal intensity. If it moves to one side or the other, it acts like a tiny lense deflecting the light to one side. By measuring the difference in intensity between the two halves of the beam, they can reconstruct the side-to-side position of the glass bead.

And what do they see? This:

That’s the trajectory for trapped beads at two different pressures. The top graphs show the position as a function of time, while the bottom graphs show the instantaneous velocity, obtained from the position data. The red points are just a hair under “normal” atmospheric pressure, while the black points are at moderately low pressure, by physics standards. These should give different values for the time between collisions, and thus somewhat different behavior for the Brownian motion.

I dunno. Those look pretty similar to me. Which is why you have to do some statistical analysis of the motion over a long time. They did 40,000,000 measurements of the position of the bead at each of these two pressures, and used them to determine the “mean squared displacement,” which is the thing that Einstein’s model of Brownian motion predicts. The results look like this:

The points are the experimental data, while the dashed lines in the upper left are the prediction of the usual long-time model introduced by Einstein. As you can see, they don’t remotely agree. The solid lines, which fit the data very nicely, are the prediction of a more sophisticated model using the Langevin equation, which allows you to describe the motion at very short times.

The key thing to notice about this graph is that the slope of the line through the experimental points over most of the range is twice as steep as the slope of the long-time prediction. This indicates that, at short times, the particle is moving ballistically– that is, it’s flying freely in between collisions.

OK, but why don’t the lines ever get near the long-time model? I mean, wouldn’t you expect it to agree with Einstein’s model if you look over longer time intervals? In an ideal world, yes. In the real world, you’re limited by the need to trap the particles. You can see from the graph that both curves kind of flatten out after 10^-4 s, at a mean square displacement of right around 1000 nm². That’s because the particles are trapped by the optical tweezers, and can’t move farther than that.

If you didn’t need the trap to keep the glass beads from falling, you would see the experimental curve sort of bend downward, and at long times– 10^-2 s, say– it would match the dashed lines pretty well. As it must, given the success of the Einstein model in describing Brownian motion in much denser fluids.

So, to recap, they measured the Brownian motion of a tiny glass bead in an entirely different fluid than Einstein was thinking of when he wrote about the subject, by using experimental techniques he couldn’t possibly have anticipated? Pretty much. Like I said, it’s a little unfair to say this “proved Einstein wrong,” but his name does get people in the media to sit up and take notice of your press release, so I can’t really fault them for framing it that way.

So, what’s the point of all this, anyway? Are they just scoring cheap points off Einstein’s celebrity? Hardly. First of all, it’s a really cool experiment in its own right. More importantly, though, the ability to track the position and velocity of the bead as closely as they do opens the possibility of using some clever tricks to control that motion, slowing the velocity of the particle and thus “cooling” the bead to very low effective temperatures. And possibly to the point where quantum effects become important.

If they can reach this quantum regime, then the measurements should diverge really dramatically from not only the long-time model used so successfully by Einstein, but also from the Langevin model that works here, which is still essentially classical. That would be extremely interesting, and potentially a way to get at the transition between the quantum behavior seen in microscopic systems and the classical behavior of objects at more ordinary length scales.

It always comes down to this macroscopic quantum behavior business with you, doesn’t it? every time we do one of these ResearchBlogging posts, it ends up having something to do with seeing quantum behavior in big objects. That’s overstating things a little bit, but it is a recurring topic. But that’s because it’s the subject of a lot of active research, mostly because it’s one of the biggest of the Big Questions we can address experimentally. Lots of people are working on this stuff because explaining how the classical world we see around us every day arises from the bizarre quantum behavior we see when we look at single atoms and electrons is one of the great mysteries of modern physics.

And, coincidentally, the place where Einstein was really wrong. Well, yeah. But that’s a topic for another day. Anyway, I hope this has explained why this experiment got the attention it did, and why Brownian motion still attracts interest after all these years.

Yeah, it was pretty OK. Thanks. No problem.

Li, T., Kheifets, S., Medellin, D., & Raizen, M. (2010). Measurement of the Instantaneous Velocity of a Brownian Particle Science, 328 (5986), 1673-1675 DOI: 10.1126/science.1189403