Tikalon Blog by Dev Gualtieri

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Maxwell, Boltzmann and Brown

September 14, 2011

In the early days of physics , many things needed to be taken on faith (i.e., belief in theory ), since direct experimental observations of the principal mechanisms were not possible. One case in point is the Maxwell-Boltzmann distribution of molecular velocities in a gas that describes the mean pressure . It isn't possible to observe the actual molecules and verify the distribution, but everything seems to work, so we push the verification into the future.

The future in that case was more than one hundred years distant, when in 2010, physicists from the Center for Nonlinear Dynamics and Department of Physics , the University of Texas at Austin (Austin, Texas), did an indirect, but satisfying, measurement that used Brownian motion .[1-2] Brownian motion is the random motion of particles in fluids that's presumably caused by their being impacted by the molecules of the fluid.

Brownian motion was first observed in 1827 by the botanist , Robert Brown , who was doing microscopic studies of pollen grains . Pollen contains extremely small, subsidiary particles, the amyloplasts and spherosomes , that Brown saw to be in a continual jittery motion.

Good scientist that he was, he repeated his observation with inorganic particles to rule-out any life-related principle, but he could not suggest an explanation. Brownian motion is still an active research study after all these years. The arXiv preprint server had 194 articles with "Brownian" in their title posted in 2010.

Brownian Motion in Three Dimensions

Brownian motion in three dimensions.

This simulation was done using the Wiener process , named for Norbert Wiener .

(Image by T.J. Sullivan, via Wikimedia Commons).

The Texas team analyzed the Brownian motion of micrometer-sized silica glass beads in air , held in place by an optical tweezer , over a wide range of pressures. They were able to measure the instantaneous velocity of a Brownian particle, not just the sloppy average you would see by eye under a microscope, called the diffusive regime.

They were able to measure the positions of the particles on a time scale small enough to observe their ballistic regime; namely, the regime between individual "hits" by the gas molecules.[1-2]

Of course, such data are sufficient to verify the Maxwell-Boltzmann distribution. The measurement is still indirect, but it's much better than a strict reliance on faith. The Texas experiments were aided by the fact that the fluid medium, air, was rarefied and had a low viscosity .

The Texas team, assisted by colleagues at the Ecole Polytechnique Fédérale de Lausanne, have since extended their observation to ballistic Brownian motion in a liquid .[3]

This was the first measurement of Brownian motion in a liquid in the ballistic regime. A liquid is more difficult for this type of experiment, since liquid viscosities are about fifty times that of air. To analyze the Brownian motion in the liquid, they captured a single particle in an optical trap and measured its position with sub-Angstrom spatial resolution in a 75 MHz bandwidth .[3]

The position measurement used a fast photodetector coupled via an optical fiber bundle, and it could resolve 20 picometers . This is about the size of a hydrogen atom . Such measurement extremes are required, since a one micrometer silica bead in water will move a nanometer every microsecond .

This is predicted by the Stokes-Einstein relation ,

D = (k_BT) / (6 π η r),

in which D, the diffusion constant , is given by k_B, the Boltzmann constant , T, the absolute temperature , η, the viscosity , and r, the particle radius.

After taking the particle inertia and the hydrodynamic effect of the fluid into account, their data were in excellent agreement with theory.[3]

Robert Brown, Botanist

Portrait by Henry William Pickersgill (1782-1875) of the Botanist, Robert Brown, as a young man.

Brown discovered Brownian motion of particles in fluids.

(Via Wikimedia Commons).

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