Ballistic motion of a Brownian particle

Mark Raizen and Tongcang Li (Physics Today, January 2015, page 56 ) describe a remarkable experimental observation of short-time ballistic motion for a single Brownian particle in an optical trap. Readers of their Quick Study may be surprised to learn that “the measurement Einstein deemed impossible” was accomplished nearly 25 years ago by Jixiang Zhu and coworkers, ¹ for an ensemble of untrapped Brownian particles. In that case, the long-time motion is diffusive, which is the actual problem considered by Albert Einstein, instead of bounded, the situation studied by Raizen and Li. In the 1992 work by Zhu and coauthors, dynamics down to the scale of 1 Å and 25 ns were probed by a dynamic light-scattering technique in which particle motions cause infinitesimal Doppler shifts in multiply scattered light that are resolved by intensity interferometry. Zhu and coworkers were thus able to capture both motion deep inside the short-time ballistic regime and the predicted long-time tail in the crossover to diffusion.

This work contains plenty of intriguing physics. Related ballistic-to-diffusive behavior can be found in the transport of electrons in conductors, phonons in solids, and photons in opaque media. But perhaps even more prominent in physics today is the nature of an intervening subdiffusive plateau that develops for disordered liquids, colloids, bubbles, grains, and so forth that are on the verge of jamming.

For the Raizen–Li case of a single harmonically bound Brownian particle, it is possible to construct a mechanical analogue. A single centimeter-scale sphere driven stochastically in a horizontal plane by a turbulent but sublevitating upward flow of air obeys not just equipartition and the Maxwell–Boltzmann speed distribution, but also the Langevin equation with colored noise satisfying the fluctuation–dissipation relation. ² Therefore, the system has truly thermal behavior, but with a huge effective temperature, and the essence of Einstein’s challenge is seen with the naked eye.