Ballastic and Diffusive motions

The impacts of heat bath on a small system embedded in it are clear on macroscopic and stationary scale, but they remain a challenging subject from the microscopic and dynamic point of view. A simple example is the Brownian motion of a single particle placed in a air or other medium. The equilibrium statistical theory was forwarded by Einstein a century ago, yet what actually take place over very short periods are still under intensive study (see previous entries). Here comes a new report [Science, 332:802(2011)]:

For many years after Einstein's contributions, it was expected that the transition from ballistic to diffusive motion would be quite sharp, corresponding to an exponential decay of the particle's memory of its earlier velocity. However, about 50 years ago, hints from computer simulations and theory started to suggest a more complex scenario. In particular, hydrodynamic vortices in the liquid created by the particle's motion lead to memory effects, and the particle's velocity decays much more slowly than exponentially, exhibiting a t^−3/2 “long-time-tail” (12). Detailed analysis by Huang et al. of data like that shown in the second figure, panel B, where the ballistic-to-diffusive transition spans more than three decades in time, has now provided a thorough verification of the full, complicated hydrodynamic theory (13, 14). Although several previous experiments had observed the breakdown of the simple diffusion picture [e.g., (15)], the present studies extend into the ballistic regime.

What next? Li et al. mention the fascinating prospect of laser cooling a trapped particle to a temperature at which quantization of the energy of this mesoscopic object could be observed (16). Huang et al. suggest extending their measurements to Brownian motion in confined regions and heterogeneous media. Here, understanding the details of prediffusive motion over subnanometer distances could well be relevant to some biological processes, such as the lock-and-key mechanism of enzyme action.

Metastable states are important in reality

Metastable states are common in nature: supercooled or superheated liquid are daily examples. These states are not the lowest-energy state, and, by thermodynamic principles, one should not expect them to be long-lived in nature. Indeed, they exist only under very stringent conditions, and very little disturbance can push the system off to a stable state around. Thermodynamically stable states dwell on one of the global minima of the free energy, around which, however, local minima might exist that are separated from the global minima by energy barriers. When a system has not reached the stable states, it will be constantly kicked by its surroundings and eventually transits from where it is in to a stable state after some period (the life of the metastable state). The interesting point is that, the life time can sometimes be very long and real transitions can hardly be seen, a situation similar to ergodicity breaking. For example, diamond has a higher energy than graphite, but it can exist for ever. The reason is because the time to make the transition is cosmologically long, due to the hugely high barrier. Another material is graphene, which should not be stable according to Wagner-Mermin-Honberg theorem. Yet, it was produced in 2004. Glasses are the third examples, in which case, transition has been frustrated by its structure. In the case of supercooled water, the transition is suppressed by distilling process.

Brownian motions under direct observation

Here is surely a very interesting report, which shall reveal a lot of details about the motions under noices. In my opinion, although many have been learned of such motions, they remain mysterious in short-time scale dynamics. Further, what about a quantum particle in noices.

Science 25 June 2010:
Vol. 328. no. 5986, pp. 1673 - 1675
DOI: 10.1126/science.1189403

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Reports
Measurement of the Instantaneous Velocity of a Brownian Particle
Tongcang Li, Simon Kheifets, David Medellin, Mark G. Raizen*

Brownian motion of particles affects many branches of science. We report on the Brownian motion of micrometer-sized beads of glass held in air by an optical tweezer, over a wide range of pressures, and we measured the instantaneous velocity of a Brownian particle. Our results provide direct verification of the energy equipartition theorem for a Brownian particle. For short times, the ballistic regime of Brownian motion was observed, in contrast to the usual diffusive regime. We discuss the applications of these methods toward cooling the center-of-mass motion of a bead in vacuum to the quantum ground motional state.

Center for Nonlinear Dynamics and Department of Physics, University of Texas at Austin, Austin, TX 78712, USA.

* To whom correspondence should be addressed. E-mail: raizen@physics.utexas.edu