Researchers in the US have done what Albert Einstein thought was impossible – measure the instantaneous velocity of a particle undergoing Brownian motion. The measurements, performed on micron-sized suspended glass beads, prove directly that a Brownian particle’s kinetic energy is independent of its size, as is the case with atoms and molecules, and suggest a way of studying the quantum properties of macroscopic particles.
In 1905 Einstein published a mathematical description of Brownian motion – the random movement of small particles held in a liquid or gas as a result of their continuous bombardment by the molecules in the fluid. This work cleared up earlier controversial studies of Brownian particles, which suggested that the particle’s velocity would approach infinity when measured over very short periods.
By combining thermodynamics and statistical mechanics, Einstein showed that the displacement of a particle continuously subject to random collisions is proportional to the square root of time and not to time, as would be the case for a particle following an undisturbed, or “ballistic”, trajectory. The experimenters had therefore been measuring the wrong quantity – velocity in this case was not simply displacement divided by time.
Impossible to follow
Einstein’s prediction was verified experimentally a few years later by the physical chemist Jean Perrin. However, Einstein knew that his formulation only applied above a certain length scale – at the smallest distances even Brownian particles should show ballistic motion. But he believed that it would be impossible in practice to track this motion, given the incredibly short timescales over which the Brownian fluctuations take place. For example, a silica sphere with a diameter of 1 µm immersed in water has a velocity that changes both in direction and magnitude every 100 ns (10-7 s), requiring that a detector system have a response time of less than about 10 ns.
Now, Mark Raizen, Tongcang Li and colleagues at the University of Texas at Austin have found a way round this problem by studying particles in air rather than a liquid. Because air is much less dense than water its molecules are farther apart and therefore the distance, and time, between Brownian collisions is much greater. Indeed, velocity of a Brownian particle changes about once every 100 µs in air.
Air, however, is not dense enough to support even the lightest of particles against the force of gravity. Instead, Raizen’s team suspended a micron-sized silicon-oxide bead in air using the radiation pressure of a pair of laser beams known as an optical tweezer. As the bead was battered to and fro by air molecules its slight displacement from the centre of the laser trap was measured by the tiny deflection of one of the beams, this beam having been split and the difference in power between the two halves then measured.
Ballistic trajectories right on target
The researchers plotted graphs of how the displacement varied with the measurement timescale. As expected, they found that on the microsecond scale this relationship obeyed the straight proportionality predicted for ballistic trajectories. They then calculated the instantaneous velocities along these ballistic trajectories (simply from dividing displacement by time) and found them to follow a Maxwell-Boltzmann distribution, as predicted by kinetic theory.
The average instantaneous velocity was found to be very close to that calculated using the equipartition of energy theorem, which states that in thermal equilibrium each degree of freedom of a particle has the same average kinetic energy – ½kT – no matter how massive the particle. The experiment therefore provides direct verification of this theory for Brownian particles, says Raizen, and proves, as he puts it, that a micron-sized glass bead has the same average kinetic energy as a single air molecule.
The researchers say that their technique can also be used to reduce the motion of a suspended particle in a vacuum, by applying a force of just the right strength in the opposite direction to the instantaneous velocity at every moment. In fact, they believe that it will be possible to cool a particle down all the way to its quantum ground state. At this point, they say, the energy equipartition theorem will no longer hold true because the existence of zero-point energy means that the kinetic energy of the particle will not approach zero even at 0 K.
‘Important step forward’
Mark Haw of the University of Strathclyde in the UK describes the work as “an important step forward” in our understanding of Brownian motion, adding that it could have important applications in systems that are strongly affected by such motion, including living cells and nanoscale machines. But he cautions that it will be difficult to reach the quantum regime. “The necessary refinement in precision may take some time,” he explains.
The research has been published in Science.
