The Colour of Brownian Motion

Brownian motion in a fluid at thermal equilibrium is typically modelled (in the optical tweezers community at least) as a continuous stochastic process with a white power spectral density... this simply means that the thermal forces (or torques) are continuously differentiable in time, and randomly drawn from a distribution which carries an equal amount of power in every equally-sized frequency window, regardless of where that window is along the the frequency axis.

This is equivalent to saying that the that the thermal forces have no memory of the position or behaviour of a particle undergoing motion in the fluid i.e. they are uncorrelated in time.



Recent work has shown experimentally that this is not in fact true: thermal forces are correlated in time, even in very simple fluids. This is seen by calculating the power spectrum of a particle's motion. We can think of the power spectrum as a Fourier decomposition of the time-domain motion. Pure white noise results in a flat power spectrum, and 'coloured' noise has more or less energy at different frequencies (blue noise has more energy at high frequencies, and pink noise more at red).

If the particle is subjected to white noise its motions have a Brownian spectrum, with more energy at low frequencies (and a characteristic dependence on the inverse square of the noise). In their experiments, Franosch et al. found that their measurements indicated that the thermal noise actually is slightly blue: it has more energy content at high frequencies than expected.

What does this mean? It means that the thermal forces are correlated with the motion of the particle! Is this surprising? Not really: in fact, it makes perfect sense. The thermal motion cause the bead to move, but the bead causes the fluid to move, creating some sort of feedback loop which carries information about the motion of the particle.

Is this the whole story? Nope: a blue spectrum, just like a white one, carries infinite power, so cannot possibly apply over the whole range of frequencies. They just happen to be nice approximations. People have already seen where these fail: if the fluid is dilute enough (near vacuum) it is perfectly possible to see the ballistic motion which occurs between thermal impulses, and its even been shown that the same thing happens in liquids!

Enjoy!