The Physics of Noise by Edoardo Milotti

Author:Edoardo Milotti
Language: eng
Format: epub, mobi
ISBN: 9781643277684
Publisher: IOP Publishing
Published: 2019-10-30T00:00:00+00:00

This split between the time development of the signal and the ensemble view amounts to an effective non-ergodicity of this noise, and the reason, it turns out, is due to one specific feature of the one-dimensional random walk, which may take arbitrarily long times to switch direction and turn back, even though this is eventually bound to happen. This means that no single finite record, whatever its length, can be long enough to show the ergodicity of the process14.

This is just one of many subtleties that must be taken into account when dealing with Brownian motion. One of these subtleties is of direct concern to physics. All the simulations of Brownian motion that we have considered so far have been carried out with the discrete version of the Wiener process.

In a discrete-time Wiener process, at each time step the random walker takes a step in a random direction. In all previous simulations, the spatial steps are taken in a random direction and have a continuous distribution of step lengths (a Gaussian distribution, to be precise). Moreover, the steps are all independent from one another. The continuous-time Wiener process is just the limiting version of the Wiener process, but its mathematical properties must be handled with great care; the interested reader can turn to one of the references in the reading list at the end of this book.

One problem with the Wiener process is the discontinuous velocity of the random walkers, which is quite unphysical because it requires accelerations that are proportional to Dirac’s delta, and correspondingly large forces. This is bad because there are no known forces with infinite amplitude spikes. To circumvent this problem we can take the approach of Paul Langevin, who introduced a remarkable equation in his paper ‘On the Theory of Brownian Motion’15, three years after Einstein’s landmark study.

In practice, the equation describes a Wiener process for the velocity (instead of the position), with a dissipative term that prevents large velocity increases.

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