Tips and Tricks - Soft Matter
When low-order expansions fail and all higher-order contributions matter—basic example of the mean squared displacement for Brownian motion
Institut für Physik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany
Accepted: 9 September 2022
Published online: 19 September 2022
Hardly any theoretically formulated realistic problem can be solved exactly. Therefore, as a standard, we resort to approximations. In this context, expansions play a major role. We are used to relying on lowest-order expansions and confining our point of view accordingly. However, one should always bear in mind that such considerations may fail at some point. Here, we address a very common example situation, namely, the motion of a Brownian particle. We know that the associated mean-squared displacement in the long term increases linearly in time. Yet, when we take the Fokker–Planck approach in combination with a low-order expansion, the direct route towards this result fails. That is, in the expansion the term linear in time vanishes. Instead, the treatment requires consideration of all higher-order contributions. Together, they restore the linear increase in time. In this way, we stress that care is always mandatory when resorting to low-order expansions, and we present in a traceable way a route to solving the considered problem.
© The Author(s) 2022
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