When low-order expansions fail and all higher-order contributions matter—basic example of the mean squared displacement for Brownian motion
European Physical Journal E, ISSN: 1292-895X, Vol: 45, Issue: 9, Page: 77
2022
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Article Description
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.
Bibliographic Details
http://www.scopus.com/inward/record.url?partnerID=HzOxMe3b&scp=85138247845&origin=inward; http://dx.doi.org/10.1140/epje/s10189-022-00232-z; http://www.ncbi.nlm.nih.gov/pubmed/36121547; https://link.springer.com/10.1140/epje/s10189-022-00232-z; https://dx.doi.org/10.1140/epje/s10189-022-00232-z; https://link.springer.com/article/10.1140/epje/s10189-022-00232-z
Springer Science and Business Media LLC
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