Multiscaling for Classical Nanosystems: Derivation of Smoluchowski and Fokker-Planck Equations

Details Multiscaling for Classical Nanosystems: Derivation of Smoluchowski and Fokker-Planck Equations Multiscaling for Classical Nanosystems: Derivation of Smoluchowski and Fokker-Planck Equations

2010-01-08

arxiv.org/abs/1001.1216v1

Physica A, 387:4053-4069, 2008

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

23 pages

Scientific paper

Using multiscale analysis and methods of statistical physics, we show that a solution to the N-atom Liouville Equation can be decomposed via an expansion in terms of a smallness parameter epsilon, wherein the long scale time behavior depends upon a reduced probability density that is a function of slow-evolving order parameters. This reduced probability density is shown to satisfy the Smoluchowski equation up to order epsilon squared for a given range of initial conditions. Furthermore, under the additional assumption that the nanoparticle momentum evolves on a slow time scale, we show that this reduced probability density satisfies a Fokker-Planck equation up to the same order in epsilon. This approach applies to a broad range of problems in the nanosciences.

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