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

Stephen Pankavich; Zeina Shreif; Peter Ortoleva

doi:10.1016/j.physa.2008.03.008

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arXiv · 1001.1216

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

Stephen Pankavich·Zeina Shreif·Peter Ortoleva

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Abstract

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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Stephen Pankavich, Zeina Shreif, Peter Ortoleva. 2010-01-08. Multiscaling for Classical Nanosystems: Derivation of Smoluchowski and Fokker-Planck Equations. https://doi.org/10.1016/j.physa.2008.03.008

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