Multiscaling for classical nanosystems: Derivation of Smoluchowski & Fokker–Planck equations |
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Authors: | S Pankavich Z Shreif P Ortoleva |
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Institution: | aDepartment of Mathematics, Indiana University, Bloomington, IN 47405, United States;bCenter for Cell and Virus Theory, Department of Chemistry, Indiana University, Bloomington, IN 47405, United States |
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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 , 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 O( 2) 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 O( 2). This approach has applications to a broad range of problems in the nanosciences. |
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Keywords: | Nanosystems All-atom multiscale analysis (AMA) Gibbs hypothesis Smoluchowski equations Fokker– Planck equations |
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