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Multiscaling for classical nanosystems: Derivation of Smoluchowski & Fokker–Planck equations

Author

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  • Pankavich, S.
  • Shreif, Z.
  • Ortoleva, P.

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 ϵ, 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(ϵ2) 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(ϵ2). This approach has applications to a broad range of problems in the nanosciences.

Suggested Citation

  • Pankavich, S. & Shreif, Z. & Ortoleva, P., 2008. "Multiscaling for classical nanosystems: Derivation of Smoluchowski & Fokker–Planck equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(16), pages 4053-4069.
    • Handle: RePEc:eee:phsmap:v:387:y:2008:i:16:p:4053-4069
    DOI: 10.1016/j.physa.2008.03.008
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    Cited by:

    1. Shreif, Z. & Ortoleva, P., 2009. "Multiscale derivation of an augmented Smoluchowski equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(5), pages 593-600.
    2. Sereda, Yuriy V. & Ortoleva, Peter J., 2013. "Variational methods for time-dependent classical many-particle systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(4), pages 628-638.

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