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

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.

23 pages

Cited by in corpus (5)

• Self-Assembly of Nanocomponents into Composite Structures: Derivation and Simulation of Langevin Equations
• Liquid-Crystal Transitions: A First Principles Multiscale Approach
• Multiscaling for Systems with a Broad Continuum of Characteristic Lengths and Times: Structural Transitions in Nanocomposites
• Scaling Behavior of Quantum Nanosystems: Emergence of Quasi-particles, Collective Modes, and Mixed Exchange Symmetry States
• Multiscale Theory of Finite Size Bose Systems: Implications for Collective and Single-Particle Excitations