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

TL;DR

This paper derives the Smoluchowski and Fokker-Planck equations for nanosystems using multiscale analysis, linking microscopic Liouville dynamics to macroscopic stochastic descriptions relevant in nanoscience.

Contribution

It introduces a systematic derivation of reduced equations for nanosystems from the N-atom Liouville equation using multiscale analysis, extending the theoretical framework.

Findings

Reduced probability density satisfies Smoluchowski equation up to order epsilon squared.

Under certain assumptions, the density also satisfies a Fokker-Planck equation.

Method applies broadly to problems in nanoscience.

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.