Multiscaling for classical nanosystems: Derivation of Smoluchowski & 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 ǫ, 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(ǫ 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(ǫ 2 ). This approach has applications to a broad range of problems in the nanosciences.

A Review of Two Multiscale Methods for the Simulation of Macromolecular Assemblies: Multiscale Perturbation and Multiscale Factorization

“…In a series of recent studies, the authors and collaborators have discovered novel multiscale techniques that probe the cross-talk among multiple scales in space and time that are inherent within such systems, yet preserve all-atom detail within the macromolecular assemblies [21][22][23][24][25][26][27][28][29][30][31][32]. Multiscale perturbation methods can be described in a general framework.…”

“…As one seeks only a few OPs ( N ), this relationship between the atomic positions and OPs cannot completely describe individual atomic motion. Previously, this was addressed by introducing residuals to capture the short-scale atomic dynamics, deriving equations for the co-evolution of the OPs and the probability distribution of atomic configurations [23,24,29,43]. However, as many systems are easily deformed by thermal stress and fluctuation, large deformations cannot be considered as coherent changes determined by merely a few OPs.…”

“…with the Liouville operator given by Equation (5) is used to arrive at a conservation law [23] for W via the chain rule. Namely, taking a time derivative and integrating by parts yields the equation:…”

“…However, one finds [19,23] that this formulation enables a novel procedure for constructing a closed equation for W when is small. Note that the expressions for Π k and π k can be determined explicitly by Equations (8) and (9).…”

“…While this aspect often makes traditional MD simulation impractical, multiscale approaches have recently been presented that allow for long-time simulation with atomic detail based on the co-evolution of slowly-varying order parameters (OPs) with the quasi-equilibrium probability density of atomic configurations [19][20][21][22][23][24][25]. In subsequent sections, two recently-discovered multiscale techniques are discussed that enable one to efficiently model and dynamically simulate evolving macromolecular systems.…”