From the Liouville to the Smoluchowski equation for a colloidal solute particle in a solvent

TL;DR

This paper derives the Smoluchowski equation for a colloidal particle from the microscopic Liouville equation, using a perturbation expansion and Monte Carlo methods to study its dynamics and approach to equilibrium.

Contribution

It presents a derivation of the Smoluchowski dynamics from the microscopic Liouville equation and introduces a Monte Carlo scheme to analyze the particle's evolution.

Findings

The Smoluchowski equation can be obtained from the Liouville equation in the high friction limit.

The Monte Carlo scheme effectively solves the evolution equation for the colloid.

The approach to equilibrium and the influence of external potential and friction are characterized.

Abstract

We show how the Smoluchowski dynamics of a colloidal Brownian particle suspended in a molecular solvent can be reached starting from the microscopic Liouvillian evolution of the full classical model in the high friction limit. The integration of the solvent degrees of freedom goes through a multiple time scale perturbation expansion which removes the secular divergences. A simple dynamical Monte Carlo scheme is then proposed to solve the resulting evolution equation for the colloid solute particle. In particular we study the approach to the equilibrium Boltzmann distribution at late times and its resilience behavior at shorter times as influenced by the steepness of the external potential and the friction coefficient around their respective minima. This is very important to understand the fate of the Brownian particle's random walk and its evolution history.