# Interdisciplinary Applied Mathematics

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The standard description of Brownian motion is based on the Langevin model, where the particles are subject to a random white-noise forcing from the thermal fluctuations. This yields the aforementioned classic Stokes-Einstein result    for    the diffusivity    of    a single    spherical    particle    that    is    independent of the    mass    of the particle.    In    a    dilute    system    the    particles    can

be considered in isolation, as is the case of many Brownian dynamics simulations of polymer chains. However, there are long-range hydrodynamic interactions between particles in a suspension under low Reynolds number conditions. These modify the mobility of a system of particles and hence the diffusivities and particle motion in response to thermal fluctuations.

The starting    point    for    most    simulation    studies    of    Brownian    motion    is

the Langevin equation written in the form of an Ito stochastic differential equation for the evolving displacements of the particle. This is essentially a Monte Carlo simulation of the suspension. An alternative is to formulate the problem as a Fokker-Planck equation for the configurational distribution function. A key element of the stochastic simulations is to characterize the vector of stochastic Brownian forces FB using the fluctuation-dissipation theorem in terms of the collective resistance tensor R for the system of particles. The fluctuating forces have zero mean and are uncorrelated in time, but the forces on individual particles are correlated on average, through the hydrodynamic interactions, with

(FB(t) • FB(t’)T) = 2kBT R 6(t — t’).

The resistance tensor R is determined by the instantaneous configuration of the particles and must be recalculated as the particles move. The fluctuating forces give rise to corresponding random displacements in the stochastic equation, which can be characterized through the corresponding mobility tensor R-1. For a system of particles the diffusion tensor is D = kBTR-1. The repeated computations of the resistance or mobility tensors can become lengthy as the system of particles becomes larger, and steps are needed to accelerate the computational steps. A simple representation of the Brownian motion using a fluctuating force FB with zero mean and the correlation specified above is

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