- #1

mikeclinton

- 7

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## Homework Statement

Demonstrate that Eq. (1.1) will convert to the Einstein relation Eq. (1.2) in the limit of t→∞ when we assume ξ=6πaμ.

Conversely, show that Eq. (1.1) will yield <x

^{2}> ~

*t*

^{2}in the limit of t→0. Confirm the consistency of the principle of equipartition of energy.

## Homework Equations

Eq. (1.1) is <x

^{2}> = (2kT/ξ)[t-(m/ξ)(1-e

^{-tξ/m})]

Eq (1.2) is (MSD)

^{2}= 2Dτ

x is the displacement of a Brownian particle moving in a viscous liquid

m is the mass of the particle

ξ is the friction coefficient. We assume it is governed by Stoke's law which states that the frictional force decelerating a spherical particle of a radius and mass ms is ξ=6πaμ, where μ the viscosity of the surrounding liquid.

MSD (or MΔ

^{2}, I don't know the HTML code for the overbar) is the mean square displacement of the particle

τ is the time interval of observation in which the particle will move as much as Δ

D is the diffusion coefficient

k is Boltzmann's constant

T is the absolute temperature I don't know where to start with this...any help will be much appreciated.