# Homework Help: Langevin and Einstein formulas of Brownian motion

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1. May 29, 2015

### mikeclinton

1. The problem statement, all variables and given/known data

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 <x2> ~ t2 in the limit of t→0. Confirm the consistency of the principle of equipartition of energy.

2. Relevant equations

Eq. (1.1) is <x2> = (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 MΔ2, 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.

2. Jun 3, 2015