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Langevin and Einstein formulas of Brownian motion

  1. May 29, 2015 #1
    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. jcsd
  3. Jun 3, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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