Question

a) The root mean square displacement for diffusion in the x-direction is <x2> = 201 Use this result to show the rms displacement for diffusion in three dimensions is 6Dt. Hint: Use the Pythagoras equation.

Answer 1

The mean square displacement for diffusion in 3 dimensions is

The mean square displacement for diffusion in 3 dimensions is < r^2 > = < x^2 + y^2 + z^2 > + < y^2 > + < y^2 > + < z^2 > This is the result of the following property of mean < A + B > = < A > + < B > GIven squareroot < x^2 > = squareroot 2Dt therefore < x^2 > = 2Dt Similarly, for the other 2 individual dimensions, the result will be < y^2 > = < z^2 > = 2Dt Therefore RMS displacement for diffusion in 3 dimensions is squareroot < r^2 > = squareroot < x^2 > + < y^2 > + < z^2 > = squareroot 2Dt + 2Dt + 2Dt = squareroot 6Dt

This is the result of the following property of mean

$<A+B>=<A>+<B>$

Given

$\sqrt{<x^2>}=\sqrt{2Dt}$

Therefore

$<x^2>=2Dt$

Similarly, for the other 2 individual dimensions, the result will be

$<y^2>=<z^2>=2Dt$

Therefore RMS displacement for diffusion in 3 dimensions is

$\sqrt{<r^2>}=\sqrt{<x^2>+<y^2>+<z^2>}=\sqrt{2Dt+2Dt+2Dt}=\sqrt{6Dt}$