Questions

1 Show in one dimension that a localised region of high concentration at time t = 0 (e.g. take a thin Gaussian) will spread out over time with a concentration profile still given by a Gaussian function. Work out the variance, and compare with the result calculated here for mean square displacement of an object having diffusion coefficient D.

2 By substituting from Equation 17 into Equation 15, verify that Equation 15 provides a solution to the equation of motion of a colloidal particle.


Answers

1 Let’s take a Gaussian form for the concentration, c,



where A is a constant. Then the time and space derivatives can be evaluated,





Equating the two expressions, as in Fick’s law (Equation 4), gives the condition



and it can be verified that a solution is



This proves that a Gaussian profile remains Gaussian with diffusion over time, and shows the width of the distribution growing with the square root of time.

2 It is useful here to recall that



This is an application of Leibniz’s integral rule, otherwise known as derivation under the integral. Using this, it can be readily seen that Equation 17 is a solution of Equation 15 by evaluating the derivative,