Thermal Physics · Problem 2

Propagation — Charles Kittel & Herbert Kroemer

Physics — Statistical Mechanics · diffusion-in-two-and-three-dimensions

Problem

Diffusion in two and three dimensions. (a) Show that the diffusion equation in two dimensions admits the solution

\theta_2(t) = (C_2/t)\exp(-r^2/4Dt) \tag{64}

and in three dimensions

\theta_3(t) = (C_3/t^{3/2})\exp(-r^2/4Dt). \tag{65}

(b) Evaluate the constants

C_2

and

C_3

. These solutions are analogous to (14) and describe the evolution of a delta function at

t = 0