6.021/Notes/2006-09-12

Review

 * Fick's first law: $$\phi(x,t)=-D\frac{\partial c(x,t)}{\partial x}$$
 * Flux is proportional to the concentration gradient
 * Continuity equation: $$-\frac{\partial\phi(x,t)}{\partial x} = \frac{\partial c(x,t)}{\partial t}$$
 * A flux gradient leads to change in concentration over time
 * Diffusion equation: $$\frac{\partial c(x,t)}{\partial t} = D\frac{\partial^2 c(x,t)}{\partial x^2}$$
 * Steady state solution: time-invariant ($$\frac{\partial c}{\partial t}=0, \frac{\partial\phi}{\partial t}=0$$)
 * $$\phi(x,t)=\phi_0$$
 * $$c(x,t) = -\frac{\phi_0}{D}X+\alpha$$
 * Equilibrium: Zero flux + time invariant
 * $$\phi(x,t)=0, C(x,t)=$$constant (uniform distribution)
 * Impulse response: Gaussian $$c(x,t)=\frac{n_0}{\sqrt{4\pi Dt}}e^{-x^2/(4Dt)}$$
 * $$x^2 \approx Dt$$