Diffusion

#Physics
For a particle that can either move left or right and makes $\displaystyle n$ steps each of size $\displaystyle \delta$

Topics

• Fick's Equations
• Diffusion Equation
• Advection Diffusion Equation

$\displaystyle {\left\langle{x(n)}\right\rangle}=0$

• Average position of a particle after $\displaystyle n$ steps is $\displaystyle 0$ due to random chance

$\displaystyle {\left\langle{x^{2}(n)}\right\rangle}=n\delta ^{2}=2Dt$

• Average of square of particle position after $\displaystyle n$ steps
• $\displaystyle D$ is the diffusive constant
• $\displaystyle t$ is the total time for $\displaystyle n$ steps

$\displaystyle {\left\langle{r^{2}}\right\rangle}=6Dt$

• Statistical independence in 3D