In Einstein's paper, he deduced that the diffusion equation is satisfied for the number of particles per unit volume ($f(x, t)$), i.e $$\frac{\partial f}{\partial t}=D\cdot \frac{\partial ^2 f}{\partial x^2}$$
The solution of this equation for $N$ particles starting at the origin is $$f(x, t)=\frac{N}{\sqrt{4\pi Dt}}\cdot e^{-\frac{x^2}{4Dt}}$$
Which is the PDF of a normal distribution (with mean $0$ and variance $\sigma^2= 2Dt$). My question is, how does this expression for the particle density tell us anything about the probability density function describing the movements of an individual particle? Why does the fact that the particle density is distributed normally imply the same for the PDF of incrementing by $\Delta$ in a small time $\tau$?
In this article, is is said that the PDF itself satisfies the above diffusion equation, which seems to not be what Einstein wrote in his paper.
I would appreciate an answer to my confusion (perhaps there is some argument for the equal treatment of the PDF and $f$?). Thanks.
