I've seen the derivation of diffusion from random walks using the following calculations:
Master equation of an unbiased random walk is $p(x,t+1) = \frac12 p(x-1,t) + \frac12 p(x+1,t)$
Let $p = p(x,t)$ and subscript denote partial derivative. Taylor expansion of both sides gives $$ p + p_t + \dots = \frac12 (p - p_x + \frac12 p_{xx} + \dots) + \frac12(p + p_x + \frac12 p_{xx} + \dots)$$ so that ignoring higher order terms and canceling $p$'s, $$p_t = \frac12 p_{xx} \;\;\;\;\; \text{(diffusion equation).}$$
I know this diffusion equation is the hydrodynamic limit -- the probability density describing the locations of $n$ iid random walks as $n\rightarrow \infty$ and if we scale the space and time somehow. But the derivation above only deals with a single particle. How do we formalize the idea that this is supposed to describe $n$ particles? For one particle, the density is discrete: if $p(x,0) = \delta_{x_0}$ is the initial condition, then $p(x,t)$ has support on discretely many points. I don't see where the jump from discrete to continuous (and where the scaling) comes in.