The Smoluchowski equation is a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers, or by positive reals, these
corresponding to the discrete or the continuous form of the equations. In dimension at least 3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a similar method to that used to derive the discrete form of the equations in Hammond and Rezakhanlou [4]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of [4]. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations.
Let $T_{n,m}=mathbb Z_ntimesmathbb Z_m$, and define a random mapping $phicolon T_{n,m}to T_{n,m}$ by $phi(x,y)=(x+1,y)$ or $(x,y+1)$ independently over $x$ and $y$ and with equal probability. We study the orbit structure of such ``quenched random wal
ks $phi$ in the limit $m,ntoinfty$, and show how it depends sensitively on the ratio $m/n$. For $m/n$ near a rational $p/q$, we show that there are likely to be on the order of $sqrt{n}$ cycles, each of length O(n), whereas for $m/n$ far from any rational with small denominator, there are a bounded number of cycles, and for typical $m/n$ each cycle has length on the order of $n^{4/3}$.
