Finite–N scaling, gelation cutoff, and matched asymptotics for Smoluchowski coagulation equation

We develop a finite-N matched-asymptotic theory for Smoluchowski coagulation. Starting from the infinite system of ODEs, we use conservation laws to obtain closed scalar evolution equations for suitable moments. For the constant kernel $a_{j,k}=K$, this leads to an exact early-time decay law for the cluster number and the linear coalescence-time scaling $T_{N}\asymp N$ when the volume scales as $V\sim N$. For the sum kernel $a_{j,k}=j+k$, the same reduction yields an exponential decay regime for the number of clusters and a logarithmic finite-N coalescence time. For the multiplicative kernel $a_{j,k}=jk$, we recover the classical finite-time blowup of the second moment and show that finite N produces a sharp gelation cutoff preceding the mean-field gelation time by a window of order $N^{-2}$. In all kernels considered, the late-time dynamics involve only finitely many clusters and contribute only lower-order corrections. The resulting structure closely parallels that of finite-N two-species annihilation: a conserved quantity reduces the dynamics to a scalar ODE, asymptotic matching at a characteristic time yields the finite-size scaling laws, and post-matching effects do not alter the leading behaviour.