Finite–N scaling, gelation cutoff, and matched asymptotics for Smoluchowski coagulation equation
Year: 2026
Type: article
Abstract: 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.
Source: Boundary Value Problems
Authors: Varun Kumar, Saiful R. Mondal
Institutions: Alliance University, King Faisal University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,890
OpenAlex ID: W7124268719