Is gelation a singularity or a flow induced instability?

TL;DR

This paper reinterprets gelation in coagulation processes as a dynamical instability characterized by spectral analysis, challenging the traditional view of gelation as a finite-time singularity.

Contribution

It introduces a stability-based framework for understanding gelation, linking it to the spectral properties of the Smoluchowski flow and providing a new perspective on gelation phenomena.

Findings

Gelation correlates with positive eigenvalues indicating instability.

Gelling kernels show persistent spectral destabilization.

Non-gelling kernels only exhibit transient effects.

Abstract

Gelation in the Smoluchowski coagulation equation is commonly interpreted as a finite-time singularity marked by mass loss or moment divergence. We instead characterize gelation as a loss of dynamical stability of the Smoluchowski flow, quantified through the time-dependent spectrum of the Jacobian along the evolving aggregation dynamics. Studying homogeneous kernels $K(i,j)=(ij)^{\alpha}$ together with the classical Smoluchowski, we show that gelation is consistently preceded by the appearance of positive real eigenvalues, indicating a loss of local dynamical stability. While non-gelling kernels exhibit only transient finite-size effects, gelling kernels display persistent spectral destabilization associated with macroscopic gel formation. Our results identify gelation as a genuine dynamical instability of the Smoluchowski flow.