Gelation as a Dynamical Instability of the Smoluchowski Flow

Gelation is commonly interpreted as a finite-time singularity in the Smoluchowski coagulation equation 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)^α 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 nongelling kernels exhibit only transient finite-size effects, gelling kernels display persistent spectral destabilization associated with macroscopic gel formation. These results place gelation within a unified dynamical framework of aggregation-driven phase transitions by identifying a spectral signature that links the kinetic runaway of gelation to the metastability of nucleation in Becker-Döring kinetics.