Is gelation a singularity or a flow induced instability?

TL;DR

The paper reinterprets gelation in the discrete Smoluchowski coagulation equation as a dynamical instability of the aggregation flow by tracking the time-dependent spectrum of the Jacobian along the evolving trajectory. For gelling kernels, notably $K_{ij}=(ij)^{\alpha}$ with $\alpha>1/2$, the authors observe persistent positive real parts in the Jacobian spectrum that peak near gelation and recede afterward, while non-gelling kernels and a diffusion-limited control show no enduring instability. They implement a finite-dimensional truncation and real-time spectral analysis to compute $\mathbf{J}(t)$ and its eigenvalues, confirming that gelation is tied to a dynamical destabilization rather than purely to diverging moments. This approach provides a robust, moment-independent diagnostic for gelation and bridges classical coagulation theory with modern dynamical-systems methods.

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 $K(i,j)=(ij)^α$ 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 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.

Figures (4)

• Figure 1: Physical and dynamical signature of gelation. Left: Structural transition from dispersed clusters (Sol) to a macroscopic network (Gel). Right: Corresponding stability landscape. The sol-state is linearly stable ($\text{Re}(\lambda_{\max}) < 0$), whereas the gelation point marks the onset of a dynamical instability ($\text{Re}(\lambda_{\max}) > 0$), causing perturbations $\delta c$ to grow exponentially.
• Figure 2: Dynamics of the non-gelling and jelling. (A) Evolution of the first moment ($M_1$) for $\alpha = 0.2$ (blue line) and $\alpha = 0.8$ (red line), and the gel fraction at $\alpha = 0.8$ (red dashed line). (B) Second moment ($M_2$) evolution, indicating polydispersity. Eigenvalue maps in the complex plane ($Im(\lambda)$ vs $Re(\lambda)$) over time (color-coded) for $\alpha=0.2$ (C) and $\alpha=0.8$ (D). (E) Real part of the maximum eigenvalue ($Re(\lambda_{max})$). The transition from positive to negative values indicates a shift in the dynamic stability of the population during the phase transition. (F) Evolution of structural parameter $H(t)$.
• Figure 3: Cluster size distributions $c_p$ at intermediate ($t=1.5$) and late ($t=5.0$) times, comparing the non-gelling ($\alpha=0.2$) and gelling ($\alpha=0.8$) regimes.
• Figure 4: Dynamics of the non-gelling Smoluchowski diffusion-limited kernel ($D_f=1.8$). (A) Evolution of the first moment $M_1$ (blue), which remains strictly conserved, and the gel fraction (orange), which stays at zero. (B) The second moment $M_2$ grows monotonically but remains finite, contrasting with the divergence observed in gelling kernels. (C) The real part of the maximum eigenvalue, $\text{Re}(\lambda_{\text{max}})$, remains vanishingly small ($\approx 0$), indicating the persistence of dynamical stability. (D) The full instantaneous spectrum of the Jacobian in the complex plane; all eigenvalues are confined to the stable left half-plane ($\text{Re}(\lambda) \le 0$). (E) Evolution of the Shannon entropy $H(t)$. (F) Snapshots of the cluster size distribution $c_p$ for intermediate ($t=1.5$) and late ($t=5.0$) times.