Instantaneous Gelation in Smoluchowski's Coagulation Equation Revisited

TL;DR

This paper investigates the effects of regularisation on instantaneous gelation in Smoluchowski's coagulation equation, revealing slow convergence to gelation time and detailed stationary states with oscillatory dynamics for specific kernels.

Contribution

It provides the first detailed analysis of regularised gelation, including analytic descriptions of stationary states and oscillations for the model kernel with monomer injection.

Findings

Gelation time decreases logarithmically with cut-off M.

Stationary cluster size distribution exhibits stretched exponential and power law decay.

Oscillations around the stationary state decay very slowly as M increases.

Abstract

We study the solutions of the Smoluchowski coagulation equation with a regularisation term which removes clusters from the system when their mass exceeds a specified cut-off size, M. We focus primarily on collision kernels which would exhibit an instantaneous gelation transition in the absence of any regularisation. Numerical simulations demonstrate that for such kernels with monodisperse initial data, the regularised gelation time decreases as M increases, consistent with the expectation that the gelation time is zero in the unregularised system. This decrease appears to be a logarithmically slow function of M, indicating that instantaneously gelling kernels may still be justifiable as physical models despite the fact that they are highly singular in the absence of a cut-off. We also study the case when a source of monomers is introduced in the regularised system. In this case a stationary state is reached. We present a complete analytic description of this regularised stationary state for the model kernel, K(m_1,m_2)=max{m_1,m_2}^v, which gels instantaneously when M tends to infinity if v>1. The stationary cluster size distribution decays as a stretched exponential for small cluster sizes and crosses over to a power law decay with exponent v for large cluster sizes. The total particle density in the stationary state slowly vanishes as (Log M^(v-1))^-1/2 when M gets large. The approach to the stationary state is non-trivial : oscillations about the stationary state emerge from the interplay between the monomer injection and the cut-off, M, which decay very slowly when M is large. A quantitative analysis of these oscillations is provided for the addition model which describes the situation in which clusters can only grow by absorbing monomers.