Instantaneous Gelation in Smoluchowski's Coagulation Equation Revisited
Robin C. Ball, Colm Connaughton, Thorwald H. M. Stein, Oleg Zaboronski
备注：
11 pages, 7 figures. Revised version extended to included a quantitative description of the oscillatory approach to the stationary state for the addition model
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.
Numerical solution of Smoluchowski coagulation equation combined with Ostwald ripening
Lomonosov Moscow State University（莫斯科国立大学）；Marchuk Institute of Numerical Mathematics of Russian Academy of Science（俄罗斯科学院马丘克数值数学研究所）；Ural Federal University（乌拉尔联邦大学）
Origin of the spontaneous oscillations in a simplified coagulation-fragmentation system driven by a source
我们考虑一个系统的聚集团的颗粒，受到凝聚和破碎过程的质量依赖率。每个单体颗粒可以聚集成更大的簇，并且每个簇可以以与聚集速率成正比的速率碎裂成单个单体。的集群密度的动力学是由一组Smoluchowski方程，我们认为，除了在恒定速率的单体源。整个动力学可以归结为求解一个独特的非线性微分方程，该方程在特定的参数范围内显示自振荡，并且对于系统中足够大的一些不同的簇。这种集体现象是由于存在波动的阻尼系数，并且与在更一般的物理系统中观察到的Liénard自振荡机制密切相关，例如van der Pol振荡器。
Impact of Initial Charge Distributions on the Kinetics of Charged Particle Coagulation
Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins（奥希金斯大学工程科学研究所）；Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile（智利大学物理与数学科学学院物理系）
