On Lushnikov's model of gelation

Scaling theory and exactly solved models in the kinetics of irreversible aggregation

“…Using then the so-called Ω-expansion devised by van Kampen, one may derive the rate equations. This leads to the discussion of corrections due to particle number fluctuations, for a thorough discussion of which the reader is referred to [70,104,105,109,11]. A considerable literature has also arisen in the literature on probability concerning such models and their connection to the Smolchowski equations.…”

“…[10] for details. 8 Specialists will recognize here the separation constant w of the papers of Ernst and van Dongen one finds as an estimate for the order of magnitude of s(t) 11) at least when p is larger than a given p 0 , the value of which depends on the details of the system under consideration. In fact, in many contexts,the typical size is defined as some such moment ratio: such are, for example, the weight average, defined as M 2 (t)/M 1 (t) and the z-average, given by M 3 (t)/M 2 (t).…”

Stochastic algorithms for studying coagulation dynamics and gelation phenomena

“…Correspondingly, gelation was interpreted s the emergence of a "superparticle" having mass proportional to the mass of the System. More general functions ψ have been considered in [4] and [17,Theorem 5]. A basic element of the convergence proofs for general kernels is uniqueness of the solution to the limiting equation.…”

Triangle Percolation in Mean Field Random Graphs—with PDE

“…A useful way to handle the Smoluchowski equation is by taking its Laplace-transform (which is essentially the same as using generating functions): the transformed differential equation becomes a well-known PDE, the Burgers equation (see [2], [3], [8]), which can be solved explicitly by using the method of characteristics.…”

“…It is easy to relate the evolution of the random graph to the mean field stochastic model of coagulation, the Marcus-Lushnikov process, which converges to the solution of the Smoluchowski coagulation equation (with multiplicative kernel), see (in historical order) [2], [3], [1], [6]. A useful way to handle the Smoluchowski equation is by taking its Laplace-transform (which is essentially the same as using generating functions): the transformed differential equation becomes a well-known PDE, the Burgers equation (see [2], [3], [8]), which can be solved explicitly by using the method of characteristics.…”