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Approach to Self-Similarity in Smoluchowski’s Coagulation Equations
, 2003
"... We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski’s equations of coagulation for the solvable kernels K(x, y) = 2, x + y and xy. In addition to the known self-similar solutions with exponential tails, there are one-parameter families of solutions with algebraic deca ..."
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Cited by 58 (8 self)
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We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski’s equations of coagulation for the solvable kernels K(x, y) = 2, x + y and xy. In addition to the known self-similar solutions with exponential tails, there are one-parameter families of solutions with algebraic decay, whose form is related to heavy-tailed distributions well-known in probability theory. For K = 2 the size distribution is Mittag-Leffler, and for K = x + y and K = xy it is a power-law rescaling of a maximally skewed α-stable Lévy distribution. We characterize completely the domains of attraction of all self-similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits.
Dust and self-similarity for the Smoluchowski coagulation equation
, 2004
"... We establish the well-posedness of the Cauchy problem for the Smoluchowski coagulation equation in the homogeneous space ˙ L 1 1 for a class of homogeneous coagulation rates of degree λ ∈ [0, 2). For any initial datum fin ∈ ˙ L 1 1 we build a weak solution which conserves the mass when λ ≤ 1 and lo ..."
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Cited by 26 (4 self)
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We establish the well-posedness of the Cauchy problem for the Smoluchowski coagulation equation in the homogeneous space ˙ L 1 1 for a class of homogeneous coagulation rates of degree λ ∈ [0, 2). For any initial datum fin ∈ ˙ L 1 1 we build a weak solution which conserves the mass when λ ≤ 1 and loses mass in finite time (gelation phenomena) when λ> 1. We then extend the existence result to a measure framework allowing dust source term. In that case we prove that the income dust instantaneously aggregates and the solution does not contain dust phase. On the other hand, we investigate the qualitative properties of self-similar solutions to the Smoluchowski’s coagulation equation when λ < 1. We prove regularity results and sharp uniform small and large size behavior for the self-similar profiles.
Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence
, 2006
"... Smoluchowski’s coagulation equation is a fundamental mean-field model of clustering dynamics. We consider the approach to self-similarity (or dynamical scaling) of the cluster size distribution for the “solvable” rate kernels K(x, y) =2,x+ y, and xy. In the case of continuous cluster size distributi ..."
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Cited by 20 (2 self)
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Smoluchowski’s coagulation equation is a fundamental mean-field model of clustering dynamics. We consider the approach to self-similarity (or dynamical scaling) of the cluster size distribution for the “solvable” rate kernels K(x, y) =2,x+ y, and xy. In the case of continuous cluster size distributions, we prove uniform convergence of densities to a self-similar solution with exponential tail, under the regularity hypothesis that a suitable moment have an integrable Fourier transform. For discrete size distributions, we prove uniform convergence under optimal moment hypotheses. Our results are completely analogous to classical local convergence theorems for the normal law in probability theory. The proofs rely on the Fourier inversion formula and the solution for the Laplace transform by the method of characteristics in the complex plane.
Clustering in coagulation-fragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and Zero-One law
"... The equilibrium distribution of a reversible coagulation-fragmentation process (CFP) and the joint distribution of components of a random combinatorial structure are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (=compon ..."
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Cited by 19 (11 self)
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The equilibrium distribution of a reversible coagulation-fragmentation process (CFP) and the joint distribution of components of a random combinatorial structure are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (=components) in the case a(k) = kp−1, k ≥ 1, p> 0, where a(k), k ≥ 1 is the parameter function that induces the invariant measure. The result obtained is compared with the ones for logarithmic random combinatorial structures (RCS’s) and for RCS’s, corresponding to the case p < 0. 1 Summary. Our main result is a central limit theorem (Theorem 2.4) for the number of groups at steady state for a class of reversible CFP’s and for the corresponding class of RCS’s. In Section 2, we provide a definition of a reversible k-CFP admitting interactions of up to k groups, as a generalization of the standard 2-CFP. The steady state of the processes considered is fully defined by a parameter function a ≥ 0 on the set of integers. It was observed by Kelly ([11], p. 183) that for all 2 ≤ k ≤ N the k-CFP’s have the same invariant measure on the set of partitions of a given
On a Class of Continuous Coagulation-Fragmentation Equations
, 1998
"... . A model for the dynamics of a system of particles undergoing simultaneously coalescence and breakup is considered, each particle being assumed to be fully identified by its size. Existence of solutions to the corresponding evolution integral partial differential equation is shown for product-type ..."
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Cited by 17 (0 self)
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. A model for the dynamics of a system of particles undergoing simultaneously coalescence and breakup is considered, each particle being assumed to be fully identified by its size. Existence of solutions to the corresponding evolution integral partial differential equation is shown for product-type coagulation kernels with a weak fragmentation. The failure of density conservation (or gelation) is also investigated in some particular cases. 1. Introduction The coagulation-fragmentation equations are a model for the dynamics of cluster growth and describe the time evolution of a system of clusters under the combined effect of coagulation and fragmentation. Each cluster is identified by its size (or volume) which is assumed to be a positive real number in the model considered in this paper. From a physical point of view the basic mechanisms taken into account are the coalescence of two clusters to form a larger one and the breakage of clusters into smaller ones. It is also assumed that t...
Lushnikov processes, Smoluchowski’s and Flory’s models, Stochastic Process
- Appl
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A pure jump Markov process associated with Smoluchowski’s coagulation equation
- in "Ann. Probab.", 2002
"... The aim of the present paper is to construct a stochastic process, whose law is the solution of the Smoluchowski’s coagulation equation. We introduce first a modified equation, dealing with the evolution of the distribu-tion Qt (dx) of the mass in the system. The advantage we take on this is that we ..."
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Cited by 11 (3 self)
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The aim of the present paper is to construct a stochastic process, whose law is the solution of the Smoluchowski’s coagulation equation. We introduce first a modified equation, dealing with the evolution of the distribu-tion Qt (dx) of the mass in the system. The advantage we take on this is that we can perform an unified study for both continuous and discrete models. The integro-partial-differential equation satisfied by {Qt}t≥0 can be interpreted as the evolution equation of the time marginals of a Markov pure jump process. At this end we introduce a nonlinear Poisson driven stochastic differential equation related to the Smoluchowski equation in the following way: if Xt satisfies this stochastic equation, then the law of Xt satisfies the modified Smoluchowski equation. The nonlinear process is richer than the Smoluchowski equation, since it provides historical information on the particles. Existence, uniqueness and pathwise behavior for the solution of this SDE are studied. Finally, we prove that the nonlinear process X can be obtained as the limit of a Marcus–Lushnikov procedure. 1. Introduction. The
EXPLOSION PHENOMENA IN STOCHASTIC COAGULATION–FRAGMENTATION MODELS
, 2005
"... First we establish explosion criteria for jump processes with an arbitrary locally compact separable metric state space. Then these results are applied to two stochastic coagulation–fragmentation models— the direct simulation model and the mass flow model. In the pure coagulation case, there is almo ..."
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Cited by 8 (1 self)
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First we establish explosion criteria for jump processes with an arbitrary locally compact separable metric state space. Then these results are applied to two stochastic coagulation–fragmentation models— the direct simulation model and the mass flow model. In the pure coagulation case, there is almost sure explosion in the mass flow model for arbitrary homogeneous coagulation kernels with exponent bigger than 1. In the case of pure multiple fragmentation with a continuous size space, explosion occurs in both models provided the total fragmentation rate grows sufficiently fast at zero. However, an example shows that the explosion properties of both models are not equivalent. 1. Introduction. Coagulation–fragmentation models are used in different application fields ranging from chemical engineering (reacting polymers, soot formation) or aerosol technology to astrophysics (formation of stars and planets). These models describe the behavior of a system of particles that
On a Boltzmann equation for elastic, inelastic and coalescing collisions, preprint 2003
"... Existence, uniqueness and qualitative behavior of the solution to a spatially homogeneous Boltzmann equation for particles undergoing elastic, inelastic and coalescing collisions are studied. Under general assumptions on the collision rates, we prove existence and uniqueness of a L1 solution. This s ..."
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Cited by 6 (3 self)
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Existence, uniqueness and qualitative behavior of the solution to a spatially homogeneous Boltzmann equation for particles undergoing elastic, inelastic and coalescing collisions are studied. Under general assumptions on the collision rates, we prove existence and uniqueness of a L1 solution. This shows in particular that the cooling effect (due to inelastic collisions) does not occur in finite time. In the long time asymptotic, we prove that the solution converges to a mass-dependent Maxwellian function (when only elastic collisions are considered), to a velocity Dirac mass (when elastic and inelastic collisions are considered) and to 0 (when elastic, inelastic and coalescing collisions are taken into account). We thus show in the latter case that the effect of coalescence is dominating in large time. Our proofs gather deterministic and
