Coagulation Fragmentation Research Articles

Numerical simulation of a transport fragmentation coagulation model

In this paper, we deal with numerical analysis of a transport fragmentation coagulation equation with power fragmentation rates and separable coagulation kernels. For numerical simulations, the model is rewritten in a conservative form and semi-discretized in space using a Laguerre pseudo-spectral method. Some error estimates are derived and efficiency of the numerical scheme is considered. The paper is concluded with several computational examples.

The singular coagulation equation with multiple fragmentation

In this paper, we prove the global existence and uniqueness of the solutions to the initial-value problem for the coagulation–fragmentation equation with singular coagulation kernel and multiple fragmentation kernel. The solution obtained in this case also satisfies the mass conservation law. The proof is based on strong convergence methods applied to suitably chosen unbounded coagulation kernels having singularities in both the coordinate axes and satisfying certain growth conditions, which can possibly reach up to a quadratic growth at infinity, and the fragmentation kernel covers a very large class of unbounded functions.

Pseudodifferential operators with non-regular operator-valued symbols

In this paper, we consider pseudodifferential operators with operator-valued symbols and their mapping properties, without assumptions on the underlying Banach space E. We show that, under suitable parabolicity assumptions, the $${W_p^k(\mathbb{R}^n, E)}$$ -realization of the operator generates an analytic semigroup. Our approach is based on oscillatory integrals and kernel estimates for them. An application to non-autonomous pseudodifferential Cauchy problems gives the existence and uniqueness of a classical solution. As an example, we include a discussion of coagulation–fragmentation processes.

Fractional Poisson processes and their representation by infinite systems of ordinary differential equations

Fractional Poisson processes, a rapidly growing area of non-Markovian stochastic processes, are useful in statistics to describe data from counting processes when waiting times are not exponentially distributed. We show that the fractional Kolmogorov–Feller equations for the probabilities at time t can be represented by an infinite linear system of ordinary differential equations of first order in a transformed time variable. These new equations resemble a linear version of the discrete coagulation–fragmentation equations, well-known from the non-equilibrium theory of gelation, cluster-dynamics and phase transitions in physics and chemistry.

Coagulation–fragmentation with mass conservation: Self-consistent mean rate model

We have developed the self-consistent mean rate approach that makes it possible to analytically solve the coagulation–fragmentation balance equations subject to the mass conservation constraint. The developed approach is not restricted to a specific form of the coagulation and fragmentation rates, thus being applicable to a variety of different coagulation–fragmentation processes. As an example of the practical applicability of the developed method we have calculated the aggregate size distribution and average aggregate diameter for the case of shear induced coagulation–fragmentation.

A one-dimensional coagulation–fragmentation process with a dynamical phase transition

We introduce a reversible Markovian coagulation–fragmentation process on the set of partitions of {1,…,L} into disjoint intervals. Each interval can either split or merge with one of its two neighbors. The invariant measure can be seen as the Gibbs measure for a homogeneous pinning model (Giacomin (2007) [10]). Depending on a parameter λ, the typical configuration can be either dominated by a single big interval (delocalized phase), or composed of many intervals of order 1 (localized phase), or the interval length can have a power law distribution (critical regime). In the three cases, the time required to approach equilibrium (in total variation) scales very differently with L. In the localized phase, when the initial condition is a single interval of size L, the equilibration mechanism is due to the propagation of two “fragmentation fronts” which start from the two boundaries and proceed by power-law jumps.

Analytic fragmentation semigroups and continuous coagulation–fragmentation equations with unbounded rates

In this paper we show that continuous fragmentation operators are sectorial for a large range of physically relevant fragmentation rates and use this fact to prove classical solvability of the combined coagulation–fragmentation equation with unbounded coagulation kernels.

Coagulation–fragmentation for a finite number of particles and application to telomere clustering in the yeast nucleus

We develop a coagulation–fragmentation model to study a system composed of a small number of stochastic objects moving in a confined domain, that can aggregate upon binding to form local clusters of arbitrary sizes. A cluster can also dissociate into two subclusters with a uniform probability. To study the statistics of clusters, we combine a Markov chain analysis with a partition number approach. Interestingly, we obtain explicit formulas for the size and the number of clusters in terms of hypergeometric functions. Finally, we apply our analysis to study the statistical physics of telomeres (ends of chromosomes) clustering in the yeast nucleus and show that the diffusion–coagulation–fragmentation process can predict the organization of telomeres.

Global classical solutions of coagulation–fragmentation equations with unbounded coagulation rates

We study discrete fragmentation coagulation equations in spaces X p , p > 1 , consisting of distributions having the p th moments finite. We show that for sufficiently regular fragmentation laws the fragmentation semigroup is analytic in X p , and fully characterize the domain of its generator. This allows for explicit characterization of the domains of the fractional powers of the generator through real interpolation. Finally, we use the linear results to show the existence of global classical solutions to fragmentation coagulation equations for a class of unbounded coagulation kernels.

Global strict solutions to continuous coagulation–fragmentation equations with strong fragmentation

We give an elementary proof of the unique, global-in-time solvability of the coagulation–(multiple) fragmentation equation with polynomially bounded fragmentation and particle production rates and a bounded coagulation rate. The proof relies on a new result concerning domain invariance for the fragmentation semigroup which is based on a simple monotonicity argument.

Moment bounds for the solutions of the Smoluchowski equation with coagulation and fragmentation

AbstractWe prove various $L^p(\mathbb{R}^d\times[0,T])$ bounds on momentswhere fm is a solution of the discrete (respectively, continuous) Smoluchowski coagulation–fragmentation equations with diffusion. In a previous paper we proved similar results for all weak solutions to the discrete Smoluchowski equation provided that there is no fragmentation and certain moments are bounded in suitable Lq-spaces initially. We prove the corresponding results in the case of the continuous Smoluchowski equation. When there is also fragmentation, we need to assume that the solution f is regular in the sense that f can be approximated by solutions to the Smoluchowski equation for which the coagulation and fragmentation coefficients are 0 when the cluster sizes are large. We also need suitable assumptions on the coagulation rates to avoid gelation. On the fragmentation rate β we assume that supn supm≤l β(m, n)/n < ∞ for every positive l, and that there exist constants a0 ≥ 0 and c0 such that β(n,m) ≤ c0(n + m)a0a0.

A kinetic model for coagulation–fragmentation

The aim of this paper is to show an existence theorem for a kinetic model of coagulation–fragmentation with initial data satisfying the natural physical bounds, and assumptions of finite number of particles and finite Lp-norm. We use the notion of renormalized solutions introduced by DiPerna and Lions (1989) [3], because of the lack of a priori estimates. The proof is based on weak-compactness methods in L1, allowed by Lp-norms propagation.

Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion

We present a new a priori estimate for discrete coagulation–fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a global L2 bound on the mass density and was previously used, for instance, in the context of reaction–diffusion equations.In this paper we demonstrate two lines of applications for such an estimate: On the one hand, it enables to simplify parts of the known existence theory and allows to show existence of solutions for generalised models involving collision-induced, quadratic fragmentation terms for which the previous existence theory seems difficult to apply. On the other hand and most prominently, it proves mass conservation (and thus the absence of gelation) for almost all the coagulation coefficients for which mass conservation is known to hold true in the space homogeneous case.

Strongly differentiable solutions of the discrete coagulation–fragmentation equation

We examine an infinite system of ordinary differential equations that models the binary coagulation and multiple fragmentation of clusters. In contrast to previous investigations, our analysis does not involve finite-dimensional truncations of the system. Instead, we treat the problem as an infinite-dimensional differential equation, posed in an appropriate Banach space, and apply perturbation results from the theory of strongly continuous semigroups of operators. The existence and uniqueness of physically meaningful solutions are established for uniformly bounded coagulation rates but with no growth restrictions imposed on the fragmentation rates.

Veneziano amplitudes, spin chains and Abelian reduction of QCD

Although QCD can be treated perturbatively in the high energy limit, lower energies require uses of nonperturbative methods such as ADS/CFT and/or Abelian reduction. These methods are not equivalent. While the first is restricted to supersymmetric Yang–Mills model with number of colors going to infinity, the second is not restricted by requirements of supersymmetry and is designed to work in the physically realistic limit of a finite number of colors. In this paper we provide arguments in favor of the Abelian reduction methods. This is achieved by further developing results of our recent works re-analyzing Veneziano and Veneziano-like amplitudes and the models associated with these amplitudes. It is shown, that the obtained new partition function for these amplitudes can be mapped exactly into that for the Polychronakos–Frahm (P–F) spin chain model recoverable from the Richardon–Gaudin (R–G) XXX spin chain model originally designed for treatments of the BCS-type superconductivity. Because of this, it is demonstrated that the obtained mapping is compatible with the method of Abelian reduction. The R–G model is recovered from the asymptotic (WKB-type) solutions of the rational Knizhnik–Zamolodchikov (K–Z) equation. Linear independence of these solutions is controlled by determinants whose explicit form (up to a constant) coincides with Veneziano (or Veneziano-like) amplitudes. In the simplest case, the determinantal conditions coincide with those discovered by Kummer in the 19th century. Kummer’s results admit physical interpretation by relating determinantal formula(s) to Veneziano-like amplitudes. Furthermore, these amplitudes can be interpreted as Poisson–Dirichlet distributions playing a central role in the stochastic theory of random coagulation–fragmentation processes. Such an interpretation is complementary to that known for the Lund model widely used for the description of coagulation–fragmentation processes in QCD.

On the invariant distribution of a one-dimensional avalanche process

We consider an interacting particle system (ηt)t≥0 with values in {0, 1}ℤ, in which each vacant site becomes occupied with rate 1, while each connected component of occupied sites become vacant with rate equal to its size. We show that such a process admits a unique invariant distribution, which is exponentially mixing and can be perfectly simulated. We also prove that for any initial condition, the avalanche process tends to equilibrium exponentially fast, as time increases to infinity. Finally, we consider a related mean-field coagulation–fragmentation model, we compute its invariant distribution and we show numerically that it is very close to that of the interacting particle system.

Gelation time in the discrete coagulation–fragmentation equations with a bilinear coagulation kernel

The discrete coagulation–fragmentation equations with a bilinear coagulation kernel, describing the evolution of clusters of particles undergoing binary reactions, are studied. We compute the gelation time and post-gelation mass for the pure coagulation equation. A method of characteristics is developed to solve numerically the partial differential equation satisfied by a moment generating function for a product coagulation kernel and a constant fragmentation rate. The accuracy of the numerical method is verified by comparison of the numerical results with an exact solution for the number density of monomers. For a given coagulation rate, a critical value of the fragmentation rate, values greater than which lead to mass conservation, is identified.

Asymptotic Analysis of Coagulation–Fragmentation Equations of Carbon Nanotube Clusters

The possibility of the existence of single-wall carbon nanotubes (SWNTs) in organic solvents in the form of clusters is discussed. A theory is developed based on abundletmodel for clusters describing the distribution function of clusters by size. The phenomena have a unified explanation in the framework of the bundlet model of a cluster, in accordance with which the free energy of an SWNT involved in a cluster is combined from two components: a volume one, proportional to the number of moleculesnin a cluster, and a surface one, proportional ton1/2. During the latter stage of the fusion process, the dynamics were governed mainly by the displacement of the volume of liquid around the fusion site between the fused clusters. The same order of magnitude for the average cluster-fusion velocity is deduced if the fusion process starts with several fusion sites. Based on a simple kinetic model and starting from the initial state of pure monomers, micellization of rod-like aggregates at high critical micelle concentration occurs in three separated stages. A convenient relation is obtained for <n> at transient stage. At equilibrium, another relation determines dimensionless binding energy α. A relation with surface dilatational viscosity is obtained.

On a coagulation and fragmentation equation with mass loss

A nonlinear integro-differential equation that models a coagulation and multiple fragmentation process in which continuous and discrete fragmentation mass loss can occur is examined using the theory of strongly continuous semigroups of operators. Under the assumptions that the coagulation kernel is constant, the fragmentation-rate function is linearly bounded, and the continuous mass-loss-rate function is locally Lipschitz, global existence and uniqueness of solutions that lose mass in accordance with the model are established. In the case when no coagulation is present and the fragmentation process is binary with constant fragmentation kernel and constant continuous mass loss, an explicit formula is given for the associated substochastic semigroup.

Coagulation–fragmentation duality, Poisson–Dirichlet distributions and random recursive trees

In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the two-parameter family of Poisson--Dirichlet distributions $\operatorname {PD}(\alpha,\theta)$ that take values in this space. We introduce families of random fragmentation and coagulation operators $\mathrm {Frag}_{\alpha}$ and $\mathrm {Coag}_{\alpha,\theta}$, respectively, with the following property: if the input to $\mathrm {Frag}_{\alpha}$ has $\operatorname {PD}(\alpha,\theta)$ distribution, then the output has $\operatorname {PD}(\alpha,\theta+1)$ distribution, while the reverse is true for $\mathrm {Coag}_{\alpha,\theta}$. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between $\operatorname {PD}(\alpha,\theta)$ and $\operatorname {PD}(\alpha\beta,\theta)$. Repeated application of the $\mathrm {Frag}_{\alpha}$ operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation--fragmentation duality.