Coagulation Fragmentation Research Articles

Travelling waves for a fast reaction limit of a discrete coagulation-fragmentation model with diffusion and proliferation.

We study traveling wave solutions for a reaction-diffusion model, introduced in the article Calvez et al. (Regime switching on the propagation speed of travelling waves of some size-structured myxobacteriapopulation models, 2023), describing the spread of the social bacterium Myxococcus xanthus. This model describes the spatial dynamics of two different cluster sizes: isolated bacteria and paired bacteria. Two isolated bacteria can coagulate to form a cluster of two bacteria and conversely, a pair of bacteria can fragment into two isolated bacteria. Coagulation and fragmentation are assumed to occur at a certain rate denoted by k. In this article we study theoretically the limit of fast coagulation fragmentation corresponding mathematically to the limit when the value of the parameter k tends to . For this regime, we demonstrate the existence and uniqueness of a transition between pulled and pushed fronts for a certain critical ratio between the diffusion coefficient of isolated bacteria and the diffusion coefficient of paired bacteria. When the ratio is below , the critical front speed is constant and corresponds to the linear speed. Conversely, when the ratio is above the critical threshold, the critical spreading speed becomes strictly greater than the linear speed.

Coagulation–Fragmentation Equilibrium for Charged Dust: Abundance of Submicron Grains Increases Dramatically in Protoplanetary Disks

Dust coagulation in protoplanetary disks is not straightforward and is subject to several slowdown mechanisms, such as bouncing, fragmentation, and radial drift to the star. Furthermore, dust grains in UV-shielded disk regions are negatively charged due to collisions with the surrounding electrons and ions, which leads to their electrostatic repulsion. For typical disk conditions, the relative velocities between micron-sized grains are small, and their collisions are strongly affected by the repulsion. On the other hand, collisions between pebble-sized grains can be too energetic, leading to grain fragmentation. The aim of the present paper is to study the combined effect of the electrostatic and fragmentation barriers on dust evolution. We numerically solve the Smoluchowski coagulation–fragmentation equation for grains whose charging occurs under conditions typical for the inner disk regions, where thermal ionization operates. We find that dust fragmentation efficiently resupplies the population of small grains under the electrostatic barrier. As a result, the equilibrium abundance of submicron grains is enhanced by several orders of magnitude compared to the case of neutral dust. For some conditions with fragmentation velocities of ∼1 m s−1, macroscopic grains are completely destroyed.

Modelling Silicosis: Dynamics of a Model with Piecewise Constant Rate Coefficients

We study the dynamics about equilibria of an infinite dimensional system of ordinary differential equations of coagulation–fragmentation–death type that was introduced recently by da Costa et al. (Eur J Appl Math 31(6):950–967, 2020) as a model for the silicosis disease mechanism. For a class of piecewise constant rate coefficients an appropriate change of variables allows for the appearance of a closed finite dimensional subsystem of the infinite-dimensional system and the analysis of the eigenvalues of the linearizations of this finite dimensional subsystem about the equilibria is then used to obtain the results on the stability of the equilibria in the original infinite dimensional model.

Dust accumulation near the magnetospheric truncation of protoplanetary discs around T Tauri stars

ABSTRACT The prevalence of short-period super-Earths that are independent of host metallicity challenges the theoretical construction of their origin. We propose that dust trapping in the global pressure bump induced by magnetospheric truncation in evolved protoplanetary discs (PPDs) around T Tauri stars offers a promising formation mechanism for super-Earths, where the host metallicity is already established. To better understand this planet-forming scenario, we construct a toy inner disc model and focus on the evolution of dust trapped in the bump, taking into account the supply from drifting pebbles and loss due to funnel flows. We develop an implicit coagulation–fragmentation code, rubble, and perform a suite of simulations to evolve the local dust-size distributions. Our study for the first time considers dust feedback effect on turbulent diffusion in this kind of model. We report that efficient dust growth and significant accumulation of dust mass are possible in less turbulent disc with sturdier solids and with faster external supply, laying out a solid foundation for further growth towards planetesimals and planetary embryos. We further find that, depending on the dominant process, solid mass may predominantly accumulate in cm-sized grains or particles in runaway growth, indicating different ways of forming planetesimals. Furthermore, these various outcomes show different efficiencies in saving dust from funnel flows, suggesting that they may be distinguishable by constraining the opacity of funnel flows. Also, these diverse dust behaviours may help to explain the observed dipper stars and rapidly varying shadows in PPDs.

Modelling silicosis: Existence, uniqueness and basic properties of solutions

We present a model for the silicosis disease mechanism following the original proposal by Tran et al. (1995), as modified recently by da Costa et al. (2020). The model consists in an infinite ordinary differential equation system of coagulation–fragmentation–death type. Results of existence, uniqueness, continuous dependence on the initial data and differentiability of solutions are proved for the initial value problem.

Growth-fragmentation-coagulation equations with unbounded coagulation kernels.

In this paper, we prove the global in time solvability of the continuous growth-fragmentation-coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth-fragmentation semigroup that allows us to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic. This article is part of the theme issue 'Semigroup applications everywhere'.

Existence and uniqueness of steady-state solution to a singular coagulation–fragmentation equation

In this article, we prove the existence of an equilibrium solution to the coagulation–fragmentation equation with •[(i)] a singular coagulation kernel in the form K(x,y)=(1+xλ+yλ)(xy)σ,where σ∈0,12 and λ−σ∈[0,1], and•[(ii)] a fragmentation kernel in the form F(x,y)=b0+b(x+y)m,where b0>0 and b,m≥0. Importantly, in the entire study, the detailed balance condition is not assumed. It is shown that the equilibrium solution of the problem under consideration is unique in the space of all continuous functions on (0,∞). The proposed results are demonstrated by a numerical illustration.

Fragmentation Coagulation Based Mixed Membership Stochastic Blockmodel

The Mixed-Membership Stochastic Blockmodel (MMSB) is proposed as one of the state-of-the-art Bayesian relational methods suitable for learning the complex hidden structure underlying the network data. However, the current formulation of MMSB suffers from the following two issues: (1), the prior information (e.g. entities' community structural information) can not be well embedded in the modelling; (2), community evolution can not be well described in the literature. Therefore, we propose a non-parametric fragmentation coagulation based Mixed Membership Stochastic Blockmodel (fcMMSB). Our model performs entity-based clustering to capture the community information for entities and linkage-based clustering to derive the group information for links simultaneously. Besides, the proposed model infers the network structure and models community evolution, manifested by appearances and disappearances of communities, using the discrete fragmentation coagulation process (DFCP). By integrating the community structure with the group compatibility matrix we derive a generalized version of MMSB. An efficient Gibbs sampling scheme with Polya Gamma (PG) approach is implemented for posterior inference. We validate our model on synthetic and real world data.

Uniqueness of solutions to the coagulation–fragmentation equation with singular kernel

The existence of a solution to an important singular coagulation equation with a multiple fragmentation kernel has been recently proved in Jpn J Ind Appl Math 35(3):1283–1302, 2018. This paper proves the uniqueness of the solution to the same problem in the function space $$\varOmega _{.,r_2} (T) = \bigcup _{\lambda >0 }\varOmega _{\lambda , r_2} (T)$$, where $$\varOmega _{\lambda , r_2} (T)$$ is the space of all continuous functions f such that $$\begin{aligned} \Vert f\Vert _{\lambda , r_2} : = \sup \limits _{0\le t \le T} \int _0^{\infty } \left( \exp (\lambda x ) + \frac{1}{x^{r_2}}\right) |f(x,t)| dx ~<~ \infty \end{aligned}$$and $$0< r_2 < 1$$.

Global solutions of continuous coagulation–fragmentation equations with unbounded coefficients

In this paper we prove the existence of global classical solutions to continuous coagulation–fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either rate. This is achieved by proving a new result on the analyticity of the fragmentation semigroup and then using its regularizing properties to prove the local and then, under a stronger assumption, the global classical solvability of the coagulation–fragmentation equation considered as a semilinear perturbation of the linear fragmentation equation. Furthermore, we show that weak solutions of the coagulation–fragmentation equation, obtained by the weak compactness method, coincide with the classical local in time solutions provided the latter exist.

Solutions with peaks for a coagulation–fragmentation equation. Part I: stability of the tails

The aim of this two-part paper is to investigate the stability properties of a special class of solutions to a coagulation–fragmentation equation. We assume that the coagulation kernel is close to the diagonal kernel, and that the fragmentation kernel is diagonal. We construct a two-parameter family of stationary solutions concentrated in Dirac masses. We carefully study the asymptotic decay of the tails of these solutions, showing that this behavior is stable. In a companion paper, we prove that for initial data which are sufficiently concentrated, the corresponding solutions approach one of these stationary solutions for large times.

Modelling silicosis: The structure of equilibria

We analyse the structure of equilibria of a coagulation–fragmentation–death model of silicosis. We present exact multiplicity results in the particular case of piecewise constant coefficients, results on existence and non-existence of equilibria in the general case, as well as precise asymptotics for the infinite series that arise in the case of power law coefficients.

Kinetic Equations for Particle Clusters Differing in Shape and the H-theorem

The question of constructing models for the evolution of clusters that differ in shape based on the Boltzmann’s H-theorem is investigated. The first, simplest kinetic equations are proposed and their properties are studied: the conditions for fulfilling the H-theorem (the conditions for detailed and semidetailed balance). These equations are to generalize the classical coagulation–fragmentation type equations for cases when not only mass but also particle shape is taken into account. To construct correct (physically grounded) kinetic models, the fulfillment of the condition of detailed balance is shown to be necessary to monitor, since it is proved that for accepted frequency functions, the condition of detailed balance is fulfilled and the H-theorem is valid. It is shown that for particular and very important cases, the H-theorem holds: the fulfillment of the Arrhenius law and the additivity of the activation energy for interacting particles are found to be essential. In addition, based on the connection of the principle of detailed balance with the Boltzmann equation for the probability of state, the expressions for the reaction rate coefficients are obtained.

Mass-conserving self-similar solutions to coagulation–fragmentation equations

Existence of mass-conserving self-similar solutions with a sufficiently small total mass is proved for a specific class of homogeneous coagulation and fragmentation coefficients. The proof combines a dynamical approach to construct such solutions for a regularized coagulation–fragmentation equation in scaling variables and a compactness method.

Quasi-Monte Carlo simulation of coagulation–fragmentation

We extend a quasi-Monte Carlo scheme designed for coagulation to the simulation of the coagulation–fragmentation equation. A number N of particles is used to approximate the mass distribution. After time discretization, three-dimensional quasi-random points decide at every time step whether the particles are undergoing coagulation or fragmentation. We prove that the scheme converges as the time step is small and N is large. In a numerical test, we show that the computed solutions are in good agreement with the exact ones, and that the error of the algorithm is smaller than the error of a corresponding Monte Carlo scheme using the same discretization parameters.

The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation

In this paper we study the discrete coagulation–fragmentation models with growth, decay and sedimentation. We demonstrate the existence and uniqueness of classical global solutions provided the linear processes are sufficiently strong. This paper extends several previous results both by considering a more general model and and also signnificantly weakening the assumptions. Theoretical conclusions are supported by numerical simulations.

Existence of mass conserving solution for the coagulation–fragmentation equation with singular kernel

In this article, we investigate the existence of a solution to the initial value problem associated to a singular coagulation–fragmentation equation. We analyze the problem with singular coagulation kernel and multiple fragmentation kernel. It is also shown that the solution satisfies the mass conservation law.

Homogenization of the discrete diffusive coagulation–fragmentation equations in perforated domains

The asymptotic behavior of the solution of an infinite set of Smoluchowski's discrete coagulation–fragmentation–diffusion equations with non-homogeneous Neumann boundary conditions, defined in a periodically perforated domain, is analyzed. Our homogenization result, based on Nguetseng–Allaire two-scale convergence, is meant to pass from a microscopic model (where the physical processes are properly described) to a macroscopic one (which takes into account only the effective or averaged properties of the system). When the characteristic size of the perforations vanishes, the information given on the microscale by the non-homogeneous Neumann boundary condition is transferred into a global source term appearing in the limiting (homogenized) equations. Furthermore, on the macroscale, the geometric structure of the perforated domain induces a correction in the diffusion coefficients.

Analytic Fragmentation Semigroups and Classical Solutions to Coagulation–fragmentation Equations — a Survey

In the paper we present a survey of recent results obtained by the author that concern discrete fragmentation–coagulation models with growth. Models like that are particularly important in mathematical biology and ecology where they describe the aggregation of living organisms. The main results discussed in the paper are the existence of classical semigroup solutions to the fragmentation–coagulation equations.

The [formula omitted]-theorem for the physico-chemical kinetic equations with discrete time and for their generalizations

In this paper the generalizations of the physico-chemical kinetics equations, including classical and quantum physico-chemical kinetics, coagulation–fragmentation equations, discrete velocity models of classical and quantum kinetic equation and, generally speaking, all discrete physico-chemical kinetics is considered. We make time discrete in these equations and prove the H-theorem.