Coagulation-fragmentation Processes Research Articles

Structured population models on Polish spaces: A unified approach including graphs, Riemannian manifolds and measure spaces to describe dynamics of heterogeneous populations

This paper presents a mathematical framework for modeling the dynamics of heterogeneous populations. Models describing local and non-local growth and transport processes appear in a variety of applications, such as crowd dynamics, tissue regeneration, cancer development and coagulation-fragmentation processes. The diverse applications pose a common challenge to mathematicians due to the multiscale nature of the structures that underlie the system’s self-organization and control. Similar abstract mathematical problems arise when formulating problems in the language of measure evolution on a multi-faceted state space. Motivated by these observations, we propose a general mathematical framework for nonlinear structured population models on abstract metric spaces, which are only assumed to be separable and complete. We exploit the structure of the space of non-negative Radon measures with the dual bounded Lipschitz distance (flat metric), which is a generalization of the Wasserstein distance, capable of addressing non-conservative problems. The formulation of models on general metric spaces allows considering infinite-dimensional state spaces or graphs and coupling discrete and continuous state transitions. This opens up exciting possibilities for modeling single-cell data, crowd dynamics or coagulation-fragmentation processes.

Double coset Markov chains

AbstractLetGbe a finite group. Let$H, K$be subgroups ofGand$H \backslash G / K$the double coset space. IfQis a probability onGwhich is constant on conjugacy classes ($Q(s^{-1} t s) = Q(t)$), then the random walk driven byQonGprojects to a Markov chain on$H \backslash G /K$. This allows analysis of the lumped chain using the representation theory ofG. Examples include coagulation-fragmentation processes and natural Markov chains on contingency tables. Our main example projects the random transvections walk on$GL_n(q)$onto a Markov chain on$S_n$via the Bruhat decomposition. The chain on$S_n$has a Mallows stationary distribution and interesting mixing time behavior. The projection illuminates the combinatorics of Gaussian elimination. Along the way, we give a representation of the sum of transvections in the Hecke algebra of double cosets, which describes the Markov chain as a mixture of Metropolis chains. Some extensions and examples of double coset Markov chains withGa compact group are discussed.

Fission-fusion dynamics and group-size-dependent composition in heterogeneous populations.

Many animal groups are heterogeneous and may even consist of individuals of different species, called mixed-species flocks. Mathematical and computational models of collective animal movement behavior, however, typically assume that groups and populations consist of identical individuals. In this paper, using the mathematical framework of the coagulation-fragmentation process, we develop and analyze a model of merge and split group dynamics, also called fission-fusion dynamics, for heterogeneous populations that contain two types (or species) of individuals. We assume that more heterogeneous groups experience higher split rates than homogeneous groups, forming two daughter groups whose compositions are drawn uniformly from all possible partitions. We analytically derive a master equation for group size and compositions and find mean-field steady-state solutions. We predict that there is a critical group size below which groups are more likely to be homogeneous and contain the abundant type or species. Despite the propensity of heterogeneous groups to split at higher rates, we find that groups are more likely to be heterogeneous but only above the critical group size. Monte Carlo simulation of the model show excellent agreement with these analytical model results. Thus, our model makes a testable prediction that composition of flocks are group-size-dependent and do not merely reflect the population level heterogeneity. We discuss the implications of our results to empirical studies on flocking systems.

Limit Shapes for Gibbs Ensembles of Partitions

We explicitly compute limit shapes for several grand canonical Gibbs ensembles of partitions of integers. These ensembles appear in models of aggregation and are also related to invariant measures of zero range and coagulation-fragmentation processes. We show, that all possible limit shapes for these ensembles fall into several distinct classes determined by the asymptotics of the internal energies of aggregates.

Stochastic coagulation-fragmentation processes with a finite number of particles and applications

Coagulation-fragmentation processes describe the stochastic association and dissociation of particles in clusters. Cluster dynamics with cluster-cluster interactions for a finite number of particles has recently attracted attention especially in stochastic analysis and statistical physics of cellular biology, as novel experimental data are now available, but their interpretation remains challenging. We derive here probability distribution functions for clusters that can either aggregate upon binding to form clusters of arbitrary sizes or a single cluster can dissociate into two sub-clusters. Using combinatorics properties and Markov chain representation, we compute steady-state distributions and moments for the number of particles per cluster in the case where the coagulation and fragmentation rates follow a detailed balance condition. We obtain explicit and asymptotic formulas for the cluster size and the number of clusters in terms of hypergeometric functions. To further characterize clustering, we introduce and discuss two mean times: one is the mean time two particles spend together before they separate and the other is the mean time they spend separated before they meet again for the first time. Finally, we discuss applications of the present stochastic coagulation-fragmentation framework in cell biology.

Transient detailed balance and product form for reaction networks

ABSTRACTThis paper explores the boundary of the set of reaction networks that have an exact transient (truncated) multidimensional Poisson or product-form distribution for the number of particles of different types. Motivated by the birth–death process, we introduce the notions of transient detailed balance and delay functions, and use these notions to obtain the novel transient product-form distribution in a coagulation-fragmentation process for polymers with a tree-like structure from that of the pure coagulation process.

Isotropic wave turbulence with simplified kernels: Existence, uniqueness, and mean-field limit for a class of instantaneous coagulation-fragmentation processes

The isotropic 4-wave kinetic equation is considered in its weak formulation using model (simplified) homogeneous kernels. Existence and uniqueness of solutions is proven in a particular setting where the kernels have a rate of growth at most linear. We also consider finite stochastic particle systems undergoing instantaneous coagulation-fragmentation phenomena and give conditions in which this system approximates the solution of the equation (mean-field limit).

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.

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.

Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths

We study random spatial permutations on Z^3 where each jump x -> \pi(x) is penalized by a factor exp(-T ||x-\pi(x)||^2). The system is known to exhibit a phase transition for low enough T where macroscopic cycles appear. We observe that the lengths of such cycles are distributed according to Poisson-Dirichlet. This can be explained heuristically using a stochastic coagulation-fragmentation process for long cycles, which is supported by numerical data.

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.

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.

The Asymptotic Behavior of Heterogeneous Coagulation-Fragmentation Processes

In this paper, we not only give a closed form of stationary distribution of heterogeneous coagulation-fragmentation process (HCFP) which models the coagulation, fragmentation and diffusion of clusters of particles on lattice, but also prove that the distribution of the number of k-cluster is asymptotically subjected to the Poisson distribution with the parameter h(k)r as the total number of particles N→∞.

Coagulation and fragmentation dynamics of inertial particles

Inertial particles suspended in many natural and industrial flows undergo coagulation upon collisions and fragmentation if their size becomes too large or if they experience large shear. Here we study this coagulation-fragmentation process in time-periodic incompressible flows. We find that this process approaches an asymptotic dynamical steady state where the average number of particles of each size is roughly constant. We compare the steady-state size distributions corresponding to two fragmentation mechanisms and for different flows and find that the steady state is mostly independent of the coagulation process. While collision rates determine the transient behavior, fragmentation determines the steady state. For example, for fragmentation due to shear, flows that have very different local particle concentrations can result in similar particle size distributions if the temporal or spatial variation in shear forces is similar.

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 Time Dynamics of Coagulation-Fragmentation Processes

We establish a characterization of coagulation-fragmentation processes, such that the induced birth and death processes depicting the total number of groups at time t≥0 are time homogeneous. Based on this, we provide a characterization of mean-field Gibbs coagulation-fragmentation models, which extends the one derived by Hendriks et al. As a by-product of our results, the class of solvable models is widened and a question posed by N. Berestycki and Pitman is answered, under restriction to mean-field models.

Coagulation and Fragmentation Models

Historically deterministic equations were used for modelling the dynamics of the density of clusters of different sizes in a medium that evolves by coagulation and/or fragmentation. An alternative stochastic approach uses the distribution of a typical particle in systems undergoing a random evolution. This approach provides tools for deriving limit equations, it easily extends to more general interactions and, finally, it shares the familiar advantages of particle-based methods for computing distributions in high dimensions. The stochastic theory is undergoing intense development, both at a theoretical level, in the construction of models and analysis of their properties, and at a computational level. Major mathematical challenges in the field of coagulation and fragmentation models are related to phase transitions due to fast coagulation (gelation) or fast fragmentation (formation of dust), and to explicit descriptions of the structure of partitions (e.g. allelic partitions induced by random mutations in models for the evolution of populations). A further direction of intensive study is the detailed analysis of particular coagulation and fragmentation processes, where special features (e.g., scaling properties) allow links to be made to Brownian motion and other Lévy processes, also to some interesting problems in random combinatorics. Major topics that have been discussed during the workshop include : 1. Spatial models of coagulation and fragmentation Models incorporating diffusion; scaling limits; derivation from particle dynamics; equilibrium measures for coagulation-fragmentation processes; formation of structured particles; diffusion limited aggregation 2. Phase transitions in coagulation and fragmentation models Gelation effects; shattering transition (appearance of dust in fragmentation); explosion phenomena; computation of the gelation time; uniqueness issues; convergence of particle systems 3. Aspects of random combinatorics Links to random graphs and trees; genealogy for certain large populations dynamics; mutation and allelic partitions; stochastic coalescents with multiple collisions ( \Lambda -coalescents) and fragmentations 4. Computational issues Approximation and numerics; analysis of algorithms; Monte Carlo issues around sensitivity in initial conditions and kernel parameters

Subcritical, critical and supercritical size distributions in random coagulation-fragmentation processes

We consider the asymptotic probability distribution of the size of a reversible random coagula-tion-fragmentation process in the thermodynamic limit. We prove that the distributions of small, medium and the largest clusters converge to Gaussian, Poisson and 0–1 distributions in the supercritical stage (post-gelation), respectively. We show also that the mutually dependent distributions of clusters will become independent after the occurrence of a gelation transition. Furthermore, it is proved that all the number distributions of clusters are mutually independent at the critical stage (gelation), but the distributions of medium and the largest clusters are mutually dependent with positive correlation coefficient in the supercritical stage. When the fragmentation strength goes to zero, there will exist only two types of clusters in the process, one type consists of the smallest clusters, the other is the largest one which has a size nearly equal to the volume (total number of units).

Critical behavior of the heterogeneous random coagulation-fragmentation processes

In this paper, we study the asymptotic distributions of the heterogeneous random coagulation-fragmentation processes (HCFP) which model the coagulation, fragmentation and diffusion of clusters of particles on a lattice. Based on a closed form of the stationary distribution for the HCFP, we prove that the mutually dependent clusters with a finite size (finite particles) in the sub-critical stage will become independent in the critical and super-critical stages, and the asymptotic distributions of the number of clusters may converge to Gaussian or Poisson distribution according to its size, to be small or large in the critical and super-critical stages.

A phase transition in the random transposition random walk

Our work is motivated by Bourque-Pevzner's simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on $n$ elements starting from the identity. Let $D_t$ be the minimum number of transpositions needed to go back to the identity element from the location at time $t$. $D_t$ undergoes a phase transition: for $0 1$, $D_cn/2 ~ u(c)n$ where u is an explicit function satisfying $u(x) < x/2$. Moreover we describe the fluctuations of $D_{cn/2}$ about its mean at each of the three stages (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the Erdős-Renyi random graph model.