Numerical approximation of a coagulation-fragmentation model for animal group size statistics

We study numerically a coagulation-fragmentation model derived by Niwa [17] and further elaborated by Degond et al. [5]. In [5] a unique equi- librium distribution of group sizes is shown to exist in both cases of continuous and discrete group size distributions. We provide a numerical investigation of these equilibria using three different methods to approximate the equilibrium: a recursive algorithm based on the work of Ma et. al. [12], a Newton method and the resolution of the time-dependent problem. All three schemes are val- idated by showing that they approximate the predicted small and large size asymptotic behaviour of the equilibrium accurately. The recursive algorithm is used to investigate the transition from discrete to continuous size distributions and the time evolution scheme is exploited to show uniform convergence to equilibrium in time and to determine convergence rates

Monte Carlo simulation of particle interactions at high dynamic range: Advancing beyond the Googol

We present a method which extends Monte Carlo studies to situations that require a large dynamic range in particle number. The underlying idea is that, in order to calculate the collisional evolution of a system, some particle interactions are more important than others and require more resolution, while the behavior of the less important, usually of smaller mass, particles can be considered collectively. In this approximation groups of identical particles, sharing the same mass and structural parameters, operate as one unit. The amount of grouping is determined by the zoom factor -- a free parameter that determines on which particles the computational effort is focused. Two methods for choosing the zoom factors are discussed: the `equal mass method,' in which the groups trace the mass density of the distribution, and the `distribution method,' which additionally follows fluctuations in the distribution. Both methods achieve excellent correspondence with analytic solutions to the Smoluchowski coagulation equation. The grouping method is furthermore applied to simulations involving runaway kernels, where the particle interaction rate is a strong function of particle mass, and to situations that include catastrophic fragmentation. For the runaway simulations previous predictions for the decrease of the runaway timescale with the initial number of particles ${\cal N}$ are reconfirmed, extending ${\cal N}$ to $10^{160}$. Astrophysical applications include modeling of dust coagulation, planetesimal accretion, and the dynamical evolution of stars in large globular clusters. The proposed method is a powerful tool to compute the evolution of any system where the particles interact through discrete events, with the particle properties characterized by structural parameters.Comment: 18 pages, 10 figures. Re-submitted to ApJ with comments of the referee include

Applied mathematical modelling with new parameters and applications to some real life problems

Some Epidemic models with fractional derivatives were proved to be well-defined, well-posed and more accurate [34, 51, 116], compared to models with the conventional derivative. An Ebola epidemic model with non-linear transmission is fully analyzed. The model is expressed with the conventional time derivative with a new parameter included, which happens to be fractional (that derivative is called the derivative). We proved that the model is well-de ned and well-posed. Moreover, conditions for boundedness and dissipativity of the trajectories are established. Exploiting the generalized Routh-Hurwitz Criteria, existence and stability analysis of equilibrium points for the Ebola model are performed to show that they are strongly dependent on the non-linear transmission. In particular, conditions for existence and stability of a unique endemic equilibrium to the Ebola system are given. Numerical simulations are provided for particular expressions of the non-linear transmission, with model's parameters taking di erent values. The resulting simulations are in concordance with the usual threshold behavior. The results obtained here may be signi cant for the ght and prevention against Ebola haemorrhagic fever that has so far exterminated hundreds of families and is still a ecting many people in West-Africa and other parts of the world. The full comprehension and handling of the phenomenon of shattering, sometime happening during the process of polymer chain degradation [129, 142], remains unsolved when using the traditional evolution equations describing the degradation. This traditional model has been proved to be very hard to handle as it involves evolution of two intertwined quantities. Moreover, the explicit form of its solution is, in general, impossible to obtain. We explore the possibility of generalizing evolution equation modeling the polymer chain degradation and analyze the model with the conventional time derivative with a new parameter. We consider the general case where the breakup rate depends on the size of the chain breaking up. In the process, the alternative version of Sumudu integral transform is used to provide an explicit form of the general solution representing the evolution of polymer sizes distribution. In particular, we show that this evolution exhibits existence of complex periodic properties due to the presence of cosine and sine functions governing the solutions. Numerical simulations are performed for some particular cases and prove that the system describing the polymer chain degradation contains complex and simple harmonic poles whose e ects are given by these functions or a combination of them. This result may be crucial in the ongoing research to better handle and explain the phenomenon of shattering. Lastly, it has become a conjecture that power series like Mittag-Le er functions and their variants naturally govern solutions to most of generalized fractional evolution models such as kinetic, di usion or relaxation equations. The question is to say whether or not this is always true! Whence, three generalized evolution equations with an additional fractional parameter are solved analytically with conventional techniques. These are processes related to stationary state system, relaxation and di usion. In the analysis, we exploit the Sumudu transform to show that investigation on the stationary state system leads to results of invariability. However, unlike other models, the generalized di usion and relaxation models are proven not to be governed by Mittag-Le er functions or any of their variants, but rather by a parameterized exponential function, new in the literature, more accurate and easier to handle. Graphical representations are performed and also show how that parameter, called ; can be used to control the stationarity of such generalized models.Mathematical SciencesPh. D. (Applied Mathematics

Classical and Quantum Mechanical Models of Many-Particle Systems

This workshop was dedicated to the presentation of recent results in the field of the mathematical study of kinetic theory and its naturalextensions (statistical physics and fluid mechanics). The main models are the Vlasov(-Poisson) equation and the Boltzmann equation, which are obtainedas limits of many-body equations (Newton’s equations in the classical case and Schrödinger’s equation in the quantum case) thanks to the mean-field and Boltzmann-Grad scalings. Numerical aspects and applications to mechanics, physics, engineering and biology were also discussed

Multi-Scale Fluctuations in Non-Equilibrium Systems: Statistical Physics and Biological Application

Understanding how fluctuations continuously propagate across spatial scales is fundamental for our understanding of inanimate matter. This is exemplified by self-similar fluctuations in critical phenomena and the propagation of energy fluctuations described by the Kolmogorov-Law in turbulence. Our understanding is based on powerful theoretical frameworks that integrate fluctuations on intermediary scales, as in renormalisation group or coupled mode theory. In striking contrast to typical inanimate systems, living matter is typically organised into a hierarchy of processes on a discrete set of spatial scales: from biochemical processes embedded in dynamic subcellular compartments to cells giving rise to tissues. Therefore, the understanding of living matter requires novel theories that predict the interplay of fluctuations on multiple scales of biological organisation and the ensuing emergent degrees of freedom. In this thesis, we derive a general theory of the multi-scale propagation of fluctuations in non-equilibrium systems and show that such processes underlie the regulation of cellular behaviour. Specifically, we draw on paradigmatic systems comprising stochastic many-particle systems undergoing dynamic compartmentalisation. We first derive a theory for emergent degrees of freedom in open systems, where the total mass is not conserved. We show that the compartment dynamics give rise to the localisation of probability densities in phase space resembling quasi-particle behaviour. This emergent quasi-particle exhibits fundamentally different response kinetics and steady states compared to systems lacking compartment dynamics. In order to investigate a potential biological function of such quasi-particle dynamics, we then apply this theory to the regulation of cell death. We derive a model describing the subcellular processes that regulate cell death and show that the quasi-particle dynamics gives rise to a kinetic low-pass filter which suppresses the response of the cell to fast fluituations in cellular stress signals. We test our predictions experimentally by quantifying cell death in cell cultures subject to stress stimuli varying in strength and duration. In closed systems, where the total mass is conserved, the effect of dynamic compartmentalisation depends on details of the kinetics on the scale of the stochastic many-particle dynamics. Using a second quantisation approach, we derive a commutator relation between the kinetic operators and the change in total entropy. Drawing on this, we show that the compartment dynamics alters the total entropy if the kinetics of the stochastic many-particle dynamics violate detailed balance. We apply this mechanism to the activation of cellular immune responses to RNA-virus infections. We show that dynamic compartmentalisation in closed systems gives rise to giant density fluctuations. This facilitates the emergence of gelation under conditions that violate theoretical gelation criteria in the absence of compartment dynamics. We show that such multi-scale gelation of protein complexes on the membranes of dynamic mitochondria governs the innate immune response. Taken together, we provide a general theory describing the multi-scale propagation of fluctuations in biological systems. Our work pioneers the development of a statistical physics of such systems and highlights emergent degrees of freedom spanning different scales of biological organisation. By demonstrating that cells manipulate how fluctuations propagate across these scales, our work motivates a rethinking of how the behaviour of cells is regulated

Population balance modelling of soot formation in laminar flames

In this thesis, a discretised population balance eqaution (PBE) with a comprehensive model of soot formation processes has been coupled with the computational fluid dynamics (CFD) to predict the soot evolution in laminar diffusion flames. Contributions have been made in terms of methodology, modelling and applications. First of all, a conservative finite volume method is proposed to discretise the PBE with regard to the coagulation process. This method rigorously calculates the double integrals arising from the coagulation terms via a geometric representation, and exactly balances the coagulation source and sink terms to conserve moments. It proves that the proposed method is able to accurately predict the distribution with a small number of sections and conserve the first moment (or any other single moment) in the coagulation process, in an extensive test of various coagulation kernels, initial distributions and 'self-preserving' distributions, by comparison with analytical solutions and direct numerical solutions of the discrete PBE. Moreoever, the method is also flexible to an arbitrary non-uniform grid. Later on, the proposed method is also coupled with the CFD program to simulate Santoro flame, a laminar ethylene diffusion flame, for the validation on its accuracy, economy and robustness. Furthermore, the simulation results have been compared with simultaneous multiple diagnostics measurements drawn from a single data source, providing guidance on soot kinetic models. Three well-established PAH-based chemical reaction mechanisms, ABF, BBP and KM2, are employed to model the inception of soot precursors and oxidants. The physical model involves the nucleation by PAH dimerisation, surface growth by HACA mechanism and PAH condensation, size-dependent coagulation. Experimental signals are directly modelled, including the line-of-sight attenuation (LOSA) for the integrated soot volume fraction, planar OH laser-induced fluorescence (OH-PLIF) and elastic light scattering (ELS) for the soot distribution. The comparisons between model predictions and experimental measurements reflect the predictive capability for soot formation in laminar diffusion flame in terms of the flame structure, soot appearance and amount of soot production. The background gas phase chemistry clearly affects the soot modelling and a sensitivity analysis suggests that coordinating the rates of nucleation and surface growth help adjust the soot production on the centreline and sooty wings. Finally, the same soot model has been extended to two studies of diffusion flames with blends oxygen-containing surrogates: (1) methyl decanoate (MD) with the addition of dibutyl ether (DBE); (2) four practical methyl ester-based real biodiesels and their blends with petroleum diesel. In the first case study, aiming to reproduce an experiment which was to investigate the effects of dibutyl ether (DBE) addition to the biodiesel surrogate (methyl decanoate, MD), a combined and reduced MD-DBE-PAH mechanism from three sub-mechanism sources has been employed in the simulation. Due to the heavy molecular weight of the biodiesel fuel, the terms of the effect of molecular weight, thermophoresis and Dufour effect in the energy equation exhibit a similar magnitude with the original diffusion term, especially in the region of high temperature and a large gradient of the average molecular weight. Predicted temperature profiles are in good comparison with the experiment in terms of position and absolute value. The swallow-tail shape of the soot region and the absolute soot production are correctly predicted by the simulation. In terms of soot suppression, the model predicts 33\% reduction of soot as the DBE addition ranges from 0\% to 40\%, in contrast to around 55\% reduction measured in the experiment. In the second study, a semi-detailed kinetic mechanism for the pyrolysis and combustion of a large variety of biodiesels fuels are considered. The model successfully captures the reduction of soot formation by addition of biodiesels, but not necessarily the rate of decrease with blending. The current investigation offers pioneer and encouraging results on modelling soot formation in biodiesel flames, which has been fewly explored.Open Acces

Aggregation-fragmentation and individual dynamics of active clusters

International audienceA remarkable feature of active matter is the propensity to self-organize. One striking instance of this ability to generate spatial structures is the cluster phase, where clusters broadly distributed in size constantly move and evolve through particle exchange, breaking or merging. Here we propose an exhaustive description of the cluster dynamics in apolar active matter. Exploiting large statistics gathered on thousands of Janus colloids, we measure the aggregation and fragmentation rates and rationalize the resulting cluster size distribution and fluctuations. We also show that the motion of individual clusters is entirely consistent with a model positing random orientation of colloids. Our findings establish a simple, generic model of cluster phase, and pave the way for a thorough understanding of clustering in active matter