Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

Chair in Applied Mathematics



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6M38Huxley BuildingSouth Kensington Campus





Publication Type

285 results found


Journal article


Journal article


Journal article

Degond P, Ferreira MA, Merino-Aceituno S, Nahon Met al., A new continuum theory for incompressible swelling materials

Swelling media (e.g. gels, tumors) are usually described by mechanicalconstitutive laws (e.g. Hooke or Darcy laws). However, constitutive relationsof real swelling media are not well known. Here, we take an opposite route andconsider a simple packing heuristics, i.e. the particles can't overlap. Wededuce a formula for the equilibrium density under a confining potential. Wethen consider its evolution when the average particle volume and confiningpotential depend on time under two additional heuristics: (i) any two particlescan't swap their position; (ii) motion should obey some energy minimizationprinciple. These heuristics determine the medium velocity consistently with thecontinuity equation. In the direction normal to the potential level sets thevelocity is related with that of the level sets while in the paralleldirection, it is determined by a Laplace-Beltrami operator on these sets. Thiscomplex geometrical feature cannot be recovered using a simple Darcy law.

Journal article

Degond P, Henkes S, Yu H, Self-Organized Hydrodynamics with nonconstant velocity

Motivated by recent experimental and computational results that show amotility-induced clustering transition in self-propelled particle systems, westudy an individual model and its corresponding Self-Organized Hydrodynamicmodel for collective behaviour that incorporates a density-dependent velocity,as well as inter-particle alignment. The modal analysis of the hydrodynamicmodel elucidates the relationship between the stability of the equilibria andthe changing velocity, and the formation of clusters. We find, in agreementwith earlier results for non-aligning particles, that the key criterion forstability is $(\rho v(\rho))'> 0$, i.e. a non-rapid decrease of velocity withdensity. Numerical simulation for both the individual and hydrodynamic modelswith a velocity function inspired by experiment demonstrates the validity ofthe theoretical results.

Journal article

Degond P, Asymptotic-Preserving Schemes for Fluid Models of Plasmas, Panoramas et Syntheses 39-40, 2013, pp. 1-90

These notes summarize a series of works related to the numericalapproximation of plasma fluid problems. We construct so-called'Asymptotic-Preserving' schemes which are valid for a large range of values(from very small to order unity) of the dimensionless parameters that appear inplasma fluid models. Specifically, we are interested in two parameters, thescaled Debye length which quantifies how close to quasi-neutrality the plasmais, and the scaled cyclotron period, which is inversely proportional to themagnetic field strength. We will largely focus on the ideas, in order to enablethe reader to apply these concepts to other situations.

Journal article

Aoki K, Charrier P, Degond P, A hierarchy of models related to nanoflows and surface diffusion, Kinetic and Related Models, 4 (2011), pp. 53-85

In last years a great interest was brought to molecular transport problems atnanoscales, such as surface diffusion or molecular flows in nano orsub-nano-channels. In a series of papers V. D. Borman, S. Y. Krylov, A. V.Prosyanov and J. J. M. Beenakker proposed to use kinetic theory in order toanalyze the mechanisms that determine mobility of molecules in nanoscalechannels. This approach proved to be remarkably useful to give new insight onthese issues, such as density dependence of the diffusion coefficient. In thispaper we revisit these works to derive the kinetic and diffusion modelsintroduced by V. D. Borman, S. Y. Krylov, A. V. Prosyanov and J. J. M.Beenakker by using classical tools of kinetic theory such as scaling andsystematic asymptotic analysis. Some results are extended to less restrictivehypothesis.

Journal article

Degond P, Dimarco G, Pareschi L, The Moment Guided Monte Carlo Method, International Journal for Numerical Methods in Fluids, 67 (2011), pp. 189-213

In this work we propose a new approach for the numerical simulation ofkinetic equations through Monte Carlo schemes. We introduce a new techniquewhich permits to reduce the variance of particle methods through a matchingwith a set of suitable macroscopic moment equations. In order to guarantee thatthe moment equations provide the correct solutions, they are coupled to thekinetic equation through a non equilibrium term. The basic idea, on which themethod relies, consists in guiding the particle positions and velocitiesthrough moment equations so that the concurrent solution of the moment andkinetic models furnishes the same macroscopic quantities.

Journal article

Ferreira MA, Despin-Guitard E, Duarte F, Degond P, Theveneau Eet al., Interkinetic nuclear movements promote apical expansion in pseudostratified epithelia at the expense of apicobasal elongation

<jats:title>Abstract</jats:title><jats:p>Pseudostratified epithelia (PSE) are a common type of columnar epithelia found in a wealth of embryonic and adult tissues such as ectodermal placodes, the trachea, the ureter, the gut and the neuroepithelium. PSE are characterized by the choreographed displacement of cells’ nuclei along the apicobasal axis according to phases of their cell cycle. Such movements, called interkinetic movements (INM) have been proposed to influence tissue expansion and shape and suggested as culprit in several congenital diseases such as CAKUT and esophageal atresia. INM rely on cytoskeleton dynamics just as adhesion, contractility and mitosis do. Therefore, longer term impairment of INM without affecting proliferation and adhesion is currently technically unachievable. Here we bypassed this hurdle by generating a 2D agent-based model of a proliferating PSE and compared its output to the growth of the chick neuroepithelium to assess the interplay between INM and these other important cell processes during growth of a PSE. We found that INM directly generates apical expansion and apical nuclear crowding. In addition, our data strongly suggest that apicobasal elongation of cells is not an emerging property of a proliferative PSE but rather requires a specific elongation program. We then discuss how such program might functionally link INM, tissue growth and differentiation.</jats:p><jats:sec><jats:title>Authors Summary</jats:title><jats:p>Pseudostratified epithelia (PSE) are a common type of epithelia characterized by the choreographed displacement of cells’ nuclei along the apicobasal axis during proliferation. These so-called interkinetic movements (INM) were proposed to influence tissue expansion and suggested as culprit in several congenital diseases. INM rely on cytoskeleton dynamics. Therefore, longer term impairment of INM without affecting proliferation and adhesion is currently t

Journal article

Degond P, Mathematical models of collective dynamics and self-organization

In this paper, we begin by reviewing a certain number of mathematicalchallenges posed by the modelling of collective dynamics and self-organization.Then, we focus on two specific problems, first, the derivation of fluidequations from particle dynamics of collective motion and second, the study ofphase transitions and the stability of the associated equilibria.

Working paper

Aceves-Sanchez P, Aymard B, Peurichard D, Kennel P, Lorsignol A, Plouraboue F, Casteilla L, Degond Pet al., A new model for the emergence of blood capillary networks

We propose a new model for the emergence of blood capillary networks. Weassimilate the tissue and extra cellular matrix as a porous medium, usingDarcy's law for describing both blood and intersticial fluid flows. Oxygenobeys a convection-diffusion-reaction equation describing advection by theblood, diffusion and consumption by the tissue. Discrete agents named capillaryelements and modelling groups of endothelial cells are created or deletedaccording to different rules involving the oxygen concentration gradient, theblood velocity, the sheer stress or the capillary element density. Oncecreated, a capillary element locally enhances the hydraulic conductivitymatrix, contributing to a local increase of the blood velocity and oxygen flow.No connectivity between the capillary elements is imposed. The coupling betweenblood, oxygen flow and capillary elements provides a positive feedbackmechanism which triggers the emergence of a network of channels of highhydraulic conductivity which we identify as new blood capillaries. We providetwo different, biologically relevant geometrical settings and numericallyanalyze the influence of each of the capillary creation mechanism in detail.All mechanisms seem to concur towards a harmonious network but the mostimportant ones are those involving oxygen gradient and sheer stress. A detaileddiscussion of this model with respect to the literature and its potentialfuture developments concludes the paper.

Journal article

Degond P, Engel M, Liu J-G, Pego RLet al., A Markov jump process modelling animal group size statistics

We translate a coagulation-framentation model, describing the dynamics ofanimal group size distributions, into a model for the population distributionand associate the \blue{nonlinear} evolution equation with a Markov jumpprocess of a type introduced in classic work of H.~McKean. In particular thisformalizes a model suggested by H.-S. Niwa [J.~Theo.~Biol.~224 (2003)] withsimple coagulation and fragmentation rates. Based on the jump process, wedevelop a numerical scheme that allows us to approximate the equilibrium forthe Niwa model, validated by comparison to analytical results by Degond et al.[J.~Nonlinear Sci.~27 (2017)], and study the population and size distributionsfor more complicated rates. Furthermore, the simulations are used to describestatistical properties of the underlying jump process. We additionally discussthe relation of the jump process to models expressed in stochastic differentialequations and demonstrate that such a connection is justified in the case ofnearest-neighbour interactions, as opposed to global interactions as in theNiwa model.

Journal article

Degond P, Diez A, Frouvelle A, Merino-Aceituno Set al., Phase transitions and macroscopic limits in a BGK model of body-attitude coordination

In this article we investigate the phase transition phenomena that occur in amodel of self-organisation through body-attitude coordination. Here, thebody-attitude of an agent is modelled by a rotation matrix in $\mathbb{R}^3$ asin [Degond, Frouvelle, Merino-Aceituno, 2017]. The starting point of this studyis a BGK equation modelling the evolution of the distribution function of thesystem at a kinetic level. The main novelty of this work is to show that in thespatially homogeneous case, self-organisation may appear or not depending onthe local density of agents involved. We first exhibit a connection betweenbody-orientation models and models of nematic alignment of polymers in higherdimensional space from which we deduce the complete description of the possibleequilibria Then, thanks to a gradient-flow structure specific to this BGKmodel, we are able to prove the stability and the convergence towards theequilibria in the different regimes. We then derive the macroscopic modelsassociated to the stable equilibria in the spirit of [Degond, Frouvelle,Merino-Aceituno, 2017] and [Degond, Frouvelle, Liu, 2015].

Working paper

Barré J, Degond P, Peurichard D, Zatorska Eet al., Modelling pattern formation through differential repulsion

Motivated by experiments on cell segregation, we present a two-species modelof interacting particles, aiming at a quantitative description of thisphenomenon. Under precise scaling hypothesis, we derive from the microscopicmodel a macroscopic one and we analyze it. In particular, we determine therange of parameters for which segregation is expected. We compare ouranalytical results and numerical simulations of the macroscopic model to directsimulations of the particles, and comment on possible links with experiments.

Journal article

Degond P, Hirstoaga S, Vignal M-H, The Vlasov model under large magnetic fields in the low-Mach number regime

This article is concerned with the kinetic modeling, by means of the Vlasovequation, of charged particles under the influence of a strong externalelectromagnetic field, i.e. when epsilon^2, the dimensionless cyclotron period,tends to zero. This leads us to split the velocity variable in the Vlasovequation into fluid and random components. The latter is supposed to have alarge magnitude of order 1/epsilon (which corresponds to the low Mach numberregime). In the limit epsilon -> 0, the resulting model is a hybrid model whichcouples a kinetic description of the microscopic random motion of the particlesto a fluid description of the macroscopic behavior of the plasma. Themicroscopic model is a first-order partial differential system for thedistribution function, which is averaged over the ultra-fast Larmor gyrationand the fast parallel motion along the magnetic field lines. The perpendicularcomponent (with respect to the magnetic field lines) of the bulk velocity isgoverned by the classical relations describing the E X B and diamagneticdrifts, while its parallel component satisfies an elliptic equation along themagnetic field lines.

Journal article

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