299 results found
Degond P, Markowich PA, 1990, On a one-dimensional steady-state hydrodynamic model for semiconductors, Applied Mathematics Letters, Vol: 3, Pages: 25-29, ISSN: 0893-9659
DEGOND P, MASGALLIC S, 1989, THE WEIGHTED PARTICLE METHOD FOR CONVECTION-DIFFUSION EQUATIONS .2. THE ANISOTROPIC CASE, MATHEMATICS OF COMPUTATION, Vol: 53, Pages: 509-525, ISSN: 0025-5718
DEGOND P, MASGALLIC S, 1989, THE WEIGHTED PARTICLE METHOD FOR CONVECTION-DIFFUSION EQUATIONS .1. THE CASE OF AN ISOTROPIC VISCOSITY, MATHEMATICS OF COMPUTATION, Vol: 53, Pages: 485-507, ISSN: 0025-5718
DEGOND P, NICLOT B, 1989, NUMERICAL-ANALYSIS OF THE WEIGHTED PARTICLE METHOD APPLIED TO THE SEMICONDUCTOR BOLTZMANN-EQUATION, NUMERISCHE MATHEMATIK, Vol: 55, Pages: 599-618, ISSN: 0029-599X
NICLOT B, DEGOND P, POUPAUD F, 1988, DETERMINISTIC PARTICLE SIMULATIONS OF THE BOLTZMANN TRANSPORT-EQUATION OF SEMICONDUCTORS, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 78, Pages: 313-349, ISSN: 0021-9991
BATT J, BERESTYCKI H, DEGOND P, et al., 1988, SOME FAMILIES OF SOLUTIONS OF THE VLASOV-POISSON SYSTEM, ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, Vol: 104, Pages: 79-103, ISSN: 0003-9527
DEGOND P, 1986, SPECTRAL THEORY OF THE LINEARIZED VLASOV-POISSON EQUATION, TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 294, Pages: 435-453, ISSN: 0002-9947
DEGOND P, 1986, GLOBAL EXISTENCE OF SMOOTH SOLUTIONS FOR THE VLASOV-FOKKER-PLANCK EQUATION IN 1 SPACE AND 2 SPACE DIMENSIONS, ANNALES SCIENTIFIQUES DE L ECOLE NORMALE SUPERIEURE, Vol: 19, Pages: 519-542, ISSN: 0012-9593
BARDOS C, DEGOND P, 1985, GLOBAL EXISTENCE FOR THE VLASOV-POISSON EQUATION IN 3 SPACE VARIABLES WITH SMALL INITIAL DATA, ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, Vol: 2, Pages: 101-118, ISSN: 0294-1449
DEGOND P, 1985, LOCAL EXISTENCE OF SMOOTH SOLUTIONS OF THE VLASOV-MAXWELL EQUATIONS, AND APPROXIMATION BY THE SOLUTIONS OF THE VLASOV-POISSON EQUATIONS, COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE, Vol: 301, Pages: 877-880, ISSN: 0764-4442
BARDOS C, NGOAN HT, DEGOND P, 1985, GLOBAL-SOLUTIONS FOR RELATIVISTIC VLASOV-POISSON EQUATIONS IN 3 SPACE VARIABLES, COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE, Vol: 301, Pages: 265-268, ISSN: 0764-4442
DEGOND P, 1985, GLOBAL EXISTENCE OF SOLUTIONS OF THE VLASOV-FOKKER-PLANCK EQUATION IN 1-DIMENSION AND 2-DIMENSION, COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE, Vol: 301, Pages: 73-76, ISSN: 0764-4442
BARDOS C, DEGOND P, 1983, GLOBAL EXISTENCE AND ASYMPTOTIC-BEHAVIOR FOR THE SOLUTION OF THE VLASOV-POISSON EQUATION, COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE, Vol: 297, Pages: 321-324, ISSN: 0764-4442
DEGOND P, 1983, APPEARANCE OF EIGEN MODES FOR THE LINEARIZED VLASOV-POISSON EQUATION, COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE I-MATHEMATIQUE, Vol: 296, Pages: 969-972, ISSN: 0764-4442
Degond P, Liu J-G, Ringhofer C, A Nash equilibrium macroscopic closure for kinetic models coupled with Mean-Field Games
We introduce a new mean field kinetic model for systems of rational agentsinteracting in a game theoretical framework. This model is inspired fromnon-cooperative anonymous games with a continuum of players and Mean-FieldGames. The large time behavior of the system is given by a macroscopic closurewith a Nash equilibrium serving as the local thermodynamic equilibrium. Anapplication of the presented theory to a social model (herding behavior) isdiscussed.
Degond P, Frouvelle A, Liu J-G, et al., Macroscopic models of collective motion and self-organization, ee, Vol: 20, Pages: 2-2013
In this paper, we review recent developments on the derivation and propertiesof macroscopic models of collective motion and self-organization. The startingpoint is a model of self-propelled particles interacting with its neighborsthrough alignment. We successively derive a mean-field model and itshydrodynamic limit. The resulting macroscopic model is the Self-OrganizedHydrodynamics (SOH). We review the available existence results and knownproperties of the SOH model and discuss it in view of its possible extensionsto other kinds of collective motion.
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.
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.
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.
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.
Degond P, Herda M, Mirrahimi S, A Fokker-Planck approach to the study of robustness in gene expression
We study several Fokker-Planck equations arising from a stochastic chemicalkinetic system modeling a gene regulatory network in biology. The densitiessolving the Fokker-Planck equations describe the joint distribution of themessenger RNA and micro RNA content in a cell. We provide theoretical andnumerical evidences that the robustness of the gene expression is increased inthe presence of micro RNA. At the mathematical level, increased robustnessshows in a smaller coefficient of variation of the marginal density of themessenger RNA in the presence of micro RNA. These results follow from explicitformulas for solutions. Moreover, thanks to dimensional analyses and numericalsimulations we provide qualitative insight into the role of each parameter inthe model. As the increase of gene expression level comes from the underlyingstochasticity in the models, we eventually discuss the choice of noise in ourmodels and its influence on our results.
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.
Aceves-Sanchez P, Aymard B, Peurichard D, et 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.
Degond P, Merino-Aceituno S, Nematic alignment of self-propelled particles in the macroscopic regime
Starting from a particle model describing self-propelled particlesinteracting through nematic alignment, we derive a macroscopic model for theparticle density and mean direction of motion. We first propose a mean-fieldkinetic model of the particle dynamics. After diffusive rescaling of thekinetic equation, we formally show that the distribution function converges toan equilibrium distribution in particle direction, whose local density and meandirection satisfies a cross-diffusion system. We show that the system isconsistent with symmetries typical of a nematic material. The derivation iscarried over by means of a Hilbert expansion. It requires the inversion of thelinearized collision operator for which we show that the generalized collisioninvariants, a concept introduced to overcome the lack of momentum conservationof the system, plays a central role. This cross diffusion system poses many newchallenging questions.
Degond P, Jin S, Zhu Y, An Uncertainty Quantification Approach to the Study of Gene Expression Robustness
We study a chemical kinetic system with uncertainty modeling a generegulatory network in biology. Specifically, we consider a system of twoequations for the messenger RNA and micro RNA content of a cell. Our target isto provide a simple framework for noise buffering in gene expression throughmicro RNA production. Here the uncertainty, modeled by random variables, entersthe system through the initial data and the source term. We obtain a sharpdecay rate of the solution to the steady state, which reveals that the biologysystem is not sensitive to the initial perturbation around the steady state.The sharp regularity estimate leads to the stability of the generalizedPolynomial Chaos stochastic Galerkin (gPC-SG) method. Based on the smoothnessof the solution in the random space and the stability of the numerical method,we conclude the gPC-SG method has spectral accuracy. Numerical experiments areconducted to verify the theoretical findings.
Barker M, Degond P, Wolfram M-T, Comparing the best reply strategy and mean field games: the stationary case
Mean field games (MFGs) and the best reply strategy (BRS) are two methods ofdescribing competitive optimisation of systems of interacting agents. Thelatter can be interpreted as an approximation of the respective MFG system. Inthis paper we present a systematic analysis and comparison of the twoapproaches in the stationary case. We provide novel existence and uniquenessresults for the stationary boundary value problems related to the MFG and BRSformulations, and we present an analytical and numerical comparison of the twoparadigms in a variety of modelling situations.
Aceves-Sanchez P, Degond P, Keaveny EE, et al., Large-scale dynamics of self-propelled particles moving through obstacles: model derivation and pattern formation
We model and study the patterns created through the interaction ofcollectively moving self-propelled particles (SPPs) and elastically tetheredobstacles. Simulations of an individual-based model reveal at least threedistinct large-scale patterns: travelling bands, trails and moving clusters.This motivates the derivation of a macroscopic partial differential equationsmodel for the interactions between the self-propelled particles and theobstacles, for which we assume large tether stiffness. The result is a coupledsystem of non-linear, non-local partial differential equations. Linearstability analysis shows that patterning is expected if the interactions arestrong enough and allows for the predictions of pattern size from modelparameters. The macroscopic equations reveal that the obstacle interactionsinduce short-ranged SPP aggregation, irrespective of whether obstacles and SPPsare attractive or repulsive.
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.
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