A model of alignment interaction for oriented particles with phase transition
|
|
- Alexina Todd
- 5 years ago
- Views:
Transcription
1 A model of alignment interaction for oriented particles with phase transition Amic Frouvelle Institut de Mathématiques de Toulouse Joint work with Jian-Guo Liu (Duke Univ.) and Pierre Degond (IMT) Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
2 Goal: macroscopic description of some animal societies Local interactions without leader Emergence of macroscopic structures Images Amic Frouvelle Benson Kua (flickr) and Alignment interaction with phase transition CIRM, July /20
3 Modeling of interacting self-propelled particles Vicsek et al. (1995). Discrete in time (interval t), alignment only, synchronous reorientation. New direction = Mean direction of neighboring particles at previous step + Noise Simulations: phase transition phenomenon, emergence of coherent structures. Degond-Motsch (2008). Time-continuous version: relaxation (with constant rate ν) towards the local mean direction. Hydrodynamic limit without phase transition phenomenon. Model presented here: making ν proportional to the local mean momentum. Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
4 Outline 1 Presentation of the model Kinetic model Hydrodynamic scaling The phase transition 2 Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
5 Individual dynamics Presentation of the model Kinetic model Hydrodynamic scaling The phase transition Particles at positions: X 1,..., X N in R n. Orientations ω 1,..., ω N in S (unit sphere). { dxk = ω k dt dω k = ν(id ω k ω k ) ω k dt + 2d(Id ω k ω k ) db k t Target direction: ω k = J k J k, J k = 1 N N K( X j X k ) ω j. j=1 Setting ν = J k ν 0, no more singularity (binary interactions): { dxk = ω k dt dω k = ν 0 (Id ω k ω k ) J k dt + 2d(Id ω k ω k ) db k t Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
6 Kinetic description Presentation of the model Kinetic model Hydrodynamic scaling The phase transition Theorem (Bolley, Cañizo, Carrillo, 2011) Probability density function f (x, ω, t), as N : t f + ω x f + ν 0 ω ((Id ω ω)j f f ) = d ω f J f (x, ω, t) = K( y x ) υ f (y, υ, t) dy dυ. y R n, υ S Tool : coupling process + estimations. d X k = ω k dt d ω k = ν 0 (Id ω k ω k ) J f N t dt + 2d(Id ω k ω k ) dbt k ft N = law( X 1, ω 1 ) = law( X k, ω k ) Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
7 Hydrodynamic scaling Presentation of the model Kinetic model Hydrodynamic scaling The phase transition Scaling, with ε 1 (and K 0 = R n K(x)dx): f ε (x, ω, t) = ν 0 K 0 f ( 1 1 dεx, ω, dε t). Mean-field rescaled and reduced equation: with (localization in space) ε( t f ε + ω x f ε ) = Q(f ε ) + O(ε 2 ), Q(f ) = ω ((Id ω ω)j f f ) + ω f, J f (x, t) = f (x, ω, t) ω dω. S Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
8 Local equilibria Presentation of the model Kinetic model Hydrodynamic scaling The phase transition Fisher von Mises distribution (orientation Ω S, concentration κ 0): For J f = κ f Ω f, we get M κω (ω) = Q(f ) = ω [ eκ ω Ω S eκ υ Ω dυ. M κf Ω f ω ( f M κf Ω f Local equilibria: ρm κω, for some Ω S. Compatibility condition for κ 0 and ρ > 0: )]. ρc(κ) = κ, where c(κ) = J MκΩ = π 0 cos θ eκ cos θ sin n 2 θ dθ π 0 eκ cos θ sin n 2. θ dθ Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
9 Presentation of the model Kinetic model Hydrodynamic scaling The phase transition Solutions to the compatibility condition, equilibria Proposition The function κ c(κ) κ is decreasing, its limit is 1 n when κ 0. Rate of convergence r(ρ) 6 ε 4 ε 2 ε n = 2 n = 3 n = Density ρ ρ n, only one solution: κ = 0. Uniform equilibrium ( stable ). ρ > n, uniform equilibrium ( unstable ) for κ = 0. Unique solution κ(ρ) > 0. Manifold of equilibria ( stable ): ρm κ(ρ)ω, Ω S. Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
10 Stochastic model and its mean-field limit Orientations ω k only. { dωk = (Id ω k ω k )J k dt + 2τ (Id ω k ω k ) db k t, J k = 1 N Nj=1 ω j. Limit as N (Bolley-Cañizo-Carrillo): t f = ω ((Id ω ω)j[f ]f ) + τ ω f = (f Ψ) + τ f, J[f ] = ω f dω, Ψ(ω) = J[f ] ω = K(ω, ω) f ( ω) d ω. S Here K = ω ω (dipolar potential). Polymers: ω ω (Onsager) or (ω ω) 2 (Maier Saupe). S Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
11 Existence, uniqueness, regularity, positivity, bounds Theorem (AF, J.-G. Liu, 2011) For an initial probability measure f 0 H s (S), (for an arbitrary s): Existence and uniqueness of a weak solution f. Global solution, in C (R + S), and f > 0 for t > 0. Instantaneous regularity estimates and uniform bounds: ( f (t) 2 H s+m C ) t m f 0 2 H s. Tool: spherical harmonics decomposition. Nonlinearity: finite number of coefficients. Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
12 Onsager free energy Free energy: F(f ) = τ S f ln f 1 2 J[f ] 2. Dissipation term: D(f ) = S f ω(τ ln f ω J[f ]) 2 0. d dt F + D = 0. The free energy F(f ) is decreasing towards F. LaSalle s principle Limit set: E = {f C (S) D(f ) = 0 and F(f ) = F }. lim t inf f (t) g H s = 0. g E Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
13 Equilibria D(f ) = 0 (no dissipation) Q(f ) = 0 (stationary solution) Equivalent conditions: f critical point of F τ ln f J[f ] ω = Cte Amounts to f = M κω, with the compatibility condition τκ = c(κ). Proposition The function κ c(κ) κ is decreasing, its limit is 1 n when κ 0. τ 1 n. Only one solution: κ = 0. Uniform equilibrium (unique minimizer of F). τ < 1 n : uniform equilibrium for κ = 0, not minimizing F. Only positive solution: κ(τ). Manifold of equilibria: M κ(τ)ω, Ω S. Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
14 Additional conservation relation Conformal Laplacian: n 1 Y l = l(l + 1)... (l + n 2)Y l, for a spherical harmonic Y l of degree l. Proposition For g H n+1 2 (S) with mean zero, S g n 1 g = 0. Norms g n 1 2 H 2 Conservation relation 1 d 2 dt f 1 2 H n 1 2 = 1 S g n 1g, n 3 g 2 H 2 = τ f 1 n 3 2 H 2 = 1 S g n 1 g: + 1 (n 2)! J[f ] 2. For τ > 1 n, it is an entropy dissipation! Global exponential convergence with rate (n 1)(τ 1 n ) towards uniform equilibrium (in any H s ). Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
15 The moving Fisher von Mises distribution ODE for J[f (t)]: J[f 0 ] 0 J[f (t)] 0 for all t. J[f (t)] We define Ω(t) = J[f (t)]. Expansion around the moving equilibrium M κ(τ)ω(t) : f = (1 + h)m κ(τ)ω(t), Notation MκΩ for the mean against this measure. Unique decomposition of the form f = (1 + α [ ω Ω(t) c(κ(τ))] + g)m κ(τ)ω(t), with g MκΩ = 0 and gω MκΩ = 0. By LaSalle s principle, h, α and g converge to 0. Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
16 Exponential convergence to a fixed equilibrium Poincaré inequality: h 2 MκΩ Λ κ (h h MκΩ ) 2 MκΩ Expansion of D(f ) et F(f ) F, in terms of α 2 et g 2 MκΩ, with h sufficiently small: exponential decay of f M κω(t) with rate r for all r < r (τ) = (c(κ) 2 + nτ 1)Λ κ. Dynamics of Ω(t): dω dt = (Id Ω Ω) g ω Ω ω M κω So we get dω dt C g 2 MκΩ Ce rt : exponential convergence of Ω to some Ω S. Interpolation + uniform bounds: exponential convergence to M κω in any H s. Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
17 Special cases Critical case τ = 1 n. Same type of method, using the decomposition f = 1 + α cos θ α2 (cos 2 θ 1 n ) α3 (cos 3 θ 3 n+2 cos θ) + g, with α = n f cos θ and cos θ = ω Ω. Expansion of D(f ) in terms of g 2 and α 6, and of F(f ) in terms of g 2 and α 4 : algebraic convergence in 1 t towards the uniform distribution. Case of no noise: τ = 0. The norm of J[f ] increases and the H n 1 2 norm of f explodes... Amic Frouvelle Alignment interaction with phase transition CIRM, July /20
A model of alignment interaction for oriented particles with phase transition
A model of alignment interaction for oriented particles with phase transition Amic Frouvelle ACMAC Joint work with Jian-Guo Liu (Duke University, USA) and Pierre Degond (Institut de Mathématiques de Toulouse,
More informationA model of alignment interaction for oriented particles with phase transition
A model of alignment interaction for oriented particles with phase transition Amic Frouvelle Archimedes Center for Modeling, Analysis & Computation (ACMAC) University of Crete, Heraklion, Crete, Greece
More informationDifferent types of phase transitions for a simple model of alignment of oriented particles
Different types of phase transitions for a simple model of alignment of oriented particles Amic Frouvelle Université Paris Dauphine Joint work with Jian-Guo Liu (Duke University, USA) and Pierre Degond
More informationDifferent types of phase transitions for a simple model of alignment of oriented particles
Different types of phase transitions for a simple model of alignment of oriented particles Amic Frouvelle CEREMADE Université Paris Dauphine Joint work with Jian-Guo Liu (Duke University, USA) and Pierre
More informationA note on phase transitions for the Smoluchowski equation with dipolar potential
A note on phase transitions for the Smoluchowski equation with dipolar potential Pierre Degond (,2), Amic Frouvelle (3), Jian-Guo Liu (4) arxiv:22.3920v [math-ph] 7 Dec 202 -Université de Toulouse; UPS,
More informationPhase Transitions, Hysteresis, and Hyperbolicity for Self-Organized Alignment Dynamics
Arch. Rational ech. Anal. 6 (05) 63 5 Digital Object Identifier (DOI) 0.007/s0005-04-0800-7 Phase Transitions, Hysteresis, and Hyperbolicity for Self-Organized Alignment Dynamics Pierre Degond, Amic Frouvelle
More informationPhase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics
Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics Pierre Degond (,), Amic Frouvelle (3), Jian-Guo Liu (4) - Université de Toulouse; UPS, INSA, UT, UT; Institut de athématiques
More informationModels of collective displacements: from microscopic to macroscopic description
Models of collective displacements: from microscopic to macroscopic description Sébastien Motsch CSCAMM, University of Maryland joint work with : P. Degond, L. Navoret (IMT, Toulouse) SIAM Analysis of
More informationAlignment processes on the sphere
Alignment processes on the sphere Amic Frouvelle CEREMADE Université Paris Dauphine Joint works with : Pierre Degond (Imperial College London) and Gaël Raoul (École Polytechnique) Jian-Guo Liu (Duke University)
More informationA new flocking model through body attitude coordination
A new flocking model through body attitude coordination Sara Merino Aceituno (Imperial College London) Pierre Degond (Imperial College London) Amic Frouvelle (Paris Dauphine) ETH Zürich, November 2016
More informationMacroscopic limits and phase transition in a system of self-propelled particles
Macroscopic limits and phase transition in a system of self-propelled particles Pierre Degond (,), Amic Frouvelle (,), Jian-Guo Liu (3) arxiv:09.404v [math-ph] ep 0 -Université de Toulouse; UP, INA, UT,
More informationHydrodynamic limit and numerical solutions in a system of self-pr
Hydrodynamic limit and numerical solutions in a system of self-propelled particles Institut de Mathématiques de Toulouse Joint work with: P.Degond, G.Dimarco, N.Wang 1 2 The microscopic model The macroscopic
More informationACMAC s PrePrint Repository
ACMAC s PrePrint Repository Macroscopic limits and phase transition in a system of self-propelled particles Pierre Degond and Amic Frouvelle and Jian-Guo Liu Original Citation: Degond, Pierre and Frouvelle,
More informationComplex systems: Self-organization vs chaos assumption
1 Complex systems: Self-organization vs chaos assumption P. Degond Institut de Mathématiques de Toulouse CNRS and Université Paul Sabatier pierre.degond@math.univ-toulouse.fr (see http://sites.google.com/site/degond/)
More informationarxiv: v1 [math.na] 14 Jan 2019
Noname manuscript No. will be inserted by the editor) Kinetic equations and self-organized band formations Quentin Griette Sebastien Motsch arxiv:191.43v1 [math.na] 14 Jan 19 the date of receipt and acceptance
More informationLate-time tails of self-gravitating waves
Late-time tails of self-gravitating waves (non-rigorous quantitative analysis) Piotr Bizoń Jagiellonian University, Kraków Based on joint work with Tadek Chmaj and Andrzej Rostworowski Outline: Motivation
More informationHypocoercivity for kinetic equations with linear relaxation terms
Hypocoercivity for kinetic equations with linear relaxation terms Jean Dolbeault dolbeaul@ceremade.dauphine.fr CEREMADE CNRS & Université Paris-Dauphine http://www.ceremade.dauphine.fr/ dolbeaul (A JOINT
More informationUne approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck
Une approche hypocoercive L 2 pour l équation de Vlasov-Fokker-Planck Jean Dolbeault dolbeaul@ceremade.dauphine.fr CEREMADE CNRS & Université Paris-Dauphine http://www.ceremade.dauphine.fr/ dolbeaul (EN
More informationand in each case give the range of values of x for which the expansion is valid.
α β γ δ ε ζ η θ ι κ λ µ ν ξ ο π ρ σ τ υ ϕ χ ψ ω Mathematics is indeed dangerous in that it absorbs students to such a degree that it dulls their senses to everything else P Kraft Further Maths A (MFPD)
More informationExponential moments for the homogeneous Kac equation
Exponential moments for the homogeneous Kac equation Maja Tasković University of Pennsylvania Young Researchers Workshop: Stochastic and deterministic methods in kinetic theory, Duke University, November
More informationFractional parabolic models arising in flocking. dynamics and fluids. Roman Shvydkoy jointly with Eitan Tadmor. April 18, 2017
Fractional April 18, 2017 Cucker-Smale model (2007) We study macroscopic versions of systems modeling self-organized collective dynamics of agents : ẋ i = v i, v i = λ N φ( x i x j )(v j v i ), N j=1 (x
More informationGlobal well-posedness and decay for the viscous surface wave problem without surface tension
Global well-posedness and decay for the viscous surface wave problem without surface tension Ian Tice (joint work with Yan Guo) Université Paris-Est Créteil Laboratoire d Analyse et de Mathématiques Appliquées
More informationOn the Boltzmann equation: global solutions in one spatial dimension
On the Boltzmann equation: global solutions in one spatial dimension Department of Mathematics & Statistics Colloque de mathématiques de Montréal Centre de Recherches Mathématiques November 11, 2005 Collaborators
More informationEntropy and irreversibility in gas dynamics. Joint work with T. Bodineau, I. Gallagher and S. Simonella
Entropy and irreversibility in gas dynamics Joint work with T. Bodineau, I. Gallagher and S. Simonella Kinetic description for a gas of hard spheres Hard sphere dynamics The system evolves under the combined
More informationYAN GUO, JUHI JANG, AND NING JIANG
LOCAL HILBERT EXPANSION FOR THE BOLTZMANN EQUATION YAN GUO, JUHI JANG, AND NING JIANG Abstract. We revisit the classical ork of Caflisch [C] for compressible Euler limit of the Boltzmann equation. By using
More informationA new flocking model through body attitude coordination
A new flocking model through body attitude coordination arxiv:1605.03509v1 [math-ph] 11 May 016 Pierre Degond 1, Amic Frouvelle, and Sara Merino-Aceituno 3 13 Department of Mathematics, Imperial College
More informationQuantum Hydrodynamics models derived from the entropy principle
1 Quantum Hydrodynamics models derived from the entropy principle P. Degond (1), Ch. Ringhofer (2) (1) MIP, CNRS and Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex, France degond@mip.ups-tlse.fr
More informationPhase Synchronization
Phase Synchronization Lecture by: Zhibin Guo Notes by: Xiang Fan May 10, 2016 1 Introduction For any mode or fluctuation, we always have where S(x, t) is phase. If a mode amplitude satisfies ϕ k = ϕ k
More informationOn the interplay between kinetic theory and game theory
1 On the interplay between kinetic theory and game theory Pierre Degond Department of mathematics, Imperial College London pdegond@imperial.ac.uk http://sites.google.com/site/degond/ Joint work with J.
More informationHeat Flows, Geometric and Functional Inequalities
Heat Flows, Geometric and Functional Inequalities M. Ledoux Institut de Mathématiques de Toulouse, France heat flow and semigroup interpolations Duhamel formula (19th century) pde, probability, dynamics
More informationTurbulence modulation by fully resolved particles using Immersed Boundary Methods
Turbulence modulation by fully resolved particles using Immersed Boundary Methods Abouelmagd Abdelsamie and Dominique Thévenin Lab. of Fluid Dynamics & Technical Flows University of Magdeburg Otto von
More informationAnomalous transport of particles in Plasma physics
Anomalous transport of particles in Plasma physics L. Cesbron a, A. Mellet b,1, K. Trivisa b, a École Normale Supérieure de Cachan Campus de Ker Lann 35170 Bruz rance. b Department of Mathematics, University
More informationarxiv: v1 [math.ap] 16 Oct 2018
Alignment of self-propelled rigid bodies: from particle systems to macroscopic equations arxiv:1810.06903v1 [math.ap] 16 Oct 2018 Pierre Degond 1, Amic Frouvelle 2, Sara Merino-Aceituno 3, and Ariane Trescases
More informationTraveling waves of a kinetic transport model for the KPP-Fisher equation
Traveling waves of a kinetic transport model for the KPP-Fisher equation Christian Schmeiser Universität Wien and RICAM homepage.univie.ac.at/christian.schmeiser/ Joint work with C. Cuesta (Bilbao), S.
More informationTheoretical Tutorial Session 2
1 / 36 Theoretical Tutorial Session 2 Xiaoming Song Department of Mathematics Drexel University July 27, 216 Outline 2 / 36 Itô s formula Martingale representation theorem Stochastic differential equations
More informationSupporting Information
Supporting Information A: Calculation of radial distribution functions To get an effective propagator in one dimension, we first transform 1) into spherical coordinates: x a = ρ sin θ cos φ, y = ρ sin
More informationBreakdown of Pattern Formation in Activator-Inhibitor Systems and Unfolding of a Singular Equilibrium
Breakdown of Pattern Formation in Activator-Inhibitor Systems and Unfolding of a Singular Equilibrium Izumi Takagi (Mathematical Institute, Tohoku University) joint work with Kanako Suzuki (Institute for
More informationExam 4 SCORE. MA 114 Exam 4 Spring Section and/or TA:
Exam 4 Name: Section and/or TA: Last Four Digits of Student ID: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may
More informationPhys 622 Problems Chapter 6
1 Problem 1 Elastic scattering Phys 622 Problems Chapter 6 A heavy scatterer interacts with a fast electron with a potential V (r) = V e r/r. (a) Find the differential cross section dσ dω = f(θ) 2 in the
More information1 Existence of Travelling Wave Fronts for a Reaction-Diffusion Equation with Quadratic- Type Kinetics
1 Existence of Travelling Wave Fronts for a Reaction-Diffusion Equation with Quadratic- Type Kinetics Theorem. Consider the equation u t = Du xx + f(u) with f(0) = f(1) = 0, f(u) > 0 on 0 < u < 1, f (0)
More informationin Bounded Domains Ariane Trescases CMLA, ENS Cachan
CMLA, ENS Cachan Joint work with Yan GUO, Chanwoo KIM and Daniela TONON International Conference on Nonlinear Analysis: Boundary Phenomena for Evolutionnary PDE Academia Sinica December 21, 214 Outline
More informationOn the leapfrogging phenomenon in fluid mechanics
On the leapfrogging phenomenon in fluid mechanics Didier Smets Université Pierre et Marie Curie - Paris Based on works with Robert L. Jerrard U. of Toronto) CIRM, Luminy, June 27th 2016 1 / 22 Single vortex
More informationProbing Fundamental Bounds in Hydrodynamics Using Variational Optimization Methods
Probing Fundamental Bounds in Hydrodynamics Using Variational Optimization Methods Bartosz Protas and Diego Ayala Department of Mathematics & Statistics McMaster University, Hamilton, Ontario, Canada URL:
More informationMathematical modelling of collective behavior
Mathematical modelling of collective behavior Young-Pil Choi Fakultät für Mathematik Technische Universität München This talk is based on joint works with José A. Carrillo, Maxime Hauray, and Samir Salem
More informationEntropic structure of the Landau equation. Coulomb interaction
with Coulomb interaction Laurent Desvillettes IMJ-PRG, Université Paris Diderot May 15, 2017 Use of the entropy principle for specific equations Spatially Homogeneous Kinetic equations: 1 Fokker-Planck:
More informationCoagulation-Fragmentation Models with Diffusion
Coagulation-Fragmentation Models with Diffusion Klemens Fellner DAMTP, CMS, University of Cambridge Faculty of Mathematics, University of Vienna Jose A. Cañizo, Jose A. Carrillo, Laurent Desvillettes UVIC
More informationNonlinear Dynamics: Synchronisation
Nonlinear Dynamics: Synchronisation Bristol Centre for Complexity Sciences Ian Ross BRIDGE, School of Geographical Sciences, University of Bristol October 19, 2007 1 / 16 I: Introduction 2 / 16 I: Fireflies
More informationin Electromagnetics Numerical Method Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD
2141418 Numerical Method in Electromagnetics Introduction to Electromagnetics I Lecturer: Charusluk Viphavakit, PhD ISE, Chulalongkorn University, 2 nd /2018 Email: charusluk.v@chula.ac.th Website: Light
More informationFrom the Newton equation to the wave equation in some simple cases
From the ewton equation to the wave equation in some simple cases Xavier Blanc joint work with C. Le Bris (EPC) and P.-L. Lions (Collège de France) Université Paris Diderot, FRACE http://www.ann.jussieu.fr/
More informationAnomalous energy transport in FPU-β chain
Anomalous energy transport in FPU-β chain Sara Merino Aceituno Joint work with Antoine Mellet (University of Maryland) http://arxiv.org/abs/1411.5246 Imperial College London 9th November 2015. Kinetic
More informationApplied Nuclear Physics (Fall 2006) Lecture 19 (11/22/06) Gamma Interactions: Compton Scattering
.101 Applied Nuclear Physics (Fall 006) Lecture 19 (11//06) Gamma Interactions: Compton Scattering References: R. D. Evans, Atomic Nucleus (McGraw-Hill New York, 1955), Chaps 3 5.. W. E. Meyerhof, Elements
More informationTHREE-BODY INTERACTIONS DRIVE THE TRANSITION TO POLAR ORDER IN A SIMPLE FLOCKING MODEL
THREE-BODY INTERACTIONS DRIVE THE TRANSITION TO POLAR ORDER IN A SIMPLE FLOCKING MODEL Purba Chatterjee and Nigel Goldenfeld Department of Physics University of Illinois at Urbana-Champaign Flocking in
More information4 Divergence theorem and its consequences
Tel Aviv University, 205/6 Analysis-IV 65 4 Divergence theorem and its consequences 4a Divergence and flux................. 65 4b Piecewise smooth case............... 67 4c Divergence of gradient: Laplacian........
More informationKinetic models of Maxwell type. A brief history Part I
. A brief history Part I Department of Mathematics University of Pavia, Italy Porto Ercole, June 8-10 2008 Summer School METHODS AND MODELS OF KINETIC THEORY Outline 1 Introduction Wild result The central
More informationNon-equilibrium mixtures of gases: modeling and computation
Non-equilibrium mixtures of gases: modeling and computation Lecture 1: and kinetic theory of gases Srboljub Simić University of Novi Sad, Serbia Aim and outline of the course Aim of the course To present
More informationLINEAR RESPONSE THEORY
MIT Department of Chemistry 5.74, Spring 5: Introductory Quantum Mechanics II Instructor: Professor Andrei Tokmakoff p. 8 LINEAR RESPONSE THEORY We have statistically described the time-dependent behavior
More informationNorm convergence of the resolvent for wild perturbations
Norm convergence of the resolvent for wild perturbations Laboratoire de mathématiques Jean Leray, Nantes Mathematik, Universität Trier Analysis and Geometry on Graphs and Manifolds Potsdam, July, 31 August,4
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More informationStabilization of a 3D Rigid Pendulum
25 American Control Conference June 8-, 25. Portland, OR, USA ThC5.6 Stabilization of a 3D Rigid Pendulum Nalin A. Chaturvedi, Fabio Bacconi, Amit K. Sanyal, Dennis Bernstein, N. Harris McClamroch Department
More informationThe Viscous Model of Quantum Hydrodynamics in Several Dimensions
January 5, 006 4: WSPC/INSTRUCTION FILE chen-dreher The Viscous Model of Quantum Hydrodynamics in Several Dimensions Li Chen Department of Mathematical Sciences, Tsinghua University, Beijing, 00084, P.R.
More informationHydrodynamic Limit with Geometric Correction in Kinetic Equations
Hydrodynamic Limit with Geometric Correction in Kinetic Equations Lei Wu and Yan Guo KI-Net Workshop, CSCAMM University of Maryland, College Park 2015-11-10 1 Simple Model - Neutron Transport Equation
More informationHow Inflation Is Used To Resolve the Flatness Problem. September 9, To Tai-Ping Liu on the occasion of his sixtieth birthday.
How Inflation Is Used To Resolve the Flatness Problem September 9, 005 Joel Smoller Blake T emple To Tai-Ping Liu on the occasion of his sixtieth birthday. Abstract We give a mathematically rigorous exposition
More informationLarge-time behaviour of Hele-Shaw flow with injection or suction for perturbed balls in R N
Large-time behaviour of Hele-Shaw flow with injection or suction for perturbed balls in R N E. Vondenhoff Department of Mathematics and Computer Science Technische Universiteit Eindhoven P.O. Box 53, 5600
More informationCHAPTER 20. Collisions & Encounters of Collisionless Systems
CHAPTER 20 Collisions & Encounters of Collisionless Systems Consider an encounter between two collisionless N-body systems (i.e., dark matter halos or galaxies): a perturber P and a system S. Let q denote
More informationStability of an abstract wave equation with delay and a Kelvin Voigt damping
Stability of an abstract wave equation with delay and a Kelvin Voigt damping University of Monastir/UPSAY/LMV-UVSQ Joint work with Serge Nicaise and Cristina Pignotti Outline 1 Problem The idea Stability
More informationExercise Set 4. D s n ds + + V. s dv = V. After using Stokes theorem, the surface integral becomes
Exercise Set Exercise - (a) Let s consider a test volum in the pellet. The substract enters the pellet by diffusion and some is created and disappears due to the chemical reaction. The two contribute to
More informationA Toy Model. Viscosity
A Toy Model for Sponsoring Viscosity Jean BELLISSARD Westfälische Wilhelms-Universität, Münster Department of Mathematics SFB 878, Münster, Germany Georgia Institute of Technology, Atlanta School of Mathematics
More informationCapacitary inequalities in discrete setting and application to metastable Markov chains. André Schlichting
Capacitary inequalities in discrete setting and application to metastable Markov chains André Schlichting Institute for Applied Mathematics, University of Bonn joint work with M. Slowik (TU Berlin) 12
More informationEntropy-dissipation methods I: Fokker-Planck equations
1 Entropy-dissipation methods I: Fokker-Planck equations Ansgar Jüngel Vienna University of Technology, Austria www.jungel.at.vu Introduction Boltzmann equation Fokker-Planck equations Degenerate parabolic
More informationSynchronization Transitions in Complex Networks
Synchronization Transitions in Complex Networks Y. Moreno 1,2,3 1 Institute for Biocomputation and Physics of Complex Systems (BIFI) University of Zaragoza, Zaragoza 50018, Spain 2 Department of Theoretical
More informationMassachusetts Institute of Technology Department of Physics. Final Examination December 17, 2004
Massachusetts Institute of Technology Department of Physics Course: 8.09 Classical Mechanics Term: Fall 004 Final Examination December 17, 004 Instructions Do not start until you are told to do so. Solve
More informationDynamical Systems & Lyapunov Stability
Dynamical Systems & Lyapunov Stability Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Ordinary Differential Equations Existence & uniqueness Continuous dependence
More informationKinetic theory of gases
Kinetic theory of gases Toan T. Nguyen Penn State University http://toannguyen.org http://blog.toannguyen.org Graduate Student seminar, PSU Jan 19th, 2017 Fall 2017, I teach a graduate topics course: same
More informationSemigroup Growth Bounds
Semigroup Growth Bounds First Meeting on Asymptotics of Operator Semigroups E.B. Davies King s College London Oxford, September 2009 E.B. Davies (KCL) Semigroup Growth Bounds Oxford, September 2009 1 /
More informationOn Some Variational Optimization Problems in Classical Fluids and Superfluids
On Some Variational Optimization Problems in Classical Fluids and Superfluids Bartosz Protas Department of Mathematics & Statistics McMaster University, Hamilton, Ontario, Canada URL: http://www.math.mcmaster.ca/bprotas
More informationPhase-field systems with nonlinear coupling and dynamic boundary conditions
1 / 46 Phase-field systems with nonlinear coupling and dynamic boundary conditions Cecilia Cavaterra Dipartimento di Matematica F. Enriques Università degli Studi di Milano cecilia.cavaterra@unimi.it VIII
More informationEntanglement Entropy in Extended Quantum Systems
Entanglement Entropy in Extended Quantum Systems John Cardy University of Oxford STATPHYS 23 Genoa Outline A. Universal properties of entanglement entropy near quantum critical points B. Behaviour of entanglement
More informationBlow-up on manifolds with symmetry for the nonlinear Schröding
Blow-up on manifolds with symmetry for the nonlinear Schrödinger equation March, 27 2013 Université de Nice Euclidean L 2 -critical theory Consider the one dimensional equation i t u + u = u 4 u, t > 0,
More informationExercise 1 (Formula for connection 1-forms) Using the first structure equation, show that
1 Stokes s Theorem Let D R 2 be a connected compact smooth domain, so that D is a smooth embedded circle. Given a smooth function f : D R, define fdx dy fdxdy, D where the left-hand side is the integral
More informationSemigroup factorization and relaxation rates of kinetic equations
Semigroup factorization and relaxation rates of kinetic equations Clément Mouhot, University of Cambridge Analysis and Partial Differential Equations seminar University of Sussex 24th of february 2014
More informationNavier-Stokes equations in thin domains with Navier friction boundary conditions
Navier-Stokes equations in thin domains with Navier friction boundary conditions Luan Thach Hoang Department of Mathematics and Statistics, Texas Tech University www.math.umn.edu/ lhoang/ luan.hoang@ttu.edu
More informationOn the bang-bang property of time optimal controls for infinite dimensional linear systems
On the bang-bang property of time optimal controls for infinite dimensional linear systems Marius Tucsnak Université de Lorraine Paris, 6 janvier 2012 Notation and problem statement (I) Notation: X (the
More information2.20 Fall 2018 Math Review
2.20 Fall 2018 Math Review September 10, 2018 These notes are to help you through the math used in this class. This is just a refresher, so if you never learned one of these topics you should look more
More informationHertz potentials in curvilinear coordinates
Hertz potentials in curvilinear coordinates Jeff Bouas Texas A&M University July 9, 2010 Quantum Vacuum Workshop Jeff Bouas (Texas A&M University) Hertz potentials in curvilinear coordinates July 9, 2010
More informationFrom a Mesoscopic to a Macroscopic Description of Fluid-Particle Interaction
From a Mesoscopic to a Macroscopic Description of Fluid-Particle Interaction Carnegie Mellon University Center for Nonlinear Analysis Working Group, October 2016 Outline 1 Physical Framework 2 3 Free Energy
More informationVALIDITY OF THE BOLTZMANN EQUATION
VALIDITY OF THE BOLTZMANN EQUATION BEYOND HARD SPHERES based on joint work with M. Pulvirenti and C. Saffirio Sergio Simonella Technische Universität München Sergio Simonella - TU München Academia Sinica
More informationConvergence to equilibrium of Markov processes (eventually piecewise deterministic)
Convergence to equilibrium of Markov processes (eventually piecewise deterministic) A. Guillin Université Blaise Pascal and IUF Rennes joint works with D. Bakry, F. Barthe, F. Bolley, P. Cattiaux, R. Douc,
More informationChapter 6. Differentially Flat Systems
Contents CAS, Mines-ParisTech 2008 Contents Contents 1, Linear Case Introductory Example: Linear Motor with Appended Mass General Solution (Linear Case) Contents Contents 1, Linear Case Introductory Example:
More informationVisibility estimates in Euclidean and hyperbolic germ-grain models and line tessellations
Visibility estimates in Euclidean and hyperbolic germ-grain models and line tessellations Pierre Calka Lille, Stochastic Geometry Workshop, 30 March 2011 Outline Visibility in the vacancy of the Boolean
More informationsystem CWI, Amsterdam May 21, 2008 Dynamic Analysis Seminar Vrije Universiteit
CWI, Amsterdam heijster@cwi.nl May 21, 2008 Dynamic Analysis Seminar Vrije Universiteit Joint work: A. Doelman (CWI/UvA), T.J. Kaper (BU), K. Promislow (MSU) Outline 1 2 3 4 Outline 1 2 3 4 Paradigm U
More informationClassical and Quantum Bianchi type I cosmology in K-essence theory
Classical and Quantum Bianchi type I cosmology in K-essence theory Luis O. Pimentel 1, J. Socorro 1,2, Abraham Espinoza-García 2 1 Departamento de Fisica de la Universidad Autonoma Metropolitana Iztapalapa,
More informationEntanglement entropy and the F theorem
Entanglement entropy and the F theorem Mathematical Sciences and research centre, Southampton June 9, 2016 H RESEARH ENT Introduction This talk will be about: 1. Entanglement entropy 2. The F theorem for
More informationAncient solutions to Ricci flow
Ancient solutions to Ricci flow Lei Ni University of California, San Diego 2012 Motivations Motivations Ricci flow equation g(t) = 2Ric(g(t)). t Motivations Ricci flow equation g(t) = 2Ric(g(t)). t An
More informationStochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno
Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.
More informationInformation geometry for bivariate distribution control
Information geometry for bivariate distribution control C.T.J.Dodson + Hong Wang Mathematics + Control Systems Centre, University of Manchester Institute of Science and Technology Optimal control of stochastic
More informationTime-varying Consumption Tax, Productive Government Spending, and Aggregate Instability.
Time-varying Consumption Tax, Productive Government Spending, and Aggregate Instability. Literature Schmitt-Grohe and Uribe (JPE 1997): Ramsey model with endogenous labor income tax + balanced budget (fiscal)
More information4. Complex Oscillations
4. Complex Oscillations The most common use of complex numbers in physics is for analyzing oscillations and waves. We will illustrate this with a simple but crucially important model, the damped harmonic
More informationPhysics Dec Time Independent Solutions of the Diffusion Equation
Physics 301 10-Dec-2004 33-1 Time Independent Solutions of the Diffusion Equation In some cases we ll be interested in the time independent solution of the diffusion equation Why would be interested in
More informationPoint Vortex Dynamics in Two Dimensions
Spring School on Fluid Mechanics and Geophysics of Environmental Hazards 9 April to May, 9 Point Vortex Dynamics in Two Dimensions Ruth Musgrave, Mostafa Moghaddami, Victor Avsarkisov, Ruoqian Wang, Wei
More informationEvolution of the Universe
Evolution of the Universe by Nikola Perkovic e-mail: perce90gm@gmail.com Institute of Physics and Mathematics, Faculty of Sciences, University of Novi Sad Abstract: This paper will provide some well based
More information