Multiscale Analysis of Many Particle Systems with Dynamical Control

 Hector Hensley
 11 months ago
 Views:
Transcription
1 Michael Herrmann Multiscale Analysis of Many Particle Systems with Dynamical Control joint work with Barbara Niethammer and Juan J.L. Velázquez Kinetic description of multiscale phenomena Archimedes Center for Modeling, Analysis & Computation, Heraklion, June 27, 2013
2 Contents Nonlocal FokkerPlanck equations driven by a dynamical constraint arise in modelling of manyparticle storage systems involve two small and independent parameters complicate dynamics due to 3 different times scales Reduced models for small parameter limit fast reaction regime by adapting Kramers formula for large deviations slow reaction regime with subtle interplay between parabolic and kinetic effects This talk formal and heuristic arguments only, rigorous proofs for fast reactions available (see preprint on arxiv) 2
3 Manyparticle storage systems e machine e cathode FePO 4 FePO 4 Li + anode FePO 4 electrolyte Lithiumion battery with powder of nanoballs air battery with rubber ballons
4 Hysteresis and phase transitions device, eperiments, pictures: Clemens Guhlke, WIAS 4
5 Hysteresis and phase transitions device, eperiments, pictures: Clemens Guhlke, WIAS phase 1 only phase miture phase 2 only 4
6 Hysteresis and phase transitions device, eperiments, pictures: Clemens Guhlke, WIAS phase 1 only phase miture phase 2 only Hysteresis diagramm voltage vs. capacity pressure vs. volume Voltage versus Li/Li + (V) Voltage hysteresis 28% Q theor Capacity (ma h g 1 ) Multiscale analysis of manyparticle batteries c ACMAC Heraklion, June 27, 2013 d 4
7 The model
8 Assumptions on dynamics H() e machine e Modelling cathode FePO 4 FePO 4 FePO 4 Li + electrolyte anode assumptions (1) energy per particle has two local minima (2) each particles minimizes its energy quickly (3) mean position is prescribed by `(t) (4) stochastic fluctuations! (5) evolution of ensemble Dreyer, Guhlke, Herrmann: Continuum Mechanics and Thermodynamics (2011) 6
9 Assumptions on dynamics H() e machine e Modelling cathode FePO 4 FePO 4 FePO 4 Li + electrolyte anode assumptions (1) energy per particle has two local minima (2) each particles minimizes its energy quickly (3) mean position is prescribed by `(t) (4) stochastic fluctuations! (5) evolution of ensemble Dreyer, Guhlke, Herrmann: Continuum Mechanics and Thermodynamics (2011) probability density of many particle system %(t, ) 2 R position = concentration or size 6
10 Mesoscopic evolution nonlocal FokkerPlanck equation = X t % % + H 0 () (t) % = X 0 Z = X dynamical constraint R %(, t)d = `(t) stable interval unstable interval stable interval 7
11 Mesoscopic evolution nonlocal FokkerPlanck equation = X t % % + H 0 () (t) % = X 0 Z = X dynamical constraint R %(, t)d = `(t) stable interval unstable interval stable interval closure relation Z for dynamical multiplier (t) = R H 0 ()%(, t)d + `(t) FokkerPlanck equation is nonlinear, nonlocal, and driven by evolving constraint 7
12 Mesoscopic evolution nonlocal FokkerPlanck equation = X t % % + H 0 () (t) % = X 0 Z = X dynamical constraint R %(, t)d = `(t) stable interval unstable interval stable interval closure relation Z for dynamical multiplier (t) = R H 0 ()%(, t)d + `(t) FokkerPlanck equation is nonlinear, nonlocal, and driven by evolving constraint similar equations Mielke, Truskinovsky: Archive for Rational Mechanics and Analysis (2012) Truskinovsky, Puglisi: Journal of the Mechanics and Physics of Solids (2005) 7
13 Three time scales standard relaation to metastable state FokkerPlanck = const convergence to equilibrium large deviations, Kramers formula ep H 2 8
14 Three time scales standard relaation to metastable state FokkerPlanck = const convergence to equilibrium large deviations, Kramers formula ep H 2 nonlocal FokkerPlanck dynamical constraint ` = O(1) 8
15 Three time scales standard relaation to metastable state FokkerPlanck = const convergence to equilibrium large deviations, Kramers formula ep H 2 nonlocal FokkerPlanck dynamical constraint ` = O(1) Goal What happens in the small parameter limit!,! 0 Are the reduced (coarsegrained) models? Herrmann, Niethammer, Velázquez: SIAM Multiscale Modeling and Simulation (2012) 8
16 Numerical Simulations
17 Simplifying assumptions `(t) =1, `(0) 1 10
18 Simplifying assumptions `(t) =1, `(0) 1 initial and final dynamics = localized peak at `(t) How is the mass transported between the stable intervals? 10
19 Simplifying assumptions `(t) =1, `(0) 1 initial and final dynamics = localized peak at `(t) How is the mass transported between the stable intervals? Macroscopic vs. ` (evolution of mean force, black curve) evolution µ vs. ` (evolution of phase fraction, gray curve) Z 0 Z +1 phase fraction µ(t) = %(, t)d + %(, t)d
20 Simulations  macroscopic outcome slow A t = 1. n = 0.05 y B t = 0.5 n = 0.05 y C t = 0.25 n = 0.05 y reactions Type I Type II D t = 0.1 n = 0.05 y E t = 0.05 n = 0.05 y F t = n = 0.05 y fast reactions G t = n = 0.2 y H t = n = 0.2 y I t = n = 0.4 y Type III Type IV 11
21 Simulations  mesoscopic view A { = 0.8 A { = 0.0 A { = 0.8 A t = 1. n = 0.05 y 12
22 Simulations  mesoscopic view A { = 0.8 A { = 0.0 A { = 0.8 A t = 1. n = 0.05 y C { = 0.8 C { = 0.9 C { = 1.7 C t = 0.25 n = 0.05 y 12
23 Simulations  mesoscopic view A A t = 1. n = 0.05 { = 0.8 A { = 0.0 A { = 0.8 y C C t = 0.25 n = 0.05 { = 0.8 C { = 0.9 C { = 1.7 y G G t = n = 0.2 { = 1.6 G { = 0.0 G { = 1.6 y Multiscale analysis of manyparticle batteries ACMAC Heraklion, June 27,
24 Scaling regimes,! 0 singlepeak evolution = a log 1/ 0 <a<a crit piecewise continuous twopeaks evolution = p 0 <p<2/3 = p 2/3 <p<1 limit of Kramers formula b =ep 2, 0 <b<b crit Kramers formula 2 log 1/!1 quasistationary limit 13
25 Fast reactions: type III =ep b 2
26 Numerical results Idea 1 always (one or) two stable peaks m ± : masses X ± : positions m + m + =1 = H 0 (X )=H 0 (X + ) 15
27 Numerical results Idea 1 always (one or) two stable peaks m ± : masses X ± : positions m + m + =1 = H 0 (X )=H 0 (X + ) transport ṁ ± =0 15
28 Numerical results Idea 1 always (one or) two stable peaks m ± : masses X ± : positions m + m + =1 = H 0 (X )=H 0 (X + ) transport phase transition ṁ ± =0 Ẋ ± =0 15
29 Numerical results Idea 1 always (one or) two stable peaks m ± : masses X ± : positions m + m + =1 = H 0 (X )=H 0 (X + ) transport phase transition ṁ ± =0 Ẋ ± =0 Idea 2 either transport or phase transition 15
30 Heuristics particles diffuse in effective potential H () =H() H () =H() h h + Effective potential depends implicitly on time!
31 Heuristics particles diffuse in effective potential H () =H() H () =H() h h + Effective potential depends implicitly on time! 0 + stochastic fluctuations = effective mass flu large deviations, Kramers formula b h± ( ) R(t) = m (t)r (t) m + (t)r + (t) r ± ( )=C ± ( )ep 2 16
32 Various subregimes graph of H' + + b H 0 ( ) = H 0 () < 0 b < < < b = b b < < 0 graph of H s graph of H s graph of H s graph of H s r 1, r + 1 m 1 r 1, r + 1 ṁ > 0 r 1, r + 1 ṁ ± =0 supercritical critical subcritical 17
33 Rateindependent limit dynamics µ = m + m + b = # {µ = 1} b = # {µ =+1} ` X ( # ) X + ( # ) 18
34 Rateindependent limit dynamics µ = m + m + b = # {µ = 1} b = # {µ =+1} ` X ( # ) X + ( # ) effective dynamics (t) µ R µ(t) µ I µ(t), C µ(t), (t), `(t) =0 Rigorous justification is possible. 18
35 Slow Reactions: types I and II a =ep
36 Overview on type II y y y transport switching transport transport splitting merging y y transport switching transport 20
37 Simplified models transport localised peaks move due to the constraint switching twopeaks ODE stable peaks enter unstable interval merging unstable peaks merge rapidly with stables ones splitting unstable peaks split rapidly into two stables ones peakwidening model to compute the net splitting time masssplitting problem to mass distribution after splitting 21
38 Twopeaks approimation Dynamical model ṁ i =0 ẋ 1 = H 0 ( 1 ) ẋ 2 = H 0 ( 2 ) = m 1 H 0 ( 1 )+m 2 H 0 ( 2 )+ ` Quasistationary limit! 0 H 0 ( 1 )=H 0 ( 2 ) m m 2 2 = ` 22
39 Twopeaks approimation Dynamical model ṁ i =0 ẋ 1 = H 0 ( 1 ) ẋ 2 = H 0 ( 2 ) = m 1 H 0 ( 1 )+m 2 H 0 ( 2 )+ ` Quasistationary limit! 0 H 0 ( 1 )=H 0 ( 2 ) m m 2 2 = ` Multiple solution branches! 22
40 Twopeaks approimation Dynamical model ṁ i =0 + 2 (t) A (t) C ẋ 1 = H 0 ( 1 ) + t ẋ 2 = H 0 ( 2 ) t 0 t 1 B t 2 = m 1 H 0 ( 1 )+m 2 H 0 ( 2 )+ ` 1 (t) Quasistationary limit! 0 H 0 ( 1 )=H 0 ( 2 ) 2 A B m m 2 2 = (t 2 ) 1 = 2 m m 2 2 = ` H 0 ( 1 )=H 0 ( 2 ) C Multiple solution branches! m m 2 2 = `(t 0 )
41 Peakwidening model y % = m 1 ˆ% + m 2 2 ` = m 1 RR ˆ% d + m
42 Peakwidening model y FPPDE for first peak + ODE for second t ˆ% ˆ% + H 0 () ˆ% % = m 1 ˆ% + m 2 2 ` = m 1 RR ˆ% d + m = H 0 ( 2 ) Z = m 1 RR H0 1 H 0 ()ˆ%d + m ()ˆ% d + m 1 H 0 ( 2 H 0 2 )+ 2 + ` ` R 23
43 Peakwidening model y FPPDE for first peak + ODE for second t ˆ% ˆ% + H 0 () ˆ% % = m 1 ˆ% + m 2 2 ` = m 1 RR ˆ% d + m = H 0 ( 2 ) Z = m 1 RR H0 1 H 0 ()ˆ%d + m ()ˆ% d + m 1 H 0 ( 2 H 0 2 )+ 2 + ` ` R position of first peak ẋ 1 = H 0 ( 1 ) 23
44 Peakwidening model y FPPDE for first peak + ODE for second t ˆ% ˆ% + H 0 () ˆ% % = m 1 ˆ% + m 2 2 ` = m 1 RR ˆ% d + m = H 0 ( 2 ) Z = m 1 RR H0 1 H 0 ()ˆ%d + m ()ˆ% d + m 1 H 0 ( 2 H 0 2 )+ 2 + ` ` R position of first peak rescaling of first peak ẋ 1 = H 0 ( 1 ) ˆ%(t, ) = 1 (t) R (t), 1(t) (t) width of first peak identify scalings! find formula for width! w(t) = (t)w (t) 23
45 Formula for width of unstable peaks epand nonlinearity (fine as long as width is small) only one reasonable choice for time and space scaling 24
46 Formula for width of unstable peaks epand nonlinearity (fine as long as width is small) only one reasonable choice for time and space scaling w(t) 1 R 2 yr as long as width is small 1 =) R(y, ) 1 p 4 ep y 2 4,W( ) p 24
47 Formula for width of unstable peaks epand nonlinearity (fine as long as width is small) only one reasonable choice for time and space scaling w(t) 1 R 2 yr as long as width is small 1 =) R(y, ) 1 p 4 ep y 2 4,W( ) p evolution of width 0 <t<t sw : w(t) =O( ) t sw <t<t sp : w(t) 1 t sp <t : w(t) 1 24
48 Formula for width of unstable peaks epand nonlinearity (fine as long as width is small) only one reasonable choice for time and space scaling w(t) 1 R 2 yr as long as width is small 1 =) R(y, ) 1 p 4 ep y 2 4,W( ) p evolution of width 0 <t<t sw : w(t) =O( ) t sw <t<t sp : w(t) 1 t sp <t : w(t) 1 Z tsp t sw H 00 1 (t) dt + a =0 can be computed by quasistationary twopeaks approimation 24
49 Mass splitting problem Simplified =0, t = t sp + s ˆ% H 0 () (s) ˆ% Equations l(s) = const = l(t sp ) ẋ 2 = (s) H 0 ( 2 ) Z (s) =m 1 H 0 ()ˆ% d + m 2 H 0 ( 2 ) R no diffusion, frozen constraint, fast time scale 25
50 Mass splitting problem Simplified =0, t = t sp + s ˆ% H 0 () (s) ˆ% Equations l(s) = const = l(t sp ) ẋ 2 = (s) H 0 ( 2 ) Z (s) =m 1 H 0 ()ˆ% d + m 2 H 0 ( 2 ) R no diffusion, frozen constraint, fast time scale Splitting = heteroclinic connection unstable steady state stable steady state (unstable manifold has codim=1 ) (from a curve of possible ones ) 25
51 Mass splitting problem Simplified =0, t = t sp + s ˆ% H 0 () (s) ˆ% Equations l(s) = const = l(t sp ) ẋ 2 = (s) H 0 ( 2 ) Z (s) =m 1 H 0 ()ˆ% d + m 2 H 0 ( 2 ) R no diffusion, frozen constraint, fast time scale Splitting = heteroclinic connection unstable steady state stable steady state (unstable manifold has codim=1 ) (from a curve of possible ones ) Asymptotic initial data (reminiscent of diffusion) ˆ%(, s) s! 1! 1 2 p ep 1 (t sw ) 4ep(2 s), = H 00 1 (t sw ) > 0 25
52 Mass splitting function Conjecture (for nonlocal but autonomous transport equation) Heteroclinic connection is welldefined and depends continuously on the parameters. Mass Splitting Function (m 1,m 2 ) 7! (µm 1,m 2 +(1 µ)m 1 ) data just before splitting data just after splitting 26
53 Mass splitting function Conjecture (for nonlocal but autonomous transport equation) Heteroclinic connection is welldefined and depends continuously on the parameters. Mass Splitting Function (m 1,m 2 ) 7! (µm 1,m 2 +(1 µ)m 1 ) data just before splitting data just after splitting 26
54 Reduced model for limit dynamics initial data A t = 1. n = 0.05 y < stable transport (in ) Flowchart is numerical integrator! switching: t = t switching stablestable transport switching: t = t switching C t = 0.25 n = 0.05 y unstable transport mass splitting problem mass update: m i = m i + [m i ] unstablestable transport splitting: t = t splitting E t = 0.05 n = 0.05 y trivial merging: t = t merging >+ stable transport (in ) merging: t = t merging final data 27
55 Summary Nonlocal FokkerPlanck equations admit different dynamical regimes Fast reaction regime = Kramers regime Kramers formula describes typeiii transitions typeiv transitions as limiting case rigorous justification is available Slow reaction regime = nonkramers regime TypeI and typeii transitions can be described by  intervals of quasistationary transport  singular times corresponding to switching, splitting, merging rigorous justification remains open More details modeling: Dreyer, Guhlke, Herrmann: Continuum Mechanics and Thermodynamics (2011) heuristics and formal epansions: Herrmann, Niethammer, Velázquez: SIAM Multiscale Modeling and Simulations (2012) rigorous analysis: Herrmann, Niethammer, Velázquez: arxiv (2012)
56 Thank you!
Effective dynamics of manyparticle systems with dynamical constraint
Michael Herrmann Effective dnamics of manparticle sstems with dnamical constraint joint work with Barbara Niethammer and Juan J.L. Velázquez Workshop From particle sstems to differential equations WIAS
More informationKramers formula for chemical reactions in the context of Wasserstein gradient flows. Michael Herrmann. Mathematical Institute, University of Oxford
eport no. OxPDE/8 Kramers formula for chemical reactions in the context of Wasserstein gradient flows by Michael Herrmann Mathematical Institute, University of Oxford & Barbara Niethammer Mathematical
More informationV. Electrostatics Lecture 24: Diffuse Charge in Electrolytes
V. Electrostatics Lecture 24: Diffuse Charge in Electrolytes MIT Student 1. PoissonNernstPlanck Equations The NernstPlanck Equation is a conservation of mass equation that describes the influence of
More informationPHYSICAL REVIEW LETTERS
PHYSICAL REVIEW LETTERS VOLUME 80 1 JUNE 1998 NUMBER 22 FieldInduced Stabilization of Activation Processes N. G. Stocks* and R. Mannella Dipartimento di Fisica, Università di Pisa, and Istituto Nazionale
More informationNew Physical Principle for MonteCarlo simulations
EJTP 6, No. 21 (2009) 9 20 Electronic Journal of Theoretical Physics New Physical Principle for MonteCarlo simulations Michail Zak Jet Propulsion Laboratory California Institute of Technology, Advance
More informationM445: Heat equation with sources
M5: Heat equation with sources David Gurarie I. On Fourier and Newton s cooling laws The Newton s law claims the temperature rate to be proportional to the di erence: d dt T = (T T ) () The Fourier law
More informationQUANTUM MODELS FOR SEMICONDUCTORS AND NONLINEAR DIFFUSION EQUATIONS OF FOURTH ORDER
QUANTUM MODELS FOR SEMICONDUCTORS AND NONLINEAR DIFFUSION EQUATIONS OF FOURTH ORDER MARIA PIA GUALDANI The modern computer and telecommunication industry relies heavily on the use of semiconductor devices.
More informationVIII.B Equilibrium Dynamics of a Field
VIII.B Equilibrium Dynamics of a Field The next step is to generalize the Langevin formalism to a collection of degrees of freedom, most conveniently described by a continuous field. Let us consider the
More informationPiecewise Smooth Solutions to the BurgersHilbert Equation
Piecewise Smooth Solutions to the BurgersHilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA emails: bressan@mathpsuedu, zhang
More informationTaylor Series and Asymptotic Expansions
Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.
More informationGene regulatory networks: A coarsegrained, equationfree approach to multiscale computation
THE JOURNAL OF CHEMICAL PHYSICS 124, 084106 2006 Gene regulatory networks: A coarsegrained, equationfree approach to multiscale computation Radek Erban a Mathematical Institute, University of Oxford,
More information14. Energy transport.
Phys780: Plasma Physics Lecture 14. Energy transport. 1 14. Energy transport. ChapmanEnskog theory. ([8], p.5175) We derive macroscopic properties of plasma by calculating moments of the kinetic equation
More informationBasic Concepts of Electrochemistry
ELECTROCHEMISTRY Electricitydriven Chemistry or Chemistrydriven Electricity Electricity: Chemistry (redox): charge flow (electrons, holes, ions) reduction = electron uptake oxidation = electron loss
More informationHandbook of Stochastic Methods
C. W. Gardiner Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences Third Edition With 30 Figures Springer Contents 1. A Historical Introduction 1 1.1 Motivation I 1.2 Some Historical
More informationStochastic equations for thermodynamics
J. Chem. Soc., Faraday Trans. 93 (1997) 17511753 [arxiv 1503.09171] Stochastic equations for thermodynamics Roumen Tsekov Department of Physical Chemistry, University of Sofia, 1164 Sofia, ulgaria The
More informationarxiv:chaodyn/ v1 30 Jan 1997
Universal Scaling Properties in Large Assemblies of Simple Dynamical Units Driven by LongWave Random Forcing Yoshiki Kuramoto and Hiroya Nakao arxiv:chaodyn/9701027v1 30 Jan 1997 Department of Physics,
More informationBlowup profiles of solutions for the exponential reactiondiffusion equation
Blowup profiles of solutions for the exponential reactiondiffusion equation Aappo Pulkkinen Department of Mathematics and Systems Analysis Aalto University School of Science and Technology Finland 4th
More information1 Introduction. 2 Diffusion equation and central limit theorem. The content of these notes is also covered by chapter 3 section B of [1].
1 Introduction The content of these notes is also covered by chapter 3 section B of [1]. Diffusion equation and central limit theorem Consider a sequence {ξ i } i=1 i.i.d. ξ i = d ξ with ξ : Ω { Dx, 0,
More informationThermodynamics for small devices: From fluctuation relations to stochastic efficiencies. Massimiliano Esposito
Thermodynamics for small devices: From fluctuation relations to stochastic efficiencies Massimiliano Esposito Beijing, August 15, 2016 Introduction Thermodynamics in the 19th century: Thermodynamics in
More informationOne Dimensional Dynamical Systems
16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with onedimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar
More informationBattery System Safety and Health Management for Electric Vehicles
Battery System Safety and Health Management for Electric Vehicles Guangxing Bai and Pingfeng Wang Department of Industrial and Manufacturing Engineering Wichita State University Content Motivation for
More informationModelling in Biology
Modelling in Biology Dr GuyBart Stan Department of Bioengineering 17th October 2017 Dr GuyBart Stan (Dept. of Bioeng.) Modelling in Biology 17th October 2017 1 / 77 1 Introduction 2 Linear models of
More informationLecture 22: 1D Heat Transfer.
Chapter 13: Heat Transfer and Mass Transport. We will focus initially on the steady state heat transfer problem. Start by looking at the transfer of thermal energy along one dimension. With no convection
More informationCOMPETITION OF FAST AND SLOW MOVERS FOR RENEWABLE AND DIFFUSIVE RESOURCE
CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 2, Number, Spring 22 COMPETITION OF FAST AND SLOW MOVERS FOR RENEWABLE AND DIFFUSIVE RESOURCE SILOGINI THANARAJAH AND HAO WANG ABSTRACT. In many studies of
More informationThe Dynamics of ReactionDiffusion Patterns
The Dynamics of ReactionDiffusion Patterns Arjen Doelman (Leiden) (Rob Gardner, Tasso Kaper, Yasumasa Nishiura, Keith Promislow, Bjorn Sandstede) STRUCTURE OF THE TALK  Motivation  Topics that won t
More informationBasic modeling approaches for biological systems. Mahesh Bule
Basic modeling approaches for biological systems Mahesh Bule The hierarchy of life from atoms to living organisms Modeling biological processes often requires accounting for action and feedback involving
More informationAnalysis of a lumped model of neocortex to study epileptiform ac
of a lumped model of neocortex to study epileptiform activity Sid Visser Hil Meijer Stephan van Gils March 21, 2012 What is epilepsy? Pathology Neurological disorder, affecting 1% of world population Characterized
More informationOn the mystery of differential negative resistance
On the mystery of differential negative resistance Sebastian Popescu, Erzilia Lozneanu and Mircea Sanduloviciu Department of Plasma Physics Complexity Science Group Al. I. Cuza University 6600 Iasi, Romania
More informationElectrochemistry objectives
Electrochemistry objectives 1) Understand how a voltaic and electrolytic cell work 2) Be able to tell which substance is being oxidized and reduced and where it is occuring the anode or cathode 3) Students
More informationLocal Phase Portrait of Nonlinear Systems Near Equilibria
Local Phase Portrait of Nonlinear Sstems Near Equilibria [1] Consider 1 = 6 1 1 3 1, = 3 1. ( ) (a) Find all equilibrium solutions of the sstem ( ). (b) For each equilibrium point, give the linear approimating
More informationParallelintime integrators for Hamiltonian systems
Parallelintime integrators for Hamiltonian systems Claude Le Bris ENPC and INRIA Visiting Professor, The University of Chicago joint work with X. Dai (Paris 6 and Chinese Academy of Sciences), F. Legoll
More informationSupplementary Figure 1 A schematic representation of the different reaction mechanisms
Supplementary Figure 1 A schematic representation of the different reaction mechanisms observed in electrode materials for lithium batteries. Black circles: voids in the crystal structure, blue circles:
More informationOpen boundary conditions in stochastic transport processes with pairfactorized steady states
Open boundary conditions in stochastic transport processes with pairfactorized steady states Hannes Nagel a, Darka Labavić b, Hildegard MeyerOrtmanns b, Wolfhard Janke a a Institut für Theoretische Physik,
More informationModeling of Material Flow Problems
Modeling of Material Flow Problems Simone Göttlich Department of Mathematics University of Mannheim Workshop on Math for the Digital Factory, WIAS Berlin May 79, 2014 Prof. Dr. Simone Göttlich Material
More informationG : Statistical Mechanics
G25.2651: Statistical Mechanics Notes for Lecture 1 Defining statistical mechanics: Statistical Mechanics provies the connection between microscopic motion of individual atoms of matter and macroscopically
More informationStochastic Thermodynamics of Langevin systems under timedelayed feedback control
JapanFrance Joint Seminar (4 August 25) New Frontiers in Nonequilibrium Physics of Glassy Materials Stochastic Thermodynamics of Langevin systems under timedelayed feedback control M.L. Rosinberg in
More informationWaves in nature, such as waves on the surface of the sea
Spectral bifurcations in dispersive wave turbulence David Cai,* Andrew J. Majda, David W. McLaughlin, and Esteban G. Tabak Courant Institute of Mathematical Sciences, New York University, New York, NY
More informationElectrochemistry. Goal: Understand basic electrochemical reactions. Half Cell Reactions Nernst Equation Pourbaix Diagrams.
Electrochemistry Goal: Understand basic electrochemical reactions Concepts: Electrochemical Cell Half Cell Reactions Nernst Equation Pourbaix Diagrams Homework: Applications Battery potential calculation
More informationResidual resistance simulation of an air spark gap switch.
Residual resistance simulation of an air spark gap switch. V. V. Tikhomirov, S.E. Siahlo February 27, 2015 arxiv:1502.07499v1 [physics.accph] 26 Feb 2015 Research Institute for Nuclear Problems, Belarusian
More informationNumerical Methods for ODEs. Lectures for PSU Summer Programs Xiantao Li
Numerical Methods for ODEs Lectures for PSU Summer Programs Xiantao Li Outline Introduction Some Challenges Numerical methods for ODEs Stiff ODEs Accuracy Constrained dynamics Stability Coarsegraining
More informationSoft turbulence in multimode lasers
PHYSICAL REVIEW A VOLUME 55, NUMBER 1 JANUARY 1997 Soft turbulence in multimode lasers D. Casini, 1 G. D Alessandro, 2 and A. Politi 1,3 1 Istituto Nazionale di Ottica, Largo E. Fermi 6, I50125 Firenze,
More informationOnsager theory: overview
Onsager theory: overview Pearu Peterson December 18, 2006 1 Introduction Our aim is to study matter that consists of large number of molecules. A complete mechanical description of such a system is practically
More informationTimeperiodic forcing of Turing patterns in the Brusselator model
Timeperiodic forcing of Turing patterns in the Brusselator model B. Peña and C. Pérez García Instituto de Física. Universidad de Navarra, Irunlarrea, 1. 31008Pamplona, Spain Abstract Experiments on temporal
More informationModels for dynamic fracture based on Griffith s criterion
Models for dynamic fracture based on Griffith s criterion Christopher J. Larsen Abstract There has been much recent progress in extending Griffith s criterion for crack growth into mathematical models
More informationCosmology holography the brain and the quantum vacuum. Antonio AlfonsoFaus. Departamento de Aerotécnia. Madrid Technical University (UPM), Spain
Cosmology holography the brain and the quantum vacuum Antonio AlfonsoFaus Departamento de Aerotécnia Madrid Technical University (UPM), Spain February, 2011. Email: aalfonsofaus@yahoo.es Abstract: Cosmology,
More informationNonequilibrium dynamics of Kinetically Constrained Models. Paul Chleboun 20/07/2015 SMFP  Heraclion
Nonequilibrium dynamics of Kinetically Constrained Models Paul Chleboun 20/07/2015 SMFP  Heraclion 1 Outline Kinetically constrained models (KCM)» Motivation» Definition Results» In one dimension There
More informationFernando O. Raineri. Office Hours: MWF 9:3010:30 AM Room 519 Tue. 3:005:00 CLC (lobby).
Fernando O. Raineri Office Hours: MWF 9:3010:30 AM Room 519 Tue. 3:005:00 CLC (lobby). P1) What is the reduction potential of the hydrogen electrode g bar H O aq Pt(s) H,1 2 3 when the aqueous solution
More informationChapter 9 Linear Momentum and Collisions
Chapter 9 Linear Momentum and Collisions The Center of Mass The center of mass of a system of particles is the point that moves as though (1) all of the system s mass were concentrated there and (2) all
More information14 Periodic phenomena in nature and limit cycles
14 Periodic phenomena in nature and limit cycles 14.1 Periodic phenomena in nature As it was discussed while I talked about the Lotka Volterra model, a great deal of natural phenomena show periodic behavior.
More informationNonlinear resistance of twodimensional electrons in crossed electric and magnetic fields
Nonlinear resistance of twodimensional electrons in crossed electric and magnetic fields Jing Qiao Zhang and Sergey Vitkalov* Department of Physics, City College of the City University of New York, New
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 ParisDauphine http://www.ceremade.dauphine.fr/ dolbeaul (A JOINT
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
More informationNonstationary Invariant Distributions and the HydrodynamicsStyle Generalization of the KolmogorovForward/Fokker Planck Equation
Accepted by Appl. Math. Lett. in 2004 1 Nonstationary Invariant Distributions and the HydrodynamicsStyle Generalization of the KolmogorovForward/Fokker Planck Equation Laboratory of Physical Electronics
More information25.2. Applications of PDEs. Introduction. Prerequisites. Learning Outcomes
Applications of PDEs 25.2 Introduction In this Section we discuss briefly some of the most important PDEs that arise in various branches of science and engineering. We shall see that some equations can
More informationIntroduction. Resonant Cooling of Nuclear Spins in Quantum Dots
Introduction Resonant Cooling of Nuclear Spins in Quantum Dots Mark Rudner Massachusetts Institute of Technology For related details see: M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 99, 036602 (2007);
More informationModeling the next battery generation: Lithiumsulfur and lithiumair cells
Modeling the next battery generation: Lithiumsulfur and lithiumair cells D. N. Fronczek, T. Danner, B. Horstmann, Wolfgang G. Bessler German Aerospace Center (DLR) University Stuttgart (ITW) Helmholtz
More informationTOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS. Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 1516, 2017
TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 1516, 2017 Abstracts of the talks Spectral stability under removal of small capacity
More informationVIII. Phase Transformations. Lecture 38: Nucleation and Spinodal Decomposition
VIII. Phase Transformations Lecture 38: Nucleation and Spinodal Decomposition MIT Student In this lecture we will study the onset of phase transformation for phases that differ only in their equilibrium
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More informationMESOSCOPIC QUANTUM OPTICS
MESOSCOPIC QUANTUM OPTICS by Yoshihisa Yamamoto Ata Imamoglu A WileyInterscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Toronto Singapore Preface xi 1 Basic Concepts
More informationSHAPE MODELLING WITH CONTOURS AND FIELDS
SHAPE MODELLING WITH CONTOURS AND FIELDS Ian Jermyn Josiane Zerubia Zoltan Kato Marie Rochery Peter Horvath Ting Peng Aymen El Ghoul INRIA University of Szeged INRIA INRIA, Szeged INRIA, CASIA INRIA Overview
More informationElectrical Transport in Nanoscale Systems
Electrical Transport in Nanoscale Systems Description This book provides an indepth description of transport phenomena relevant to systems of nanoscale dimensions. The different viewpoints and theoretical
More informationGAUSS CIRCLE PROBLEM
GAUSS CIRCLE PROBLEM 1. Gauss circle problem We begin with a very classical problem: how many lattice points lie on or inside the circle centered at the origin and with radius r? (In keeping with the classical
More informationLognormal Moment Closures for Biochemical Reactions
Lognormal Moment Closures for Biochemical Reactions Abhyudai Singh and João Pedro Hespanha Abstract In the stochastic formulation of chemical reactions, the dynamics of the the first M order moments of
More informationIntroduction to multiscale modeling and simulation. Almost every system is multiscale.
Introduction to multiscale modeling and simulation Giovanni Samaey, Scientific Computing Dept. of Computer Science, K.U.Leuven Lecture 1: Course introduction Almost every system is multiscale. We are interested
More informationScaling Analysis of Energy Storage by Porous Electrodes
Scaling Analysis of Energy Storage by Porous Electrodes Martin Z. Bazant May 14, 2012 1 Theoretical Capacity The maximum theoretical capacity occurs as E i 0, E p 0 E a 1, where E i, E p, and E a are the
More informationOn the backbone exponent
On the backbone exponent Christophe Garban Université Lyon 1 joint work with JeanChristophe Mourrat (ENS Lyon) Cargèse, September 2016 C. Garban (univ. Lyon 1) On the backbone exponent 1 / 30 Critical
More informationParameter estimation in linear Gaussian covariance models
Parameter estimation in linear Gaussian covariance models Caroline Uhler (IST Austria) Joint work with Piotr Zwiernik (UC Berkeley) and Donald Richards (Penn State University) Big Data Reunion Workshop
More informationSimulation of Coulomb Collisions in Plasma Accelerators for Space Applications
Simulation of Coulomb Collisions in Plasma Accelerators for Space Applications D. D Andrea 1, W.Maschek 1 and R. Schneider 2 Vienna, May 6 th 2009 1 Institut for Institute for Nuclear and Energy Technologies
More informationThe correlation between stochastic resonance and the average phasesynchronization time of a bistable system driven by colourcorrelated noises
Chin. Phys. B Vol. 19, No. 1 (010) 01050 The correlation between stochastic resonance and the average phasesynchronization time of a bistable system driven by colourcorrelated noises Dong XiaoJuan(
More informationMETEOR PROCESS. Krzysztof Burdzy University of Washington
University of Washington Collaborators and preprints Joint work with Sara Billey, Soumik Pal and Bruce E. Sagan. Math Arxiv: http://arxiv.org/abs/1308.2183 http://arxiv.org/abs/1312.6865 Mass redistribution
More informationDensity Functional Modeling of Nanocrystalline Materials
Density Functional Modeling of Nanocrystalline Materials A new approach for modeling atomic scale properties in materials Peter Stefanovic Supervisor: Nikolas Provatas 70 / Part 17 February 007 Density
More informationVII. Porous Media Lecture 36: Electrochemical Supercapacitors
VII. Porous Media Lecture 36: Electrochemical Supercapacitors MIT Student (and MZB) 1. Transmission Line Model for Linear Response Last time, we took the supercapacitor limit of a general porous medium
More informationTravelling waves. Chapter 8. 1 Introduction
Chapter 8 Travelling waves 1 Introduction One of the cornerstones in the study of both linear and nonlinear PDEs is the wave propagation. A wave is a recognizable signal which is transferred from one part
More informationThermal Systems. What and How? Physical Mechanisms and Rate Equations Conservation of Energy Requirement Control Volume Surface Energy Balance
Introduction to Heat Transfer What and How? Physical Mechanisms and Rate Equations Conservation of Energy Requirement Control Volume Surface Energy Balance Thermal Resistance Thermal Capacitance Thermal
More informationReachable sets for autonomous systems of differential equations and their topological properties
American Journal of Applied Mathematics 2013; 1(4): 4954 Published online October 30, 2013 (http://www.sciencepublishinggroup.com/j/ajam) doi: 10.11648/j.ajam.20130104.13 Reachable sets for autonomous
More informationSpace Plasma Physics Thomas Wiegelmann, 2012
Space Plasma Physics Thomas Wiegelmann, 2012 1. Basic Plasma Physics concepts 2. Overview about solar system plasmas Plasma Models 3. Single particle motion, Test particle model 4. Statistic description
More informationBatteries crash simulation with LSDYNA. LithiumIon cell. In collaboration with J. Marcicki et al, Ford Research and Innovation Center, Dearborn, MI
Batteries crash simulation with LSDYNA LithiumIon cell In collaboration with J. Marcicki et al, Ford esearch and Innovation Center, Dearborn, MI Battery crash simulation (general) Car crash Mechanical
More informationFrom Bohmian Mechanics to Bohmian Quantum Gravity. Antonio Vassallo Instytut Filozofii UW Section de Philosophie UNIL
From Bohmian Mechanics to Bohmian Quantum Gravity Antonio Vassallo Instytut Filozofii UW Section de Philosophie UNIL The Measurement Problem in Quantum Mechanics (1) The wavefunction of a system is complete,
More informationRedOx Chemistry. with. Dr. Nick
RedOx Chemistry with Dr. Nick What is RedOx Chemistry? The defining characteristic of a RedOx reaction is that electron(s) have completely moved from one atom / molecule to another. The molecule receiving
More informationSolutions to Math 53 First Exam April 20, 2010
Solutions to Math 53 First Exam April 0, 00. (5 points) Match the direction fields below with their differential equations. Also indicate which two equations do not have matches. No justification is necessary.
More informationAna María Cetto, Luis de la Peña and Andrea Valdés Hernández
Ana María Cetto, Luis de la Peña and Andrea Valdés Hernández Instituto de Física, UNAM EmQM13, Vienna, 36 October 2013 1 Planck's law as a consequence of the zeropoint field Equilibrium radiation field
More informationON LIE GROUP CLASSIFICATION OF A SCALAR STOCHASTIC DIFFERENTIAL EQUATION
Journal of Nonlinear Mathematical Physics ISSN: 14029251 (Print) 17760852 (Online) Journal homepage: http://www.tandfonline.com/loi/tnmp20 ON LIE GROUP CLASSIFICATION OF A SCALAR STOCHASTIC DIFFERENTIAL
More informationKinetic Monte Carlo (KMC)
Kinetic Monte Carlo (KMC) Molecular Dynamics (MD): highfrequency motion dictate the timestep (e.g., vibrations). Time step is short: picoseconds. Direct Monte Carlo (MC): stochastic (nondeterministic)
More informationMath 256: Applied Differential Equations: Final Review
Math 256: Applied Differential Equations: Final Review Chapter 1: Introduction, Sec 1.1, 1.2, 1.3 (a) Differential Equation, Mathematical Model (b) Direction (Slope) Field, Equilibrium Solution (c) Rate
More informationAsymptotics of rare events in birth death processes bypassing the exact solutions
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 9 (27) 6545 (2pp) doi:.88/9538984/9/6/6545 Asymptotics of rare events in birth death processes bypassing
More informationInversesquare law between time and amplitude for crossing tipping thresholds
Inversesquare law between time and amplitude for crossing tipping thresholds Paul Ritchie Earth System Science, College of Life and Environmental Sciences, Harrison Building, University of Exeter, Exeter,
More informationDetermination of Planck s constant and work function of metals using photoelectric effect
Determination of Planck s constant and work function of metals using photoelectric effect Objective I. To determine Planck s constant h from the stopping voltages measured at different frequencies (wavelengths)
More informationIntroduction to Dynamical Systems Basic Concepts of Dynamics
Introduction to Dynamical Systems Basic Concepts of Dynamics A dynamical system: Has a notion of state, which contains all the information upon which the dynamical system acts. A simple set of deterministic
More informationKINETIC DESCRIPTION OF MAGNETIZED TECHNOLOGICAL PLASMAS
KINETIC DESCRIPTION OF MAGNETIZED TECHNOLOGICAL PLASMAS Ralf Peter Brinkmann, Dennis Krüger Fakultät für Elektrotechnik und Informationstechnik Lehrstuhl für Theoretische Elektrotechnik Magnetized low
More informationHIGH FRICTION LIMIT OF THE KRAMERS EQUATION : THE MULTIPLE TIME SCALE APPROACH. Lydéric Bocquet
HIGH FRICTION LIMIT OF THE KRAMERS EQUATION : THE MULTIPLE TIME SCALE APPROACH Lydéric Bocquet arxiv:condmat/9605186v1 30 May 1996 Laboratoire de Physique, Ecole Normale Supérieure de Lyon (URA CNRS 1325),
More informationAvailable online at ScienceDirect. Physics Procedia 57 (2014 ) 77 81
Available online at www.sciencedirect.com ScienceDirect Physics Procedia 57 (204 ) 77 8 27th Annual CSP Workshops on Recent Developments in Computer Simulation Studies in Condensed Matter Physics, CSP
More informationPhysics 228. Momentum and Force Kinetic Energy Relativistic Mass and Rest Mass Photoelectric Effect Energy and Momentum of Photons
Physics 228 Momentum and Force Kinetic Energy Relativistic Mass and Rest Mass Photoelectric Effect Energy and Momentum of Photons Lorentz Transformations vs. Rotations The Lorentz transform is similar
More informationBrownian Motion and The Atomic Theory
Brownian Motion and The Atomic Theory Albert Einstein Annus Mirabilis Centenary Lecture Simeon Hellerman Institute for Advanced Study, 5/20/2005 Founders Day 1 1. What phenomenon did Einstein explain?
More informationResidencetime distributions as a measure for stochastic resonance
W e ie rstra ß In stitu t fü r A n g e w a n d te A n a ly sis u n d S to ch a stik Period of Concentration: Stochastic Climate Models MPI Mathematics in the Sciences, Leipzig, 23 May 1 June 2005 Barbara
More informationUnique equilibrium states for geodesic flows in nonpositive curvature
Unique equilibrium states for geodesic flows in nonpositive curvature Todd Fisher Department of Mathematics Brigham Young University Fractal Geometry, Hyperbolic Dynamics and Thermodynamical Formalism
More informationConvectiondriven dynamos in the limit of rapid rotation
Convectiondriven dynamos in the limit of rapid rotation Michael A. Calkins Jonathan M. Aurnou (UCLA), Keith Julien (CU), Louie Long (CU), Philippe Marti (CU), Steven M. Tobias (Leeds) *Department of Physics,
More informationCourse. Print and use this sheet in conjunction with MathinSite s Maclaurin Series applet and worksheet.
Maclaurin Series Learning Outcomes After reading this theory sheet, you should recognise the difference between a function and its polynomial epansion (if it eists!) understand what is meant by a series
More informationChapter 2 Thermodynamics
Chapter 2 Thermodynamics 2.1 Introduction The First Law of Thermodynamics is a statement of the existence of a property called Energy which is a state function that is independent of the path and, in the
More informationChapter 13 Rates of Reactions
Chapter 13 Rates of Reactions Chemical reactions require varying lengths of time for completion, depending on the characteristics of the reactants and products. The study of the rate, or speed, of a reaction
More information