Multiscale Analysis of Many Particle Systems with Dynamical Control

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Multiscale Analysis of Many Particle Systems with Dynamical Control"

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 Fokker-Planck equations driven by a dynamical constraint arise in modelling of many-particle 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 Many-particle storage systems e machine e cathode FePO 4 FePO 4 Li + anode FePO 4 electrolyte Lithium-ion battery with powder of nano-balls 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 many-particle 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 Fokker-Planck 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 Fokker-Planck 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) Fokker-Planck equation is nonlinear, nonlocal, and driven by evolving constraint 7

12 Mesoscopic evolution nonlocal Fokker-Planck 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) Fokker-Planck 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 meta-stable state Fokker-Planck = const convergence to equilibrium large deviations, Kramers formula ep H 2 8

14 Three time scales standard relaation to meta-stable state Fokker-Planck = const convergence to equilibrium large deviations, Kramers formula ep H 2 nonlocal Fokker-Planck dynamical constraint ` = O(1) 8

15 Three time scales standard relaation to meta-stable state Fokker-Planck = const convergence to equilibrium large deviations, Kramers formula ep H 2 nonlocal Fokker-Planck dynamical constraint ` = O(1) Goal What happens in the small parameter limit!,! 0 Are the reduced (coarse-grained) 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 many-particle batteries ACMAC Heraklion, June 27,

24 Scaling regimes,! 0 single-peak evolution = a log 1/ 0 <a<a crit piecewise continuous two-peaks 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 quasi-stationary 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 Rate-independent limit dynamics µ = m + m + b = # {µ = 1} b = # {µ =+1} ` X ( # ) X + ( # ) 18

34 Rate-independent 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 two-peaks ODE stable peaks enter unstable interval merging unstable peaks merge rapidly with stables ones splitting unstable peaks split rapidly into two stables ones peak-widening model to compute the net splitting time mass-splitting problem to mass distribution after splitting 21

38 Two-peaks 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 )+ ` Quasi-stationary limit! 0 H 0 ( 1 )=H 0 ( 2 ) m m 2 2 = ` 22

39 Two-peaks 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 )+ ` Quasi-stationary limit! 0 H 0 ( 1 )=H 0 ( 2 ) m m 2 2 = ` Multiple solution branches! 22

40 Two-peaks 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) Quasi-stationary 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 Peak-widening model y % = m 1 ˆ% + m 2 2 ` = m 1 RR ˆ% d + m

42 Peak-widening model y FP-PDE 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 Peak-widening model y FP-PDE 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 Peak-widening model y FP-PDE 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 quasi-stationary two-peaks 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 well-defined 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 well-defined 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 stable-stable 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 ] unstable-stable 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 Fokker-Planck equations admit different dynamical regimes Fast reaction regime = Kramers regime Kramers formula describes type-iii transitions type-iv transitions as limiting case rigorous justification is available Slow reaction regime = non-kramers regime Type-I and type-ii transitions can be described by - intervals of quasi-stationary 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 many-particle systems with dynamical constraint

Effective dynamics of many-particle systems with dynamical constraint Michael Herrmann Effective dnamics of man-particle sstems with dnamical constraint joint work with Barbara Niethammer and Juan J.L. Velázquez Workshop From particle sstems to differential equations WIAS

More information

Kramers formula for chemical reactions in the context of Wasserstein gradient flows. Michael Herrmann. Mathematical Institute, University of Oxford

Kramers 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 information

V. Electrostatics Lecture 24: Diffuse Charge in Electrolytes

V. Electrostatics Lecture 24: Diffuse Charge in Electrolytes V. Electrostatics Lecture 24: Diffuse Charge in Electrolytes MIT Student 1. Poisson-Nernst-Planck Equations The Nernst-Planck Equation is a conservation of mass equation that describes the influence of

More information

PHYSICAL REVIEW LETTERS

PHYSICAL REVIEW LETTERS PHYSICAL REVIEW LETTERS VOLUME 80 1 JUNE 1998 NUMBER 22 Field-Induced Stabilization of Activation Processes N. G. Stocks* and R. Mannella Dipartimento di Fisica, Università di Pisa, and Istituto Nazionale

More information

New Physical Principle for Monte-Carlo simulations

New Physical Principle for Monte-Carlo simulations EJTP 6, No. 21 (2009) 9 20 Electronic Journal of Theoretical Physics New Physical Principle for Monte-Carlo simulations Michail Zak Jet Propulsion Laboratory California Institute of Technology, Advance

More information

M445: Heat equation with sources

M445: 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 information

QUANTUM MODELS FOR SEMICONDUCTORS AND NONLINEAR DIFFUSION EQUATIONS OF FOURTH ORDER

QUANTUM 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 information

VIII.B Equilibrium Dynamics of a Field

VIII.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 information

Piecewise Smooth Solutions to the Burgers-Hilbert Equation

Piecewise Smooth Solutions to the Burgers-Hilbert Equation Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang

More information

Taylor Series and Asymptotic Expansions

Taylor 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 information

Gene regulatory networks: A coarse-grained, equation-free approach to multiscale computation

Gene regulatory networks: A coarse-grained, equation-free approach to multiscale computation THE JOURNAL OF CHEMICAL PHYSICS 124, 084106 2006 Gene regulatory networks: A coarse-grained, equation-free approach to multiscale computation Radek Erban a Mathematical Institute, University of Oxford,

More information

14. Energy transport.

14. Energy transport. Phys780: Plasma Physics Lecture 14. Energy transport. 1 14. Energy transport. Chapman-Enskog theory. ([8], p.51-75) We derive macroscopic properties of plasma by calculating moments of the kinetic equation

More information

Basic Concepts of Electrochemistry

Basic Concepts of Electrochemistry ELECTROCHEMISTRY Electricity-driven Chemistry or Chemistry-driven Electricity Electricity: Chemistry (redox): charge flow (electrons, holes, ions) reduction = electron uptake oxidation = electron loss

More information

Handbook of Stochastic Methods

Handbook 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 information

Stochastic equations for thermodynamics

Stochastic equations for thermodynamics J. Chem. Soc., Faraday Trans. 93 (1997) 1751-1753 [arxiv 1503.09171] Stochastic equations for thermodynamics Roumen Tsekov Department of Physical Chemistry, University of Sofia, 1164 Sofia, ulgaria The

More information

arxiv:chao-dyn/ v1 30 Jan 1997

arxiv:chao-dyn/ v1 30 Jan 1997 Universal Scaling Properties in Large Assemblies of Simple Dynamical Units Driven by Long-Wave Random Forcing Yoshiki Kuramoto and Hiroya Nakao arxiv:chao-dyn/9701027v1 30 Jan 1997 Department of Physics,

More information

Blow-up profiles of solutions for the exponential reaction-diffusion equation

Blow-up profiles of solutions for the exponential reaction-diffusion equation Blow-up profiles of solutions for the exponential reaction-diffusion equation Aappo Pulkkinen Department of Mathematics and Systems Analysis Aalto University School of Science and Technology Finland 4th

More information

1 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. 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 information

Thermodynamics for small devices: From fluctuation relations to stochastic efficiencies. Massimiliano Esposito

Thermodynamics 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 information

One Dimensional Dynamical Systems

One Dimensional Dynamical Systems 16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with one-dimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar

More information

Battery System Safety and Health Management for Electric Vehicles

Battery 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 information

Modelling in Biology

Modelling in Biology Modelling in Biology Dr Guy-Bart Stan Department of Bioengineering 17th October 2017 Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October 2017 1 / 77 1 Introduction 2 Linear models of

More information

Lecture 22: 1-D Heat Transfer.

Lecture 22: 1-D 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 information

COMPETITION OF FAST AND SLOW MOVERS FOR RENEWABLE AND DIFFUSIVE RESOURCE

COMPETITION 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 information

The Dynamics of Reaction-Diffusion Patterns

The Dynamics of Reaction-Diffusion Patterns The Dynamics of Reaction-Diffusion Patterns Arjen Doelman (Leiden) (Rob Gardner, Tasso Kaper, Yasumasa Nishiura, Keith Promislow, Bjorn Sandstede) STRUCTURE OF THE TALK - Motivation - Topics that won t

More information

Basic modeling approaches for biological systems. Mahesh Bule

Basic 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 information

Analysis of a lumped model of neocortex to study epileptiform ac

Analysis 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 information

On the mystery of differential negative resistance

On 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 information

Electrochemistry objectives

Electrochemistry 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 information

Local Phase Portrait of Nonlinear Systems Near Equilibria

Local 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 information

Parallel-in-time integrators for Hamiltonian systems

Parallel-in-time integrators for Hamiltonian systems Parallel-in-time 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 information

Supplementary Figure 1 A schematic representation of the different reaction mechanisms

Supplementary 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 information

Open boundary conditions in stochastic transport processes with pair-factorized steady states

Open boundary conditions in stochastic transport processes with pair-factorized steady states Open boundary conditions in stochastic transport processes with pair-factorized steady states Hannes Nagel a, Darka Labavić b, Hildegard Meyer-Ortmanns b, Wolfhard Janke a a Institut für Theoretische Physik,

More information

Modeling of Material Flow Problems

Modeling 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 7-9, 2014 Prof. Dr. Simone Göttlich Material

More information

G : Statistical Mechanics

G : 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 information

Stochastic Thermodynamics of Langevin systems under time-delayed feedback control

Stochastic Thermodynamics of Langevin systems under time-delayed feedback control Japan-France Joint Seminar (-4 August 25) New Frontiers in Non-equilibrium Physics of Glassy Materials Stochastic Thermodynamics of Langevin systems under time-delayed feedback control M.L. Rosinberg in

More information

Waves in nature, such as waves on the surface of the sea

Waves 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 information

Electrochemistry. Goal: Understand basic electrochemical reactions. Half Cell Reactions Nernst Equation Pourbaix Diagrams.

Electrochemistry. 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 information

Residual resistance simulation of an air spark gap switch.

Residual 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.acc-ph] 26 Feb 2015 Research Institute for Nuclear Problems, Belarusian

More information

Numerical Methods for ODEs. Lectures for PSU Summer Programs Xiantao Li

Numerical 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 Coarse-graining

More information

Soft turbulence in multimode lasers

Soft 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, I-50125 Firenze,

More information

Onsager theory: overview

Onsager 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 information

Time-periodic forcing of Turing patterns in the Brusselator model

Time-periodic forcing of Turing patterns in the Brusselator model Time-periodic forcing of Turing patterns in the Brusselator model B. Peña and C. Pérez García Instituto de Física. Universidad de Navarra, Irunlarrea, 1. 31008-Pamplona, Spain Abstract Experiments on temporal

More information

Models for dynamic fracture based on Griffith s criterion

Models 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 information

Cosmology holography the brain and the quantum vacuum. Antonio Alfonso-Faus. Departamento de Aerotécnia. Madrid Technical University (UPM), Spain

Cosmology holography the brain and the quantum vacuum. Antonio Alfonso-Faus. Departamento de Aerotécnia. Madrid Technical University (UPM), Spain Cosmology holography the brain and the quantum vacuum Antonio Alfonso-Faus Departamento de Aerotécnia Madrid Technical University (UPM), Spain February, 2011. E-mail: aalfonsofaus@yahoo.es Abstract: Cosmology,

More information

Non-equilibrium dynamics of Kinetically Constrained Models. Paul Chleboun 20/07/2015 SMFP - Heraclion

Non-equilibrium dynamics of Kinetically Constrained Models. Paul Chleboun 20/07/2015 SMFP - Heraclion Non-equilibrium 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 information

Fernando O. Raineri. Office Hours: MWF 9:30-10:30 AM Room 519 Tue. 3:00-5:00 CLC (lobby).

Fernando O. Raineri. Office Hours: MWF 9:30-10:30 AM Room 519 Tue. 3:00-5:00 CLC (lobby). Fernando O. Raineri Office Hours: MWF 9:30-10:30 AM Room 519 Tue. 3:00-5:00 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 information

Chapter 9 Linear Momentum and Collisions

Chapter 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 information

14 Periodic phenomena in nature and limit cycles

14 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 information

Nonlinear resistance of two-dimensional electrons in crossed electric and magnetic fields

Nonlinear resistance of two-dimensional electrons in crossed electric and magnetic fields Nonlinear resistance of two-dimensional 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 information

Hypocoercivity for kinetic equations with linear relaxation terms

Hypocoercivity 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 information

Applied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.

Applied 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 information

Nonstationary Invariant Distributions and the Hydrodynamics-Style Generalization of the Kolmogorov-Forward/Fokker Planck Equation

Nonstationary Invariant Distributions and the Hydrodynamics-Style Generalization of the Kolmogorov-Forward/Fokker Planck Equation Accepted by Appl. Math. Lett. in 2004 1 Nonstationary Invariant Distributions and the Hydrodynamics-Style Generalization of the Kolmogorov-Forward/Fokker Planck Equation Laboratory of Physical Electronics

More information

25.2. Applications of PDEs. Introduction. Prerequisites. Learning Outcomes

25.2. 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 information

Introduction. Resonant Cooling of Nuclear Spins in Quantum Dots

Introduction. 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 information

Modeling the next battery generation: Lithium-sulfur and lithium-air cells

Modeling the next battery generation: Lithium-sulfur and lithium-air cells Modeling the next battery generation: Lithium-sulfur and lithium-air cells D. N. Fronczek, T. Danner, B. Horstmann, Wolfgang G. Bessler German Aerospace Center (DLR) University Stuttgart (ITW) Helmholtz

More information

TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS. Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017

TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS. Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017 TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017 Abstracts of the talks Spectral stability under removal of small capacity

More information

VIII. Phase Transformations. Lecture 38: Nucleation and Spinodal Decomposition

VIII. 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 information

1 Lyapunov theory of stability

1 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 information

MESOSCOPIC QUANTUM OPTICS

MESOSCOPIC QUANTUM OPTICS MESOSCOPIC QUANTUM OPTICS by Yoshihisa Yamamoto Ata Imamoglu A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Toronto Singapore Preface xi 1 Basic Concepts

More information

SHAPE MODELLING WITH CONTOURS AND FIELDS

SHAPE 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 information

Electrical Transport in Nanoscale Systems

Electrical Transport in Nanoscale Systems Electrical Transport in Nanoscale Systems Description This book provides an in-depth description of transport phenomena relevant to systems of nanoscale dimensions. The different viewpoints and theoretical

More information

GAUSS CIRCLE PROBLEM

GAUSS 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 information

Lognormal Moment Closures for Biochemical Reactions

Lognormal 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 information

Introduction to multiscale modeling and simulation. Almost every system is multiscale.

Introduction 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 information

Scaling Analysis of Energy Storage by Porous Electrodes

Scaling 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 information

On the backbone exponent

On the backbone exponent On the backbone exponent Christophe Garban Université Lyon 1 joint work with Jean-Christophe Mourrat (ENS Lyon) Cargèse, September 2016 C. Garban (univ. Lyon 1) On the backbone exponent 1 / 30 Critical

More information

Parameter estimation in linear Gaussian covariance models

Parameter 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 information

Simulation of Coulomb Collisions in Plasma Accelerators for Space Applications

Simulation 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 information

The correlation between stochastic resonance and the average phase-synchronization time of a bistable system driven by colour-correlated noises

The correlation between stochastic resonance and the average phase-synchronization time of a bistable system driven by colour-correlated noises Chin. Phys. B Vol. 19, No. 1 (010) 01050 The correlation between stochastic resonance and the average phase-synchronization time of a bistable system driven by colour-correlated noises Dong Xiao-Juan(

More information

METEOR PROCESS. Krzysztof Burdzy University of Washington

METEOR 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 information

Density Functional Modeling of Nanocrystalline Materials

Density 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 1-7 February 007 Density

More information

VII. Porous Media Lecture 36: Electrochemical Supercapacitors

VII. 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 information

Travelling waves. Chapter 8. 1 Introduction

Travelling 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 information

Thermal Systems. What and How? Physical Mechanisms and Rate Equations Conservation of Energy Requirement Control Volume Surface Energy Balance

Thermal 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 information

Reachable sets for autonomous systems of differential equations and their topological properties

Reachable sets for autonomous systems of differential equations and their topological properties American Journal of Applied Mathematics 2013; 1(4): 49-54 Published online October 30, 2013 (http://www.sciencepublishinggroup.com/j/ajam) doi: 10.11648/j.ajam.20130104.13 Reachable sets for autonomous

More information

Space Plasma Physics Thomas Wiegelmann, 2012

Space 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 information

Batteries crash simulation with LS-DYNA. Lithium-Ion cell. In collaboration with J. Marcicki et al, Ford Research and Innovation Center, Dearborn, MI

Batteries crash simulation with LS-DYNA. Lithium-Ion cell. In collaboration with J. Marcicki et al, Ford Research and Innovation Center, Dearborn, MI Batteries crash simulation with LS-DYNA Lithium-Ion cell In collaboration with J. Marcicki et al, Ford esearch and Innovation Center, Dearborn, MI Battery crash simulation (general) Car crash Mechanical

More information

From 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 From Bohmian Mechanics to Bohmian Quantum Gravity Antonio Vassallo Instytut Filozofii UW Section de Philosophie UNIL The Measurement Problem in Quantum Mechanics (1) The wave-function of a system is complete,

More information

RedOx Chemistry. with. Dr. Nick

RedOx 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 information

Solutions to Math 53 First Exam April 20, 2010

Solutions 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 information

Ana María Cetto, Luis de la Peña and Andrea Valdés Hernández

Ana 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, 3-6 October 2013 1 Planck's law as a consequence of the zero-point field Equilibrium radiation field

More information

ON LIE GROUP CLASSIFICATION OF A SCALAR STOCHASTIC DIFFERENTIAL EQUATION

ON LIE GROUP CLASSIFICATION OF A SCALAR STOCHASTIC DIFFERENTIAL EQUATION Journal of Nonlinear Mathematical Physics ISSN: 1402-9251 (Print) 1776-0852 (Online) Journal homepage: http://www.tandfonline.com/loi/tnmp20 ON LIE GROUP CLASSIFICATION OF A SCALAR STOCHASTIC DIFFERENTIAL

More information

Kinetic Monte Carlo (KMC)

Kinetic Monte Carlo (KMC) Kinetic Monte Carlo (KMC) Molecular Dynamics (MD): high-frequency motion dictate the time-step (e.g., vibrations). Time step is short: pico-seconds. Direct Monte Carlo (MC): stochastic (non-deterministic)

More information

Math 256: Applied Differential Equations: Final Review

Math 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 information

Asymptotics of rare events in birth death processes bypassing the exact solutions

Asymptotics 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:.88/953-8984/9/6/6545 Asymptotics of rare events in birth death processes bypassing

More information

Inverse-square law between time and amplitude for crossing tipping thresholds

Inverse-square law between time and amplitude for crossing tipping thresholds Inverse-square 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 information

Determination 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 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 information

Introduction to Dynamical Systems Basic Concepts of Dynamics

Introduction 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 information

KINETIC DESCRIPTION OF MAGNETIZED TECHNOLOGICAL PLASMAS

KINETIC 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 information

HIGH 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 HIGH FRICTION LIMIT OF THE KRAMERS EQUATION : THE MULTIPLE TIME SCALE APPROACH Lydéric Bocquet arxiv:cond-mat/9605186v1 30 May 1996 Laboratoire de Physique, Ecole Normale Supérieure de Lyon (URA CNRS 1325),

More information

Available online at ScienceDirect. Physics Procedia 57 (2014 ) 77 81

Available 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 information

Physics 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 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 information

Brownian Motion and The Atomic Theory

Brownian 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 information

Residence-time distributions as a measure for stochastic resonance

Residence-time 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 information

Unique equilibrium states for geodesic flows in nonpositive curvature

Unique 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 information

Convection-driven dynamos in the limit of rapid rotation

Convection-driven dynamos in the limit of rapid rotation Convection-driven 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 information

Course. Print and use this sheet in conjunction with MathinSite s Maclaurin Series applet and worksheet.

Course. 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 information

Chapter 2 Thermodynamics

Chapter 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 information

Chapter 13 Rates of Reactions

Chapter 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