Multiscale Analysis of Many Particle Systems with Dynamical Control


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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!
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