Multiscale Analysis of Many Particle Systems with Dynamical Control

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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

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

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

Hysteresis and phase transitions device, eperiments, pictures: Clemens Guhlke, WIAS 4

Hysteresis and phase transitions device, eperiments, pictures: Clemens Guhlke, WIAS phase 1 only phase miture phase 2 only 4

Hysteresis and phase transitions device, eperiments, pictures: Clemens Guhlke, WIAS phase 1 only phase miture phase 2 only 4.2 4.0 Hysteresis diagramm voltage vs. capacity pressure vs. volume Voltage versus Li/Li + (V) 3.8 3.6 3.4 3.2 3.0 Voltage hysteresis 28% Q theor 2.8 2.6 0 20 40 60 80 100 120 140 160 Capacity (ma h g 1 ) Multiscale analysis of many-particle batteries c ACMAC Heraklion, June 27, 2013 d 4

The model

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

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

Mesoscopic evolution nonlocal Fokker-Planck equation = X + @ t % = @ 2 @ % + H 0 () (t) % = X 0 Z = X dynamical constraint R %(, t)d = `(t) stable interval unstable interval stable interval 7

Mesoscopic evolution nonlocal Fokker-Planck equation = X + @ t % = @ 2 @ % + 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

Mesoscopic evolution nonlocal Fokker-Planck equation = X + @ t % = @ 2 @ % + 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

Three time scales standard relaation to meta-stable state Fokker-Planck = const convergence to equilibrium large deviations, Kramers formula ep H 2 8

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

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

Numerical Simulations

Simplifying assumptions `(t) =1, `(0) 1 10

Simplifying assumptions `(t) =1, `(0) 1 initial and final dynamics = localized peak at `(t) How is the mass transported between the stable intervals? 10

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 1 0 10

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 = 0.001 n = 0.05 y fast reactions G t = 0.001 n = 0.2 y H t = 0.00001 n = 0.2 y I t = 0.0001 n = 0.4 y Type III Type IV 11

Simulations - mesoscopic view A { = -0.8 A { = -0.0 A { = 0.8 A t = 1. n = 0.05 y 12

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

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 = 0.001 n = 0.2 { = -1.6 G { = -0.0 G { = 1.6 y Multiscale analysis of many-particle batteries ACMAC Heraklion, June 27, 2013 12

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

Fast reactions: type III =ep b 2

Numerical results Idea 1 always (one or) two stable peaks m ± : masses X ± : positions m + m + =1 = H 0 (X )=H 0 (X + ) 15

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

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

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

Heuristics particles diffuse in effective potential H () =H() H () =H() h h + Effective potential depends implicitly on time! + 0 16

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

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

Rate-independent limit dynamics µ = m + m + b = # {µ = 1} b = # {µ =+1} ` X ( # ) X + ( # ) 18

Rate-independent limit dynamics µ = m + m + b = # {µ = 1} b = # {µ =+1} ` X ( # ) X + ( # ) effective dynamics (t) 2 @ µ R µ(t) + @ µ I µ(t), C µ(t), (t), `(t) =0 Rigorous justification is possible. 18

Slow Reactions: types I and II a =ep

Overview on type II y y y transport switching transport transport splitting merging y y transport switching transport 20

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

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 1 1 + m 2 2 = ` 22

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 1 1 + m 2 2 = ` Multiple solution branches! 22

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 1 1 + m 2 2 = (t 2 ) 1 = 2 m 1 1 + m 2 2 = ` H 0 ( 1 )=H 0 ( 2 ) C Multiple solution branches! m 1 1 + m 2 2 = `(t 0 ) 0 1 22

Peak-widening model y % = m 1 ˆ% + m 2 2 ` = m 1 RR ˆ% d + m 2 2 23

Peak-widening model y FP-PDE for first peak + ODE for second peak @ t ˆ% = @ 2 @ ˆ% + H 0 () ˆ% % = m 1 ˆ% + m 2 2 ` = m 1 RR ˆ% d + m 2 2 2 = 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

Peak-widening model y FP-PDE for first peak + ODE for second peak @ t ˆ% = @ 2 @ ˆ% + H 0 () ˆ% % = m 1 ˆ% + m 2 2 ` = m 1 RR ˆ% d + m 2 2 2 = 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

Peak-widening model y FP-PDE for first peak + ODE for second peak @ t ˆ% = @ 2 @ ˆ% + H 0 () ˆ% % = m 1 ˆ% + m 2 2 ` = m 1 RR ˆ% d + m 2 2 2 = 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

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

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

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

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

Mass splitting problem Simplified =0, t = t sp + s @ 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

Mass splitting problem Simplified =0, t = t sp + s @ 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

Mass splitting problem Simplified =0, t = t sp + s @ 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

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

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

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

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 1212.3128 (2012)

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