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 Berlin, 21 Februar 2012
Contents Man-particle storage sstems Modelling (thermodnamics group at WIAS) Nonlocal Fokker-Planck equations with two small parameters Asmptotic Analsis (this talk) Effective ODEs for small parameter limits Fast reaction regime via Kramers formula for large deviations Slow reaction regime via regular transport and singular events From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 2
Man-particle storage sstems 1. Interconnected rubber balloons pictures taken b Clemens Guhlke (WIAS) 2. Lithium-ion batteries Ke features cathode e FePO 4 FePO 4 machine Li + e anode fast relaation to local equilibrium free energ of single-particle sstem is double-well potential moment of the man-particle sstem is controlled (dnamical constraint) FePO 4 electrolte From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 3
First guess for model H() simple gradient flow dnamical constraint nonlocal multiplier ẋ i (t) = (t) H 0 i (t) N N 1 X i=1 (t) =N 1 i (t) =`(t) X N i=1 H 0 i (t) + `(t) Problem Macroscopic evolution is ill-posed! Remed Take into account entropic effects! Quenched Disorder Mielke & Truskinovsk (ARMA 2012) Boltzmann Entrop non-local Fokker-Planck eqautions From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 4
Nonlocal Fokker-Planck equations more details in Dreer, Guhlke, Herrmann (CMAT 2011) relaation time entrop dnamical multiplier Z @ t % = @ 2 @ % + H 0 () (t) % R %(, t)d = `(t) dnamical constraint Z = X + (t) = R H 0 ()%(, t)d + `(t) = X = X 0 stable interval unstable interval stable interval From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 5
3 Time scales @ t % = @ 2 @ % + H 0 () (t) % Goal Understand small parameter dnamics!,! 0 Different times scales: relaation time of single particle sstem (relaation to metastable state) 4H chemical reactions (Kramers formula) (relaation to equilibrium) ep 2 dnamical constraint ` = O(1) From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 6
Overview on different scaling regimes
Simple initial value problems simplifing assumptions ` >0 `(0) <, %(, 0) `(0) () macroscopic output t 7! `(t), (t), stress-strain relation Z t 7! `(t),µ(t) phase-fraction - strain relation mean force (t) = H 0 ()%(, t)d RZ 0 = (t) ` Z +1 phase fraction µ(t) = 1 %(, t)d + 0 %(, t)d From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 8
Numerical simulations - macroscopic view slow A t = 1. n = 0.05 B t = 0.5 n = 0.05 C t = 0.25 n = 0.05 reactions Tpe I Tpe II D t = 0.1 n = 0.05 E t = 0.05 n = 0.05 F t = 0.001 n = 0.05 fast reactions G t = 0.001 n = 0.2 H t = 0.00001 n = 0.2 I t = 0.0001 n = 0.4 Tpe III Tpe IV From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 9
Numerical simulations - microscopic view A A t = 1. n = 0.05 { = -0.8 A { = -0.0 A { = 0.8 C C t = 0.25 n = 0.05 { = -0.8 C { = 0.9 C { = 1.7 G G t = 0.001 n = 0.2 { = -1.6 G { = -0.0 G { = 1.6 From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 10
Scaling regimes for parameters,! 0 single-peak evolution = a log 1/ 0 <a<a crit piecewise continuous two-peaks evolution = p 0 <p<2/3 open problem = p 2/3 <p<1 limit of Kramers formula b =ep 2, 0 <b<b crit Kramers formula 2 log 1/!1 quasi-stationar limit From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 11
Kramers formula and Tpe-II transitions =ep b 2
Kramers formula H () =H() particles can cross the energ barrier h h + due to stochastic fluctuations (large deviations, tunneling) 0 + Kramers formula provides mass flu between wells time scale = ep +4H 2 = ep min{h,h+ } 2 Observation =ep( b/ 2 ) For there eists, such that b (1) mass flu is of order 1 provided that (t) = b + 2 (t) (2) small fluctuations of are sufficient to satisf the constraint From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 13
Inner and outer epansions Idea (1) for each we have three positions with, 0, + H 0 ( /0/+ )= m ± (t) (2) two narrow peaks with masses at ± (t) m (t) m + (t) inner epansion outer epansion mass flu due to Kramers 2 @ % + H 0 0 for ± (t) ()% = (t), (outer epansion) R(t) for 0 (t). (inner epansion) From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 14
Matching of inner and outer epansions Outer epansion 8 >< H (t) () %(, t) >: µ (t)ep 2 H (t) () for < 0 (t), µ + (t)ep 2 for > 0 (t). Z ±1 H (t) ( ± (t)) m ± (t) =± 0 (t) %(, t)d c ± (t)µ ± (t) ep 2 Inner epansion % 0 (t) ±, t ep H (t) 0 (t) ± 2 C(t) R(t) 2 ep H (t) 0 (t) 2! Matching conditions result from equating the time-dependent pre-factors! From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 15
Kramers formula for mass flu R(t) = m (t)r (t) m + (t)r + (t) r ± (t) =c ± (t)ep b h ± (t) =H (t) ( 0 (t)) h± (t) 2 H (t) ( ± (t)) Observation For each there eists such that (t) < b =) r (t) 1 (t) = b + 2 (t) =) r (t) ep( (t)) (t) > b =) r (t) 1 Strateg Adjust according to dnamical constraint From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 16
Main result for fast reactions F t = 0.001 n = 0.05 Main result. Suppose that the dnamical constraint and the initial data satisfies (4) and (5), and that and are coupled b b =ep 2 (11) for some constant b (0, h crit ). Then there eists a constant b (0, ) such that 1. the dnamical multiplier satisfies G t = 0.001 n = 0.2 (t) 0 H (t) for t<t 1, b for t 2 <t<t 2, H (t) for t>t 2 where t 1 and t 2 are uniquel determined b (t 1 )=X ( b ) and (t 2 )=X + ( b ), 2. the state of the sstem satisfies (, t) 0 m (t) X ( (t)) ()+m + (t) X+ ( (t))(). H t = 0.00001 n = 0.2 where m + (t) =1 m (t) and 1 for t<t 1, X + ( b ) (t) m (t) = X + ( b ) X ( b ) for t 1 <t<t 2, 0 for t>t 2. Moreover, the assertions remain true 1. with b =0if ep hcrit 2, 2. with b = if 2 3 but > ep b2 for all b>0. From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 17
Slow reaction limit: Tpe-I/II transitions =ep a
Overview - states for increasing constraints Single-peak configurations stable unstable stable Two-peaks configurations transport due to constraint switching events splitting events stable-stable unstable-stable merging events From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 19
Overview - Tpe-I phase transitions transport switching transport merging transport From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 20
Overview - Tpe-II phase transitions transport switching transport transport splitting merging transport switching transport From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 21
Overview - Simplified models transport localised peaks move due to the constraint switching two-peaks ODE stable peaks enter unstable interval merging unstable peaks merge rapidl with stables ones splitting unstable peaks split rapidl into two stables ones peak-widening model mass-splitting problem From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 22
two-peaks approimation transport, switching, and merging of peaks
Two-peaks approimation to FP Dnamical model ẋ 1 = H 0 ( 1 ) ẋ 2 = H 0 ( 2 ) m 1 + m 2 =1 ṁ i =0 = m 1 H 0 ( 1 )+m 2 H 0 ( 2 )+ ` Quasi-stationar limit! 0 H 0 ( 1 )=H 0 ( 2 ) m 1 1 + m 2 2 = ` Multiple solution branches! Which ones are selected b dnamics? From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 24
Two-peaks approimation to FP + + 2 (t) switching A (t) C 2 A B m 1 1 + m 2 2 = (t 2 ) 1 = 2 t t 0 t 1 B t 2 1 (t) merging H 0 ( 1 )=H 0 ( 2 ) m 1 1 + m 2 2 = `(t 0 ) 0 C 1 From particle sstems to differential equations WIAS Berlin, 21 Februar 2012
entropic effects widening and splitting of unstable peaks peak-widening model: width of unstable peaks blows up (almost) instantaneousl, determines splitting time mass splitting problem: sstem forms (almost) instantaneousl two stable peaks determines jump of the sstem
Peak-widening model % = m 1 ˆ% + m 2 2 @ t ˆ% = @ 2 @ ˆ% + H 0 () ˆ% 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 ` = m 1 RR ˆ% d + m 2 2 position of peak ẋ 1 = H 0 ( 1 ) width of peak w(t) = (t)w (t) ˆ%(, t) =: 1 (t) R ( 1 t) (t) 2, (t)! From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 27
Formula for width of unstable peaks define scaling factors epand nonlinearit (fine if width is small) w(t) 1 =) @ R = @ 2 R as long as width is small 1 =) R(, ) 1 p 4 ep 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 b quasi-stationar two-peaks approimation From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 28
Mass splitting problem ansatz =0, t = t sp + s @ s ˆ% = @ H 0 () (s) ˆ% 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 asmptotic initial data ˆ%(, s) s! 1! 1 2 p ep 1 (t sw ) 4ep(2 s), = H 00 1 (t sw ) > 0 From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 29
Mass splitting problem Conjecture s =+1 s = 1 Data at depend continuousl on data at. Mass Splitting Function (m 1,m 2 ) 7! (µm 1,m 2 +(1 µ)m 1 ) µ = M(`, m 1 ) data just before splitting data just after splitting From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 30
Main result for slow reactions initial data A t = 1. n = 0.05 < stable transport (in ) switching: t = t switching stable-stable transport switching: t = t switching C t = 0.25 n = 0.05 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 trivial merging: t = t merging >+ stable transport (in ) merging: t = t merging final data From particle sstems to differential equations WIAS Berlin, 21 Februar 2012 31
Summar Fast reaction regime Kramers formula describes Tpe-II transitions Tpe-I transitions as limiting case Slow reaction regime Tpe-I and Tpe-II transitions can be described b - intervals of quasi-stationar transport - singular times corresponding to switching, splitting, merging Splitting events require to solve Mass Splitting Problem Open problems Find rigorous proofs! Fill the gap in the scaling regimes! From particle sstems to differential equations WIAS Berlin, 21 Februar 2012
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