Micro-Macro Transition in the Wasserstein Metric

1. Micro-Macro Transition in the Wasserstein Metric Martin Burger Institute for Computational and Applied Mathematics European Institute for Molecular Imaging (EIMI) Center for Nonlinear Science (CeNoS) Westfälische Wilhelms-Universität Münsterjoint work with Marco Di Francesco, Daniela Morale, Axel Voigt

2. Introduction • Transition from microscopic stochastic particle models to macroscopic mean field equations is a classical topic in statistical mechanics and applied analysis (McKean-Vlasov limit) • Rigorous results are hard and amazingly few (first results on Vlasov in the 70s, first results on Vlasov-Poisson in the 90s .. ) Micro-Macro Transition in the Wasserstein Metric WPI, August 08

3. Introduction X k ( ) d d d X r F X X W t + - ¾ = k k N j t 6 k j = • Consider for simplicity the friction-dominated case (relevant in biology and many other application fields) • N particles, at locations Xk • FNmodels interaction, Wk are independent Brownian motions Micro-Macro Transition in the Wasserstein Metric WPI, August 08

4. Mean Field Limit @ 1 ¡ ½ ( ) ( ) F N F ( ) p p r r F ¢ = N + ¤ ¢ ½ ½ ¾ ½ = @ t • Classical mean-field limit under the scaling • Formal limit is nonlocal transport(-diffusion) equation for the particle density Micro-Macro Transition in the Wasserstein Metric WPI, August 08

5. Non-local Transport Equations @ ½ ( ) r r F 0 + ¤ ¢ ½ ½ = @ t • Diffusive limit easier due to regularity (+ simple uniqueness proof) • Consider s = 0, nonlocal transport equation • How to prove existence and uniqueness ? Micro-Macro Transition in the Wasserstein Metric WPI, August 08

6. Non-local Transport Equations • Existence the usual way (diffusive limit) • Uniqueness not obvious • Correct long-time behaviour (= same as microscopic particles) ? Micro-Macro Transition in the Wasserstein Metric WPI, August 08

7. Non-local Transport Equations Z Z [ ] ( ) ( ) ( ) d d E F ¡ ¡ ½ x y ½ x ½ y x y = • Solution to this problem via Gradient-Flow formulation in the Wasserstein metric • McCann, Otto, Toscani,Villani, Carrillo, .. • Ambrosio-Gigli-Savare 05 • Energy functional • Uniqueness straight-forward Micro-Macro Transition in the Wasserstein Metric WPI, August 08

8. Non-local Transport Equations • Concentration to Dirac measure at center of mass for concave potential (convex energy) • Carrillo-Toscani • For potentials with global support, local concavity of F at zero suffices for concentration • For potentials with local support, concentration to different Dirac measures (distance larger than interaction range) can happen mb-DiFrancesco 07 Micro-Macro Transition in the Wasserstein Metric WPI, August 08

9. Aggregation • Gaussian aggregation kernel Micro-Macro Transition in the Wasserstein Metric WPI, August 08

10. Aggregation • Gaussian kernel, rescaled density Micro-Macro Transition in the Wasserstein Metric WPI, August 08

11. Aggregation • Finite support kernel, rescaled density Micro-Macro Transition in the Wasserstein Metric WPI, August 08

12. Micro-Macro Transition • Classical techniques for micro-macro transition: - a-priori compactness + weak convergence (weak* convergence in this case) • - Analysis via trajectories, characteristics for smooth potential Braun-Hepp 77, Neunzert 77 • Generalization of trajectory-approach to Wasserstein metricDobrushin 79, reviewed in Golse 02 Micro-Macro Transition in the Wasserstein Metric WPI, August 08

13. Micro-Macro Transition N 1 X N ± ¹ = X j N j 1 = • Key observation: empirical density • is a measure-valued solution of the nonlocal transport equation • Dobrushin proved stability estimate for measure-valued solutions in the Wasserstein metric Micro-Macro Transition in the Wasserstein Metric WPI, August 08

14. Micro-Macro Transition • Implies quantitative estimates for convergence in Wasserstein metric, only in dependence of (distribution of) initial values, Lipschitz-constant L of interaction force, and final time T • Recent results for convex interaction allow to eliminate dependence on L and T, hence the micro-macro transition does not change in the long-time limit Micro-Macro Transition in the Wasserstein Metric WPI, August 08

15. Open Cases • Singular interaction kernels: models for charged particles, chemotaxis (Poisson) • Non-smooth interaction kernels: models for opinion-formation Hegselmann-Krause 03, Bollt-Porfiri-Stilwell 07 • Different scaling ofinteraction with N: aggregation models with local repulsion • Mogilner-EdelsteinKeshet 99, Capasso-Morale-Ölschläger 03, Bertozzi et al 04-07 .. Micro-Macro Transition in the Wasserstein Metric WPI, August 08

16. Local Repulsion 1 1 1 1 ¡ ¡ ¡ ¡ ¢ ¡ ( ) ( ) ( ) l F N F N F i + p m ² p ² 1 ² p = = N A R N N N N 1 ! • Local repulsion modeled by second term with opposite behaviour and different scaling • Aggregation kernel FA (locally concave) and repulsion kernel FR (locally convex) • Repulsive force range larger than individual particle size (moderate limit) Micro-Macro Transition in the Wasserstein Metric WPI, August 08

17. Local Repulsion @ ½ ( ( ) ) r r F 0 + ¡ ¤ ¢ ½ ½ ° ½ = @ t • Repulsion kernel concentrates to a Dirac distribution in the many particle limit • Continuum limit is nonlocal transport equation with nonlinear diffusion • Similar analysis as a gradient flow in the Wasserstein metric. Stationary states not completely concentrated, but local peaksmb-Capasso-Morale 06mb-DiFrancesco 07 Micro-Macro Transition in the Wasserstein Metric WPI, August 08

18. Local Repulsion • Rigorous analysis of the micro-macro transition is still open, except for smooth solutions Capasso-Morale-Ölschläger 03 • Recent stability estimates in the Wasserstein metric should help • Additional problems since empirical density has no meaning in the continuum limit Micro-Macro Transition in the Wasserstein Metric WPI, August 08

19. Local Repulsion • Nonlocal aggregation + nonlinear diffusion Micro-Macro Transition in the Wasserstein Metric WPI, August 08

20. Stepped Surfaces • Stepped surfaces arise in many applications, in particular in surface growth by epitaxy • Growth in several layers, on each layer nucleation and horizontal growth • Computational complexity too large for many layers • Continuum limit described by height function Micro-Macro Transition in the Wasserstein Metric WPI, August 08

21. Stepped Surfaces From Caflisch et. Al. 1999 Micro-Macro Transition in the Wasserstein Metric WPI, August 08

22. Epitaxial Nanostructures • SiGe/Si Quantum Dots (Bauer et. al. 99)Nucleation and Growth driven by elastic misfit Single Grain Final Morphology Micro-Macro Transition in the Wasserstein Metric WPI, August 08

23. Calcite Crystallization • Insulin Crystal Ward, Science, 2005 Micro-Macro Transition in the Wasserstein Metric WPI, August 08

24. Formation of Basalt Columns ´ Giant‘s Causeway Panska Skala (Northern Ireland) (Czech Republic) See: http://physics.peter-kohlert.de/grinfeld.html Micro-Macro Transition in the Wasserstein Metric WPI, August 08

25. Step Interaction Models • To understand continuum limit, start with simple 1D models • Steps are described by their position Xi and their sign si (+1 for up or -1 for down) • Height of a step equals atomic distance a • Step height function Micro-Macro Transition in the Wasserstein Metric WPI, August 08

26. Step Interaction Models • Energy models for step interaction, e.g. nearest neighbour only • Scaling of height to maximal value 1, relative scale b between x and z, monotone steps Micro-Macro Transition in the Wasserstein Metric WPI, August 08

27. Step Interaction Models • Simplest dynamics by direct step interaction • Gradient flow structure for X Micro-Macro Transition in the Wasserstein Metric WPI, August 08

28. Gradient Flow Structure • Gradient flow obtained as limit of time-discrete problems (d N = L2-metric) • Introduce piecewise linear function w N on [0,1] with values Xk at z=k/N Micro-Macro Transition in the Wasserstein Metric WPI, August 08

29. Gradient Flow Structure • Energy equals • Metric equals • P is projection operator from piecewise linear to piecewise constant Micro-Macro Transition in the Wasserstein Metric WPI, August 08

30. Gradient Flow Structure • Time-discrete formulation Micro-Macro Transition in the Wasserstein Metric WPI, August 08

31. Continuum Limit • Energy • Metric • Gradient Flow Micro-Macro Transition in the Wasserstein Metric WPI, August 08

32. Continuum Height Function • Function w is inverse of height function u • Continuum equation by change of variables • Transport equation in the limit, gradient flow in the Wasserstein metric of probability measures (u equals distribution function) Micro-Macro Transition in the Wasserstein Metric WPI, August 08

33. Continuum Height Function • Function w is inverse of height function u • Energy • Continuum equation by change of variables Micro-Macro Transition in the Wasserstein Metric WPI, August 08

34. Continuum Height Function • Transport equation in the limit, gradient flow in the Wasserstein metric of probability measures (u equals distribution function) • Rigorous convergence to continuum: standard numerical analysis problem • Max / Min of the height function do not change (obvious for discrete, maximum principle for continuum). Large flat areas remain flat Micro-Macro Transition in the Wasserstein Metric WPI, August 08

35. Non-monotone Step Trains • Treatment with inverse function not possible • Models can still be formulated as metric gradinent flow on manifolds of measures • Manifold defined by structure of the initial value (number of hills and valleys) Micro-Macro Transition in the Wasserstein Metric WPI, August 08

36. BCF Models • In practice, more interesting class are BCF-type models(Burton-Cabrera-Frank 54) • Micro-scale simulations by level set methods etc (Caflisch et. al. 1999-2003) • Simplest BCF-model Micro-Macro Transition in the Wasserstein Metric WPI, August 08

37. Chemical Potential • Chemical potential is the difference between adatom density and equilibrium density • From equilibrium boundary conditions for adatoms • From adatom diffusion equation (stationary) Micro-Macro Transition in the Wasserstein Metric WPI, August 08

38. Continuum Limit • Two additional spatial derivatives lead to formal 4-th order limit (Pimpinelli-Villain 97, Krug 2004, Krug-Tonchev-Stoyanov-Pimpinelli 2005) • 4-th order equations destroy various properties of the microscale model (flat regions stay never flat, global max / min not conserved ..) • Is this formal limit correct ? Micro-Macro Transition in the Wasserstein Metric WPI, August 08

39. Continuum Limit • Formal 4-th order limit Micro-Macro Transition in the Wasserstein Metric WPI, August 08

40. Gradient Flow Formulation • Reformulate BCF-model as gradient flow • Analogous as above, we only need to change metric • P appropriate projection operator Micro-Macro Transition in the Wasserstein Metric WPI, August 08

41. Gradient Flow Structure • Time-discrete formulation • Minimization over manifold for suitable deformation T Micro-Macro Transition in the Wasserstein Metric WPI, August 08

42. Continuum Limit • Manifold constraint for continuous time for a velocity V • Modified continuum equations Micro-Macro Transition in the Wasserstein Metric WPI, August 08

43. Continuum Limit • 4th order vs. modified 4th order Micro-Macro Transition in the Wasserstein Metric WPI, August 08

44. Generalizations • Various generalizations are immediate by simple change of the metric: deposition, adsorption, time-dependent diffusion • Not yet: limit with Ehrlich-Schwoebel barrier • Not yet: nucleation Micro-Macro Transition in the Wasserstein Metric WPI, August 08

45. Generalizations Can this approach change also the understanding of fourth- or higher-order equations when derived from microscopic particle models ?(Cahn-Hilliard, thin-film, … ) Micro-Macro Transition in the Wasserstein Metric WPI, August 08

46. Download and Contact Papers and talks at www.math.uni-muenster.de/u/burger Email martin.burger@uni-muenster.de TODAY 3pm talk by Mary Wolfram on numerical simulation of related problems Micro-Macro Transition in the Wasserstein Metric WPI, August 08