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 Münster joint work with Marco Di Francesco, Daniela Morale, Axel Voigt

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 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 MetricWPI, August 08 Introduction Consider for simplicity the friction-dominated case (relevant in biology and many other application fields) N particles, at locations X k F N models interaction, W k are independent Brownian motions d X k = X j 6 = k rF N ( X k - X j ) d t + ¾ d W k t

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

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Non-local Transport Equations Diffusive limit easier due to regularity (+ simple uniqueness proof) Consider  = 0, nonlocal transport equation How to prove existence and uniqueness t + r ¢ ( ½ rF ¤ ½ ) = 0

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 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 MetricWPI, August 08 Non-local Transport Equations 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 E [ ½ ] = ¡ ZZ F ( x ¡ y ) ½ ( x ) ½ ( y ) d x d y

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 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 MetricWPI, August 08 Aggregation Gaussian aggregation kernel

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Aggregation Gaussian kernel, rescaled density

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Aggregation Finite support kernel, rescaled density

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 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 metric Dobrushin 79, reviewed in Golse 02

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Micro-Macro Transition 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 ¹ N = 1 N N X j = 1 ± X j

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 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 MetricWPI, August 08 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 of interaction with N: aggregation models with local repulsion Mogilner-EdelsteinKeshet 99, Capasso-Morale- Ölschläger 03, Bertozzi et al

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

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Local Repulsion 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 peaks mb-Capasso-Morale 06mb-DiFrancesco t + r ¢ ( ½ r ( F ¤ ½ ¡ °½ )) = 0

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 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 MetricWPI, August 08 Local Repulsion Nonlocal aggregation + nonlinear diffusion

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

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 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 MetricWPI, August 08 Calcite Crystallization Insulin Crystal Ward, Science, 2005

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Formation of Basalt Columns ´ Giant‘s Causeway Panska Skala (Northern Ireland) (Czech Republic) See:

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

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

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

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Gradient Flow Structure Gradient flow obtained as limit of time- discrete problems (d N = L 2 -metric) Introduce piecewise linear function w N on [0,1] with values X k at z=k/N

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Gradient Flow Structure Energy equals Metric equals  is projection operator from piecewise linear to piecewise constant

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Gradient Flow Structure Time-discrete formulation

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Continuum Limit Energy Metric Gradient Flow

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

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 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 MetricWPI, August 08 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 MetricWPI, August 08 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 ) Simplest BCF-model

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 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 MetricWPI, August 08 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 MetricWPI, August 08 Continuum Limit Formal 4-th order limit

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Gradient Flow Formulation Reformulate BCF-model as gradient flow Analogous as above, we only need to change metric  appropriate projection operator

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Gradient Flow Structure Time-discrete formulation Minimization over manifold for suitable deformation T

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Continuum Limit Manifold constraint for continuous time for a velocity V Modified continuum equations

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 Continuum Limit 4th order vs. modified 4th order

Micro-Macro Transition in the Wasserstein MetricWPI, August 08 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 MetricWPI, August 08 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 MetricWPI, August 08 Papers and talks at TODAY 3pm talk by Mary Wolfram on numerical simulation of related problems Download and Contact