Micro-Macro Transition in the Wasserstein Metric

Martin Burger

Institute for Computational and Applied MathematicsEuropean Institute for Molecular Imaging (EIMI)

Center for Nonlinear Science (CeNoS)

Westfälische Wilhelms-Universität Münster

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

8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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 .. )

Introduction Consider for simplicity the friction-dominated case (relevant in biology and many other application fields)

N particles, at locations Xk

FN models interaction, Wk are independent Brownian motions

dX k =X

j 6=kr FN (X k - X j )dt +¾dWk

Mean Field Limit Classical mean-field limit under the scaling

Formal limit is nonlocal transport(-diffusion) equation for the particle density

FN (p) = N ¡ 1F (p)

@½@t +r ¢(½r F ¤½) =¾¢½

Non-local Transport Equations Diffusive limit easier due to regularity (+ simple uniqueness proof)

Consider = 0, nonlocal transport equation

How to prove existence and uniqueness ?

@½@t +r ¢(½r F ¤½) = 0

Non-local Transport Equations Existence the usual way (diffusive limit)

Uniqueness not obvious

Correct long-time behaviour (= same as microscopic particles) ?

Non-local Transport Equations Solution to this problem via Gradient-Flow formulation in the Wasserstein metricMcCann, Otto, Toscani,Villani, Carrillo, ..Ambrosio-Gigli-Savare 05

Energy functional

Uniqueness straight-forward

E [½]= ¡Z Z

F (x ¡ y)½(x)½(y) dx dy

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

Aggregation

Gaussian aggregation kernel

Aggregation

Gaussian kernel, rescaled density

Aggregation

Finite support kernel, rescaled density

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 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 = 1N

j =1±X j

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

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 repulsionMogilner-EdelsteinKeshet 99, Capasso-Morale-Ölschläger 03, Bertozzi et al 04-07 ..

Local Repulsion 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)

FN (p) = N ¡ 1¡FA (p) + ²¡ 1N FR (²¡ 1N p)

limN ! 1

N²¡ 1N = 1

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 peaksmb-Capasso-Morale 06 mb-DiFrancesco 07

@½@t +r ¢(½r (F ¤½¡ °½)) = 0

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

Local Repulsion

Nonlocal aggregation + nonlinear diffusion

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

Stepped Surfaces

From Caflisch et. Al. 1999

Epitaxial Nanostructures SiGe/Si Quantum Dots (Bauer et. al. 99) Nucleation and Growth driven by elastic misfit

Single Grain Final Morphology

Calcite Crystallization

Insulin Crystal

Ward, Science, 2005

Formation of Basalt Columns

´Giant‘s Causeway

Panska Skala (Northern Ireland) (Czech Republic) See: http://physics.peter-kohlert.de/grinfeld.html

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

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

Step Interaction Models Simplest dynamics by direct step

interaction

Gradient flow structure for X

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

Gradient Flow Structure Energy equals

Metric equals

is projection operator from piecewise linear to piecewise constant

Gradient Flow Structure Time-discrete formulation

Continuum Limit Energy

Metric

Gradient Flow

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)

Continuum Height Function Function w is inverse of height function u Energy

Continuum equation by change of variables

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

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)

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

Chemical Potential Chemical potential is the difference

between adatom density and equilibrium density

From equilibrium boundary conditions for adatoms

From adatom diffusion equation (stationary)

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 ?

Continuum Limit Formal 4-th order limit

Gradient Flow Formulation Reformulate BCF-model as gradient flow

Analogous as above, we only need to change metric

appropriate projection operator

Gradient Flow Structure Time-discrete formulation

Minimization over manifold

for suitable deformation T

Continuum Limit Manifold constraint for continuous time

for a velocity V

Modified continuum equations

Continuum Limit 4th order vs. modified 4th order

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

Generalizations

Can this approach change also the understanding of fourth- or higher-order equations when derived from microscopic particle models ?(Cahn-Hilliard, thin-film, … )

Papers and talks at

www.math.uni-muenster.de/u/burger

martin.burger@uni-muenster.de

TODAY 3pm talk by Mary Wolfram on numerical

simulation of related problems

Download and Contact

Micro-Macro Transition in the Wasserstein Metric

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