micro-macro transition in the wasserstein metric
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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) - PowerPoint PPT PresentationTRANSCRIPT
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 .. )
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
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
t
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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 ?
@½@t +r ¢(½r F ¤½) = 0
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Non-local Transport Equations Existence the usual way (diffusive limit)
Uniqueness not obvious
Correct long-time behaviour (= same as microscopic particles) ?
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Aggregation
Gaussian aggregation kernel
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Aggregation
Gaussian kernel, rescaled density
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Aggregation
Finite support kernel, rescaled density
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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 = 1N
NX
j =1±X j
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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 repulsionMogilner-EdelsteinKeshet 99, Capasso-Morale-Ölschläger 03, Bertozzi et al 04-07 ..
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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 peaksmb-Capasso-Morale 06 mb-DiFrancesco 07
@½@t +r ¢(½r (F ¤½¡ °½)) = 0
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Local Repulsion
Nonlocal aggregation + nonlinear diffusion
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Stepped Surfaces
From Caflisch et. Al. 1999
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Epitaxial Nanostructures SiGe/Si Quantum Dots (Bauer et. al. 99) Nucleation and Growth driven by elastic misfit
Single Grain Final Morphology
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Calcite Crystallization
Insulin Crystal
Ward, Science, 2005
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Formation of Basalt Columns
•
´Giant‘s Causeway
Panska Skala (Northern Ireland) (Czech Republic) See: http://physics.peter-kohlert.de/grinfeld.html
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Step Interaction Models Simplest dynamics by direct step
interaction
Gradient flow structure for X
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Gradient Flow Structure Energy equals
Metric equals
is projection operator from piecewise linear to piecewise constant
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Gradient Flow Structure Time-discrete formulation
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Continuum Limit Energy
Metric
Gradient Flow
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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)
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Continuum Height Function Function w is inverse of height function u Energy
Continuum equation by change of variables
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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)
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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. 1999-2003)
Simplest BCF-model
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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)
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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 ?
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Continuum Limit Formal 4-th order limit
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Gradient Flow Formulation Reformulate BCF-model as gradient flow
Analogous as above, we only need to change metric
appropriate projection operator
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Gradient Flow Structure Time-discrete formulation
Minimization over manifold
for suitable deformation T
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Continuum Limit Manifold constraint for continuous time
for a velocity V
Modified continuum equations
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Continuum Limit 4th order vs. modified 4th order
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, 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, … )
8.8.2007 Micro-Macro Transition in the Wasserstein Metric WPI, August 08
Papers and talks at
www.math.uni-muenster.de/u/burger
TODAY 3pm talk by Mary Wolfram on numerical
simulation of related problems
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