Stochastic Loewner chains for DLA-like growth

Stochastic Loewner chains for DLA-like growth

Ilya A. Gruzberg

Ohio State University

Based on unfinished work with

M. Mineev-Weinstein (IIP, Natal, Brazil)

D. Leshchiner (Yandex, Moscow, Russia)

Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016


Laplacian growth

• Pattern formation and growth controlled by a Laplacian field

• Examples:

• Viscous fingering: pressure

• Electrodeposition and dielectric breakdown model:

electric potential

• Crystal growth: diffusive field and/or temperature

• Diffusion limited aggregation (DLA): probability density

of aggregating particles

• Lots of applications from oil extraction to art and jewelry


Viscous fingering: flow in a Hele-Shaw cell





http://n-e-r-v-o-u-s.com/projects/albums/laplacian-growth-2d/


Dielectric breakdown and electrodeposition

Bert Hickman - http://www.capturedlightning.com

http://www.capturedlightning.com/


Diffusion-limited aggregation

T. Witten, L. Sander, 1981

• Show DLA applet from http://apricot.polyu.edu.hk/~lam/dla/dla.html


Diffusion-limited aggregation

T. Witten, L. Sander, 1981

• Complicated fractals with multifractal “charge” distribution


Viscous fingering vs. DLA (experiment vs. numerics)

O. Praud and H. L. Swinney, 2005

• Are these patterns “the same”?

• The above authors answered “yes” based on numerically obtained

multifractal spectrum of harmonic measure


Viscous fingering vs. DLA

• Are these patterns

“the same”?

• To formulate the question precisely, we need:

• Well-defined models for both types of processes,

for example, continuous Loewner chains

• Embed the models into a family that interpolates

between LG and DLA

• Study fractal properties: multifractal spectrum of

the harmonic measure


Harmonic measure on a curve

• Probability that a Brownian particle

hits a portion of the curve

• Electrostatic analogy: charge on the

portion of the curve (total charge one)

• Related to local behavior of electric field:

potential near wedge of angle


Harmonic measure on a curve

• Electric field of a charged cluster

Multifractal spectrum

• Lumpy charge distribution on a cluster boundary

• Non-linear is the hallmark of a multifractal

• Multifractal spectrum of harmonic measure is known exactly for

conformally-invariant critical curves (SLE)

• Only numerically known for DLA

• Cover the curve by small discs

of radius

• Charges (probabilities) inside discs

• Moments



Deterministic Laplacian growth

• Basic equations

– Darcy law

– Incompressibility of oil

– Zero viscosity of water

– Continuity

– Sink at infinity

Vn

D(t)

G(t)


Continuous Loewner chains

• Growing domain is described by a conformal map

growth density


Continuous Loewner chains

• Equivalent description: the motion of the boundary

• Point on the boundary

• Unit normal

• Normal velocity

L. A. Galin, P. Ya. Polubarinova-Kochina, 1945


Examples

• Radial (multiple) SLE:

• Hele-Shaw flow without surface tension:

• Integrable model with finite time singularities

• Hele-Shaw flow with a finite surface tension

• Dielectric breakdown:


Integrability of Laplacian growth

• LG conserves exterior harmonic moments of the interface

• Many different families of

explicit solutions

• Some of these become singular

in finite time: cusp formation

• Regularization: surface tension

or finite size of particles (“quantizaton”)

S. Richardson, S. Howison

B. Shraiman, D. Bensimon

S. Dawson, M. Mineev-Weinstein

Ar. Abanov, A. Zabrodin

B. Shraiman, D. Bensimon, Phys. Rev. A 30 (1984)


Integrability of Laplacian growth

I. Krichever, A. Marshakov, M. Mineev-Weinstein, P. Wiegmann, A. Zabrodin

• Dispersionless limit of 2D Toda hierarchy

• Harmonic moments are the times of the commuting flows

• Relation to random matrices and quantum Hall effect


Discrete Loewner chains

• Iterated conformal maps M. Hastings and L. Levitov, 1998

• adjusted to produce bumps of (roughly) equal area

• are random from uniform distribution



M. Stepanov and

L. Levitov, 2001



M. Stepanov and L. Levitov, 2001


Discrete Loewner chains: noise reduction

M. Stepanov and

L. Levitov, 2001

• “Flat” particles


Discrete Loewner chains: noise reduction

M. Stepanov and L. Levitov, 2001

• Thick and smooth branches resembling viscous fingers


Viscous fingering vs. DLA: recap

• Are these patterns

“the same”?

• To formulate the question precisely, we need:

• Well-defined models for both types of processes,

for example, continuous Loewner chains

• Embed the models into a family that interpolates

between LG and DLA

• Study stochastic fractal properties: spectrum of

harmonic measure

Interpolating models


• Divide the boundary of the cluster into segments

• Drop many building blocks with rate (area per unit time)

• Drop blocks per time interval

• Each block has area , so that

• Growth step is specified by , the number of blocks sticking

to segment on the boundary:

• Division of the boundary, probabilities of configurations ,

and the shape of the blocks can be treated differently

Model I: definition


• Divide the boundary of the cluster into segments

that are images of uniform segments on the unit circle

• The lengths of segments in the plane are controlled by

harmonic measure, and the heights of blocks – by the area

• Probability of configuration is given by multinomial

or Poisson

Model I: thermodynamic limit


• In thermodynamic limit

trade

and replace sums by integrals

• Then can relate , the map , and the Loewner density

• Growth process is described by the stochastic Loewrner chain

Model I: action and integral over scenarios


• In the same thermodynamic limit get

and an “action” for the stochastic process

• A particular realization of has the weight

• Averages of obesrvables (e.g. integral means) are integrals over

“scenatios”

Model I: some results


• The action has the unique saddle point

corresponding to LG

• The saddle point is infinitely deep in the limit , so that

and Model I becomes LG

• For finite the model naturally interpolates between LG

and stochastic DLA-like growth

• Exterior harmonic moments (integrals of motions for LG)

are conserved in the mean in general

• Can expand near the saddle point and treat noise in

perturbation theory

Model I: a shortcoming


• Blocks are not uniform in shape

• Can be wide and short or narrow and tall

• The narrow blocks may dominate late stages of growth

• Model I may be in the same class as the non-random Laplacian

needle growth model considered by Makarov and Carleson

• This motivates Model II

Model II: definition


• Divide the boundary of the cluster into uniform segments

of length . Their number grows in time

• Drop “square” blocks

• Now the probabilities of attachment are controlled by

harmonic measure

• Probability of configuration is given by multinomial

Model II: thermodynamic limit


• In thermodynamic limit

get the same stochastic Loewner chain

• The action is much more complicated

• Simplifies in the further limit

• Model II interpolates between LG and DLA

Integrability and noise


• Noise as a regularizer for LG singularities

• More generally: stochastic perturbations of integrable systems

• Harmonic oscillator

• Explicit solution of a Cauchy problem

– Generally is not available for integrable equations

Open issues


• Stochastic perturbations of integrable systems

• Effective solutions of Cauchy problems

• For LG there is an implicit solution in terms of the Schwarz

function

• More explicit for some finite-dimensional reductions

• Perhaps, can use these as approximations

• Relate (fluctuating) harmonic moments

and the multifractal spectrum

• Compute the spectrum

3D DLA


http://math.mit.edu/~chr/research/3d-dla.htm

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Stochastic Loewner chains for DLA-like growth