Stochastic Loewner chains for DLA-like growth
Transcript of 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
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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Viscous fingering: flow in a Hele-Shaw cell
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Viscous fingering: flow in a Hele-Shaw cell
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Viscous fingering: flow in a Hele-Shaw cell
http://n-e-r-v-o-u-s.com/projects/albums/laplacian-growth-2d/
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Dielectric breakdown and electrodeposition
Bert Hickman - http://www.capturedlightning.com
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Diffusion-limited aggregation
T. Witten, L. Sander, 1981
• Show DLA applet from http://apricot.polyu.edu.hk/~lam/dla/dla.html
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Diffusion-limited aggregation
T. Witten, L. Sander, 1981
• Complicated fractals with multifractal “charge” distribution
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Deterministic Laplacian growth
• Basic equations
– Darcy law
– Incompressibility of oil
– Zero viscosity of water
– Continuity
– Sink at infinity
Vn
D(t)
G(t)
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Continuous Loewner chains
• Growing domain is described by a conformal map
growth density
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
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:
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
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)
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Discrete Loewner chains
M. Stepanov and
L. Levitov, 2001
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Discrete Loewner chains
M. Stepanov and L. Levitov, 2001
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Discrete Loewner chains: noise reduction
M. Stepanov and
L. Levitov, 2001
• “Flat” particles
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
Discrete Loewner chains: noise reduction
M. Stepanov and L. Levitov, 2001
• Thick and smooth branches resembling viscous fingers
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
• 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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
• 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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
• 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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
• 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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
• 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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
• 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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
• 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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
• 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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
• 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
Non-equilibrium dynamics of stochastic and quantum integrable systems, KITP, February 19, 2016
• 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