Exploring Geometry Dependence ofKardar-Parisi-Zhang Interfaces

Kazumasa A. Takeuchi

(Tokyo Tech)

Joint work withYohsuke T. Fukai (Univ. Tokyo & Tokyo Tech)

* This is a simplified ppt intended for public opening

Growing Interfaces & KPZ

(1+1)d Kardar-Parisi-Zhang (KPZ) universality class KPZ equation:

Exact solutions unveiled universal fluctuation properties of h.Deep connection to random matrix theory / integrable systems, etc.

We study it by experiments on liquid-crystal turbulence & simulations.

[Yunker et al. Nature 2011, PRL 2013]

particle deposition (“coffee-ring effect”) proliferating cancer cells paper combustion

[Huergo et al. PRE 2012]

[Maunuksela et al. 1997-]

: rescaled variable: parameters

Liquid-Crystal Turbulence

Growing turbulence of nematic liquid crystal driven by an electric field

Speed x5,

local radiush(x,t)

x

local heighth(x,t)

x

metastable turbulence (defect-less)

will shootUV laser here

We generated both circular and flat interfaces (~1000 times)and measured fluctuations of h.

stableturbulence

(defect-filled)

[KaT et al., Sci. Rep. 1, 34 (2011); J. Stat. Phys. 147, 853 (2012)]

Previous Results

h

h

• Both circular & flat cases show the same KPZ exponent.• Circular (flat) interfaces show GUE (GOE) Tracy-Widom distribution,

in agreement with earlier results for solvable models.• KPZ class splits into different universality subclasses!

[KaT et al., Sci. Rep. 1, 34 (2011); J. Stat. Phys. 147, 853 (2012)]

circular flat

Fluctuation amplitudecircular flat

GUETracy-Widom

GOETracy-Widom

Distribution ofheight fluctuations

circular : flat :

Why Different Distributions?

PNG (= polynuclear growth) modelTime evolution:(1) random local nucleation(2) deterministic lateral expansion (at speed 1)

Circular caseAssume nucleations are restricted to

Equivalent to“point-to-point problemof directed polymer”

GUE

[Prähofer & Spohn, PRL 84, 4882 (2000)]

Flat caseNo constraints on nucleations

line-to-point problem

nucleation

stepsmirror imagetime reversalsymmetry

GOE

KPZ Universality Subclasses

• Initial condition : point or infinitely narrow wedge

• Distribution : GUE Tracy-Widom distribution• Spatial correlation : correlation of Airy2 process (power-law decay)

Circular Subclass

• Initial condition : straight line• Distribution : GOE Tracy-Widom distribution• Spatial correlation : correlation of Airy1 process (super-exponential decay)

Flat Subclass

Note) a few other subclasses exist, especially the stationary subclass.

But what happens for more general initial conditions?[notable progress: variational formula by Quastel & Remenik, arXiv:1606.09228]

Same KPZ exponents, but different distribution & correlation functions.

[See review by I. Corwin, Rand. Mat. Theor. Appl. 1, 1130001]

A Simple Question

What happens as a function of the initial curvature?

circular subclassflat subclass

Naive guess: Circular subclass may arise for any nonzero initial curvature

but no direct evidence was ever shown.

in-growing

In-Growing Case

How to Control the Initial Condition?

Use a hologram generated by a spatial light modulator (SLM)

system

phase modulation by SLM

at focal point (= system)lens = Fourier transformer

Target intensity profilecan be designed

by tuning the phase shift

video speed 5x1mmcontrolled parameter:

initial radius

laser

Arbitrary initial conditioncan be generated!

Experimental Result

Distribution is analyzed by skewness & kurtosis

skewness kurtosis

GOE-TW(flat)

GUE-TW(circ)

GOE-TW(flat)

GUE-TW(circ)

No sign of the circular subclass!Flat subclass (GOE Tracy-Widom) is seen except for the last moments.

larger initialcurvature larger initial

curvature

Numerics: Eden Model

Evolution rule1. Choose a particle randomly2. Add a new neighbor in random direction

unless it overlaps with existing particles.3. Repeat it. [Takeuchi, J. Stat. Mech. 2012]

: initial circumference

As in the experiment, flat subclass is seen

except for the last moments

skewness & kurtosis

larger curv.

larger curv.

Variance and Spatial Correlation

Rescaled variance

Spatial correlation function

Experiment

GOE-TW(flat)

resc

aled

rescaled length

Airy2(circular)

Airy1(flat)Eden

vari

ance

GUE-TW(circular)GUE-TW(circular)

resc

aled

rescaled length

Airy2(circular)

Airy1(flat)

GOE-TW(flat)

GUE-TW(circular)

Flat subclass is checked with individual cumulant & spatial correlation

vari

ance

Toward the Collapse

In-growing interfaces collapse at finite time

time rescaling

mean

variance

Deviation from GOE-TW is controlled by the interface collapse.

Eden

Shade: param. estimation error

Bars:statistical error

experiment

timerescaling

timerescaling

Universal?

Summary

In-growing flat, then collapse. No sign of circular subclass.

Behavior is parameterized by . Universal scaling functions?

Need to reconsider the coffee-ring experiment.[Yunker et al., PRL 110, 035501 (2013)]

KPZ interfaces growing inward from a circle.“in-growing ≠ out-growing” Sign of the curvature matters!

circular subclassflat subclassin-growing

Authorsclaimed GUE-TW.

Fukai & KaT, arXiv:1611.00650Ref: (in-growing)