Sk alainvari ans dinamika t agul o t erbenglu.elte.hu/~statfiz/eloadasok/2013-04-17-Somfai.pdfcan be...

Skalainvarians dinamikatagulo terben

Somfai EllakUniversity of Warwick, Wigner FK SZFI

Adnan AliStefan GrosskinskyRobin Ball

University of Warwick

Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.

Introduction

Scale invariant trajectories – random walk, (fractional) Brownianmotion

dX ∼ dtγ (eg. γ = 1/2)evetually slower than ballistic

in 1D: all trajectories meet (with probability 1)

Our question:

what if space expands faster than 〈|dX |〉 ?

Introduction – Expanding space

Cosmology

physicsforme.wordpress.com

Thin sheets

Klein, Efrati, Sharon, Science (2007)

Growing substrate

stanford.edu/group/brainsinsilicon

Indirect: domain walls

Hallatschek, Hersen, Ramanathan, Nelson; PNAS (2007)

Cosmology

Thin sheets

Growing substrate

Cosmology

Thin sheets

Growing substrate

Cosmology

Thin sheets

Growing substrate

Introduction – Genetic drift and range expansion

Domain boundaries

domain boundaries grow perpendicular to surfaceXh is superdiffusive due to surface roughness

M(h) := 〈X 2h 〉 ≈ σ2h2γ , γ = 2/3

Domain boundaries

M(h) := 〈X 2h 〉 ≈ σ2h2γ , γ = 2/3

Range expansion

fixed, finite size geometry: fixation (absorbing state in finite time)

range expansion: promotes diversity and segregation

Mapping

understand radial growthwith help of same dynamics in fixed size rectangular space

polar-like coordinate transformation:linear: 0 ≤ Xh < L, 0 ≤ h <∞radial: 0 ≤ Yr < 2πr , r0 ≤ r <∞, 2πr0 = L

Mapping

increment:

dYr =dr

r0Xh +

r0dXh = Yr

r+ dYr ,

r0dXh = dYr

preserving local structure:

dXh ∼ (dh)γ , dYr ∼ (dr)γγ = 1/2 diffusive fluctuationsγ = 2/3 KPZ domain boundaries

=( r0r

Mapping

increment:

dYr =dr

r0Xh +

r0dXh = Yr

r+ dYr ,

r0dXh = dYr

=( r0r

Mapping

increment:

dYr =dr

r0Xh +

r0dXh = Yr

r+ dYr ,

r0dXh = dYr

=( r0r

Mapping

increment:

dYr =dr

r0Xh +

r0dXh = Yr

r+ dYr ,

r0dXh = dYr

=( r0r

Mapping

integrating differential equation:

h(r) =

γ1−γ

(1− (r0/r)

1−γγ

), γ 6= 1

r0 ln(r/r0) , γ = 1

local interaction:{ r0rYr

}dist.={Xh(r)

}for all r ≥ r0

properties:h(r) ≈ r − r0 for r close to r0hγ(∞) = limr→∞ h(r) = γ

1−γ r0 < ∞

1 2 3 4 5 60

r′ = r/r0

h′ =

γ=2/3

γ=1/3

γ=1/2

(∞)/r0

Mapping

integrating differential equation:

h(r) =

γ1−γ

(1− (r0/r)

1−γγ

), γ 6= 1

r0 ln(r/r0) , γ = 1

local interaction:{ r0rYr

}dist.={Xh(r)

}for all r ≥ r0

properties:h(r) ≈ r − r0 for r close to r0hγ(∞) = limr→∞ h(r) = γ

1−γ r0 < ∞

1 2 3 4 5 60

r′ = r/r0

h′ =

γ=2/3

γ=1/3

γ=1/2

(∞)/r0

Results

number of domains:

h, r−r0

⟨ N ⟩

× ⟨ NF(h) ⟩

° ⟨ NR(r) ⟩

h, h(r)

⟨ N ⟩

° ⟨ NR(r) ⟩

× ⟨ NF(h) ⟩

Results

Different processes:

Levy flight:jump size distribution: P(Xh+ − Xh = x) ∼ C |x |−(1+α) (α > 0)Markovianγ = max{1/α, 1/2}

fractional Brownian motion:covariances: 〈Xh+∆hXh〉 ∼ (h + ∆h)2γ + h2γ − (∆h)2γ

non-Markovian

Results

Levy flight (γ = max{1/α, 1/2}) + absorption:number of surviving trajectories N,

mean square distance: DF (h)2 =N(h)∑i=1

(i+1)h − X

⟨ N ⟩

Levy(a)

α=3/2

× ⟨ NF(h) ⟩

° ⟨ NR(r) ⟩

α=5/2

⟨ D2 ⟩

Levy(b)

(∞)° ⟨ (r

R(r)/r)2 ⟩

× ⟨ D2F(h) ⟩

α=3/2α=5/2

Results

fractional Brownian motion:number of surviving trajectories N,

mean square distance: DF (h)2 =N(h)∑i=1

(i+1)h − X

⟨ N ⟩

(∞)h

2/3(∞)γ=2/3

γ=1/2

(c)fBm

× ⟨ NF(h) ⟩

° ⟨ NR(r) ⟩

γ=1/3

⟨ D2 ⟩

γ=2/3γ=1/2

(d) fBm

× ⟨ DF2(h) ⟩

° ⟨ (r0D

R(r)/r)2⟩

γ=1/3

Results – finite particle size

fixed size d in expanding space corresponds todecreasing size r0

r(h)d in fixed space

⟨ N ⟩

Bm 2+1d=0.3d=0.5

° ⟨ NR ⟩

+ ⟨ NF,uncorrected

× ⟨ NF ⟩

Results – branching-coalescing Brownian motionbranching rate RR in expanding space vs.branching rate RF in fixed space:

∆R(dr)/dr

∆F (dh)/dh=

dr=( r0r

⟨ N⟩/r

° ⟨ NR(r) ⟩/r

branching−coalescing Bm

× ⟨ NF(h) ⟩/r(h)

RR=0.1

Conclusions – mapping

locally scale invariant growth processesmapped from homogenously expanding space to fixed space

can be used eg. to handle 2D radial growth,asymptotic state in radial growth corresponds to finite time in fixed space

can be extended to include:

I finite particle size,I branching-coalescing,I higher dimensions etc.

Bacteria vs. yeast

Biology

ignore many biological details (shape, growth direction, etc.)

consider: correlations due to reproduction time

consider: overall geometry

Bacteria (E. Coli)

emc.maricopa.edu

Yiest (S. cerevisiae)

visualphotos.com

off-lattice Eden growth model

one-parameter family, δ ∈ [0, 1]: reproduction time T with delay 1− δdistribution: T ∼ 1− δ + Exp(1/δ)normalized average: 〈T 〉 = 1variation coefficient: σ(T )/〈T 〉 = δ

Surface – KPZ scaling

on large scale:

∂th(x , t) = v0 + ν∂2xh +

2(∂xh)2 + Dη(x , t)

scaling:x → x ′ = bx , t → t ′ = bz t, h→ h′ = bαhstatistical scale invariance: h(x , t) ∼ h′(x ′, t ′)Family-Vicsek scaling: w(L, t) :=

√〈(h − 〈h〉x)2〉x ∼ Lαf (t/Lz)

ahol f (u) ∼

{uβ, if u � 1

const, if u � 1

exponents: α + z = 2, z = α/β1D: α = 1/2, β = 1/3, z = 3/2.

Surface – KPZ scaling

w(L, t) ∼

{t1/3, for t � L3/2

L1/2, for t � L3/2

Domain boundaries

M(h) := 〈X 2h 〉 ≈ σ2h2γ , γ = 2/3

Correlations

Partial synchronization leads to intrinsic vertical correlations.

N(t) growth events with height ∆hi : hN(t) =∑N(t)

i=1 ∆hi

var[hN(t)] = 〈∆hi 〉2var[N(t)] + 〈N(t)〉var[∆hi ]

var[hN(t)] = t〈∆hi 〉2(δ2 + ε2)!

= O(1)where correlation coefficient due to geometric effects:ε =

√var[∆hi ]/〈∆hi 〉

intrinsic vertical correlation: τ ∼ 1δ2+ε2

lateral correlation length: ξ‖(t) ∼ (t/τ)1/z

mean square displacementM(h) := 〈[X (h)− X (0)]2〉 ≈ σ2

δh2γ ∼ ξ2

‖(h)

γ = 2/3 and σ2δ ∝ (δ2 + ε2)4/3

Correlations

Prefactor of mean square displacement: σ2δ ∝ (δ2 + ε2)4/3

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

σ2 δ

0.2765(δ2+0.3958)4/3

S. cerevisiae (yeast)

Replace δ-family with realistic reproduction times.

T H∆=0.248L

T H∆=0.190L

0.0 0.5 1.0 1.5 2.0

t�XT\

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60.08

δσ δ2

δ−familyMDM and D

Conclusions – partially synchronized growth

extension of off-lattice Eden modelreproduction time has variation coefficient δ

stays in KPZ universality class

changes in patterns are due to changing prefactors – quantified

works for realistic reproduction times (S. cerevisiae)

Sk alainvari ans dinamika t agul o t erbenglu.elte.hu/~statfiz/eloadasok/2013-04-17-Somfai.pdfcan be...

Documents

Transcript of Sk alainvari ans dinamika t agul o t erbenglu.elte.hu/~statfiz/eloadasok/2013-04-17-Somfai.pdfcan be...

Population Growth

ELLAKTOR Group: Positioned for Growth

17 Machine Learning Radial Basis Functions

Eikonal Equations for Null Radial Geodesics in the ...

The Solow growth model

Macroeconomics Chapter 51 Conditional Convergence and Long- Run Economic Growth C h a p t e r 5.

Growth Monitoring

αPD-1-mesoCAR-T cells partially inhibit the growth of ...

Growth of function 2008. 1. 28 : 1 Chapter 3. Growth of function.

Chemical vapor deposition growth of carbon …static.sif.it/SIF/resources/public/files/congr11/mc/...Chemical Vapour Deposition (CVD) T= 500 C Characterizations: T= 700 C SEM XPS 1000

Rotation and angular momentum - Uni Siegen€¦ · Relation between linear and angular variables tangential acceleration radial acceleration. ... gives a relation between the radial

CHAPTER SEVEN Economic Growth Ipioneer.netserv.chula.ac.th/~msompraw/chap7_Slide.pdf · CHAPTER SEVEN Economic Growth I ... CHAPTER 7 Economic Growth I slide 2 The importance of economic

Governance and Growth of Cooperatives in Microfinance · Governance and Growth of ... Spl i,t+1 =α+ β1 ×GOV i +β2 ×SIZ E ... anais.perilleux@yale.edu 34. Title: ciriec-vienna2012-Anais-Perilleux

Trade Growth and Inequality

Induced Growth of Calcium Carbonate Using an Atomic Force ...jtd/presentations/ann/showcase.pdf · spontaneous growth can still be produced. The scanning induced growth rate does

Population Review. Exponential growth N t+1 = N t + B – D + I – E ΔN = B – D + I – E For a closed population ΔN = B – D.

GE 40 ES-2RS - bearings-online.co.uk · 02/06/2020 · GE 40 ES-2RS Radial spherical plain bearings Bearing data Tolerances, Radial internal clearance Radial location of bearings

Radial positive deﬁnite functions and Schoenberg … · arXiv:1502.07179v1 [math.CA] 25 Feb 2015 Radial positive deﬁnite functions and Schoenberg matrices with negative eigenvalues

Automobile Industry - history, evolution & growth

Harrod Domar Growth Model Solow Growth Model … · Solow Growth Model Endogenous Growth Model. Harrod-Domar Model ... depreciation, labor force growth and development