text: ∗ † M. Majka and P. F. G´ora ‡ § The Gaussian chain model is the classical description of a polymeric chain, which provides the analytical results regarding end-to-end distance, the distribution of segments around the mass cen- ter of a chain, coarse grained interactions between two chains and eﬀective interactions in binary mixtures. This hierarchy of results can be calculated thanks to the alpha stability of the Gaussian distribution. In this paper we show that it is possible to generalize the model of Gaussian chain to the entire class of alpha stable distributions, obtaining the analogous hierarchy of results ex- pressed by the analytical closed-form formulas in the Fourier space. This allows us to establish the alpha-stable chain model. We begin with reviewing the applications of Levy ﬂights in the context of polymer sciences, which include: chains with heavy-tailed distributions of persistence length, poly- mers adsorbed to the surface and the chains driven by a noise with power-law spatial correlations. Further, we derive the distribution of segments around the mass center of the alpha-stable chain and the coarse-grained interaction potential between two chains is constructed. These results are employed to discuss the model of binary mixture consisting of the alpha-stable chains. On what follows, we establish the spinodal decomposition condition generalized to the particles described by the shape of alpha-stable distributions. This condition is ﬁnally applied to analyze the on-surface phase separation of adsorbed polymers, which are known to be described with heavy tailed statistics. PACS numbers: 36.20.-r, 05.40.Fb, 68.35.Dv, 82.35.Gh I. INTRODUCTION context in which the alpha stable distributions are rele- vant for the polymer sciences. Since the application of While the theory of Flory provides an accurate de- alpha-stable distributions (or Levy walks/ﬂights) in this scription of the ideal polymeric chains [1], factors such context is not an entirely new concept, in Section we re- as complex environment interactions, adsorption or de- view the relevant literature. In addition, we provide our signed chemical composition can lead to signiﬁcant devi- own simulations of a model polymeric chain under the ations from this model. The Flory approach is based on spatially correlated noise, which establish another con- the Gaussian chain model, in which the conformation of text for our considerations. a chain is equivalent to the trajectory of a particle under- The main part of this paper is focused on deriving and going the thermal Brownian motion [1]. In this model the analyzing the diﬀerent aspects of the alpha-stable chain chain is characterized by the Gaussian distribution of the model. In Section we introduce the model itself. In nearest neighbor distances, a fact that leads to the entire Section the distribution of nodes around the mass center hierarchy of analytical results. In particular, the Gaus- of a chain is calculated and in Section the coarse-grained sian shape propagates to such characteristics as: end- model of interaction between two chains is established. to-end distance distribution [1], distribution of segments All of these results are analytical and closed-form in the around the mass center of the chain [2] and the coarse- Fourier space. grained interaction potential between two chains in terms Another goal of this paper is to analyze the stability of the distance between their mass centers [3]. Deriving of binary mixtures composed of the alpha-stable chains. all of these characteristics is possible due to a single fact: Understanding the behavior of binary mixtures is a vital the Gaussian distribution is alpha-stable. Since there ex- problem in industry, medicine, wet nano-technology and ist the entire class of alpha-stable, heavy-tailed distribu- biophysics. The stability of solution is governed by eﬀec- tions [4], this suggests that a natural and equally proliﬁc tive interactions [5], predicting which is a classical prob- generalization of the Gaussian chain model can be based lem of soft matter physics [6]. In the context of binary on the alpha-stable distributions. Indeed, in this paper mixtures of Gaussian particles, Bolhuis et al. found via we discuss the alpha-stable chain model and calculate all simulations that the interaction between particles of one of the characteristics analogous to the Gaussian model. species has also a Gaussian proﬁle, but with an addition The ﬁrst goal of this paper is to establish the physical of a shallow attractive tail [7]. Similar results were pre- dicted half-analytically via closure-relations techniques in [8, 9]. On the other hand, a simpler, but entirely analyt- ∗ ical method has been recently proposed by the present † authors, in [10]. Therein, we have studied the stabil- maciej.majka@uj.edu.pl ‡ ity of Gaussian particles mixtures and our results proved § similar to the spinodal decomposition analysis [11, 12], ##LTLine##

text: M.

text: Majka

text: and

text: P.

text: F.

text: G´ora

text: The

text: Gaussian

text: chain

text: model

text: is

text: the

text: classical

text: description

text: of

text: a

text: polymeric

text: chain,

text: which

text: provides

text: the

text: analytical

text: results

text: regarding

text: end-to-end

text: distance,

text: the

text: distribution

text: of

text: segments

text: around

text: the

text: mass

text: cen-

text: ter

text: of

text: a

text: chain,

text: coarse

text: grained

text: interactions

text: between

text: two

text: chains

text: and

text: eﬀective

text: interactions

text: in

text: binary

text: mixtures.

text: This

text: hierarchy

text: of

text: results

text: can

text: be

text: calculated

text: thanks

text: to

text: the

text: alpha

text: stability

text: of

text: the

text: Gaussian

text: distribution.

text: In

text: this

text: paper

text: we

text: show

text: that

text: it

text: is

text: possible

text: to

text: generalize

text: the

text: model

text: of

text: Gaussian

text: chain

text: to

text: the

text: entire

text: class

text: of

text: alpha

text: stable

text: distributions,

text: obtaining

text: the

text: analogous

text: hierarchy

text: of

text: results

text: ex-

text: pressed

text: by

text: the

text: analytical

text: closed-form

text: formulas

text: in

text: the

text: Fourier

text: space.

text: This

text: allows

text: us

text: to

text: establish

text: the

text: alpha-stable

text: chain

text: model.

text: We

text: begin

text: with

text: reviewing

text: the

text: applications

text: of

text: Levy

text: ﬂights

text: in

text: the

text: context

text: of

text: polymer

text: sciences,

text: which

text: include:

text: chains

text: with

text: heavy-tailed

text: distributions

text: of

text: persistence

text: length,

text: poly-

text: mers

text: adsorbed

text: to

text: the

text: surface

text: and

text: the

text: chains

text: driven

text: by

text: a

text: noise

text: with

text: power-law

text: spatial

text: correlations.

text: Further,

text: we

text: derive

text: the

text: distribution

text: of

text: segments

text: around

text: the

text: mass

text: center

text: of

text: the

text: alpha-stable

text: chain

text: and

text: the

text: coarse-grained

text: interaction

text: potential

text: between

text: two

text: chains

text: is

text: constructed.

text: These

text: results

text: are

text: employed

text: to

text: discuss

text: the

text: model

text: of

text: binary

text: mixture

text: consisting

text: of

text: the

text: alpha-stable

text: chains.

text: On

text: what

text: follows,

text: we

text: establish

text: the

text: spinodal

text: decomposition

text: condition

text: generalized

text: to

text: the

text: particles

text: described

text: by

text: the

text: shape

text: of

text: alpha-stable

text: distributions.

text: This

text: condition

text: is

text: ﬁnally

text: applied

text: to

text: analyze

text: the

text: on-surface

text: phase

text: separation

text: of

text: adsorbed

text: polymers,

text: which

text: are

text: known

text: to

text: be

text: described

text: with

text: heavy

text: tailed

text: statistics.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: context

text: in

text: which

text: the

text: alpha

text: stable

text: distributions

text: are

text: rele-

text: vant

text: for

text: the

text: polymer

text: sciences.

text: Since

text: the

text: application

text: of

text: While

text: the

text: theory

text: of

text: Flory

text: provides

text: an

text: accurate

text: de-

text: alpha-stable

text: distributions

text: (or

text: Levy

text: walks/ﬂights)

text: in

text: this

text: scription

text: of

text: the

text: ideal

text: polymeric

text: chains

text: factors

text: such

text: context

text: is

text: not

text: an

text: entirely

text: new

text: concept,

text: in

text: Section

text: we

text: re-

text: as

text: complex

text: environment

text: interactions,

text: adsorption

text: or

text: de-

text: view

text: the

text: relevant

text: literature.

text: In

text: addition,

text: we

text: provide

text: our

text: signed

text: chemical

text: composition

text: can

text: lead

text: to

text: signiﬁcant

text: devi-

text: own

text: simulations

text: of

text: a

text: model

text: polymeric

text: chain

text: under

text: the

text: ations

text: from

text: this

text: model.

text: The

text: Flory

text: approach

text: is

text: based

text: on

text: spatially

text: correlated

text: noise,

text: which

text: establish

text: another

text: con-

text: the

text: Gaussian

text: chain

text: model,

text: in

text: which

text: the

text: conformation

text: of

text: text

text: for

text: our

text: considerations.

text: a

text: chain

text: is

text: equivalent

text: to

text: the

text: trajectory

text: of

text: a

text: particle

text: under-

text: The

text: main

text: part

text: of

text: this

text: paper

text: is

text: focused

text: on

text: deriving

text: and

text: going

text: the

text: thermal

text: Brownian

text: motion

text: In

text: this

text: model

text: the

text: analyzing

text: the

text: diﬀerent

text: aspects

text: of

text: the

text: alpha-stable

text: chain

text: chain

text: is

text: characterized

text: by

text: the

text: Gaussian

text: distribution

text: of

text: the

text: model.

text: In

text: Section

text: we

text: introduce

text: the

text: model

text: itself.

text: In

text: nearest

text: neighbor

text: distances,

text: a

text: fact

text: that

text: leads

text: to

text: the

text: entire

text: Section

text: the

text: distribution

text: of

text: nodes

text: around

text: the

text: mass

text: center

text: hierarchy

text: of

text: analytical

text: results.

text: In

text: particular,

text: the

text: Gaus-

text: of

text: a

text: chain

text: is

text: calculated

text: and

text: in

text: Section

text: the

text: coarse-grained

text: sian

text: shape

text: propagates

text: to

text: such

text: characteristics

text: as:

text: end-

text: model

text: of

text: interaction

text: between

text: two

text: chains

text: is

text: established.

text: to-end

text: distance

text: distribution

text: distribution

text: of

text: segments

text: All

text: of

text: these

text: results

text: are

text: analytical

text: and

text: closed-form

text: in

text: the

text: around

text: the

text: mass

text: center

text: of

text: the

text: chain

text: and

text: the

text: coarse-

text: Fourier

text: space.

text: grained

text: interaction

text: potential

text: between

text: two

text: chains

text: in

text: terms

text: Another

text: goal

text: of

text: this

text: paper

text: is

text: to

text: analyze

text: the

text: stability

text: of

text: the

text: distance

text: between

text: their

text: mass

text: centers

text: Deriving

text: of

text: binary

text: mixtures

text: composed

text: of

text: the

text: alpha-stable

text: chains.

text: all

text: of

text: these

text: characteristics

text: is

text: possible

text: due

text: to

text: a

text: single

text: fact:

text: Understanding

text: the

text: behavior

text: of

text: binary

text: mixtures

text: is

text: a

text: vital

text: the

text: Gaussian

text: distribution

text: is

text: alpha-stable.

text: Since

text: there

text: ex-

text: problem

text: in

text: industry,

text: medicine,

text: wet

text: nano-technology

text: and

text: ist

text: the

text: entire

text: class

text: of

text: alpha-stable,

text: heavy-tailed

text: distribu-

text: biophysics.

text: The

text: stability

text: of

text: solution

text: is

text: governed

text: by

text: eﬀec-

text: tions

text: this

text: suggests

text: that

text: a

text: natural

text: and

text: equally

text: proliﬁc

text: tive

text: interactions

text: predicting

text: which

text: is

text: a

text: classical

text: prob-

text: generalization

text: of

text: the

text: Gaussian

text: chain

text: model

text: can

text: be

text: based

text: lem

text: of

text: soft

text: matter

text: physics

text: In

text: the

text: context

text: of

text: binary

text: on

text: the

text: alpha-stable

text: distributions.

text: Indeed,

text: in

text: this

text: paper

text: mixtures

text: of

text: Gaussian

text: particles,

text: Bolhuis

text: et

text: al.

text: found

text: via

text: we

text: discuss

text: the

text: alpha-stable

text: chain

text: model

text: and

text: calculate

text: all

text: simulations

text: that

text: the

text: interaction

text: between

text: particles

text: of

text: one

text: of

text: the

text: characteristics

text: analogous

text: to

text: the

text: Gaussian

text: model.

text: species

text: has

text: also

text: a

text: Gaussian

text: proﬁle,

text: but

text: with

text: an

text: addition

text: The

text: ﬁrst

text: goal

text: of

text: this

text: paper

text: is

text: to

text: establish

text: the

text: physical

text: of

text: a

text: shallow

text: attractive

text: tail

text: Similar

text: results

text: were

text: pre-

text: dicted

text: half-analytically

text: via

text: closure-relations

text: techniques

text: in

text: On

text: the

text: other

text: hand,

text: a

text: simpler,

text: but

text: entirely

text: analyt-

text: ical

text: method

text: has

text: been

text: recently

text: proposed

text: by

text: the

text: present

text: authors,

text: in

text: Therein,

text: we

text: have

text: studied

text: the

text: stabil-

text: maciej.majka@uj.edu.pl

text: ity

text: of

text: Gaussian

text: particles

text: mixtures

text: and

text: our

text: results

text: proved

text: similar

text: to

text: the

text: spinodal

text: decomposition

text: analysis

text: ##LTLine##


text: )n(nl γ )γ(D 4 required to cover the points in a given set, n, changes D(γ) for the uniform Cantor set with the nearest−neighbor method as the size of the partitions (cid:15) changes. The scaling rela- 0.7 tion can be extracted for a fractal set as the size of the 0.65 partitions decreases, namely, 0.6 0.55 − ln n((cid:15)) D = lim (10) (cid:15)→0 ln (cid:15) 0.5 0.45 Due to the simplicity of the method, it is widely used 0.4 among researchers. However, it has been pointed out 0.35 by many that this method and, more generally, meth- ods that involve partitions of the same size such as the 0.3 −4 −2 0 2 4 Correlation method, do not work well for q < 1.18 A γ heuristic explanation is given below to understand this result. In Eq. (2), we can see that if q > 1, the con- FIG. 2. In this graph, the Dimension Function D(γ) for the tribution from relatively large pi is emphasized, and if Uniform Cantor set was computed as the slope of the best-ﬁt q < 1, the contribution form relatively sm|al|l pi plays the line to the corresponding data set which is partially plotted dominant role. The larger the value of q , the greater in Fig. 1. D(γ) diverges from the analytical result which is the eﬀective discrimination. Therefore, the fact that the log 2/ log 3 for negative γ. method does not produce a good result for D with q < 1 q means that the sparse regions of the set are not well- represented in the ﬁnite representation of the Cantor set. The approach called the “nearest neighbor method” Since a true fractal possesses an inﬁnite number of points was ﬁrst introduced by Badii and Politi.10 This method or elements, any ﬁnite set may not be large enough to is essentially based on their observation that represent the true Cantor set in relatively sparse regions. ∼ − 1 In some instances, a ﬁnite representation of a fractal may < δ > n D0 (11) be thought of as a subset of a corresponding fractal as in the H´enon map.19 As the size of the cells diminishes, the where < δ > denotes the mean distance from each point to its nearest neighbor among n randomly chosen points truncated ﬁnite sample no longer statistically represents from a given test set and, as discussed earlier, The value the sparse regions of a true fractal. Under the same con- D is just the box-counting dimension. By naturally ex- dition, the dense regions are aﬀected less by the ﬁnite size 0 tending the premise, the Dimension Function D(γ) can eﬀect. Since numerical methods always have to deal with be computed by using the moments of order γ of the dis- a ﬁnite sample, diﬀerent methods need to be considered tribution function P(δ, n) generated by an ensemble of n to ﬁnd an accurate result for q < 1. randomly chosen poin(cid:90)ts: B. Nearest Neighbor Method ∞ ≡ ≡ − < δγ > Mγ(n) δγP(δ, n)dδ = Kn Dγ(γ) (12) The scaling of γ ln(n) vs. −log(Mγ(n)) for the uniform Cantor set 0 60 γ=−5 where K is some function of n and γ which asymp- γ=−4 40 γ=−3 totically remains bounded as n becomes large. Here, the γ=−2 meaning of γ should be clear; the dense region of a given 20 γ=−1 γ=0.1 set generates smaller values of δ, the distance to the near- γ=1 est neighbor, and vice versa. The proof of a more general 0 γ=2 γ=3 relation is provided by van de Walter and Schram.11 It −20 γ=4 follows that the Dimension Function D(γ) can be ob- γ=5 tained by: −40 −60 − γ ln n −150 −100 −50 0 50 100 D(γ) = lim (13) →∞ −log(Mγ(n)) n ln Mγ(n) − FIG. 1. For the uniform Cantor Set, γ ln(n) vs. ln(Mγ(n)) The function K generally depends on n and γ but K is plotted for each γ as n is increased. According to Eq. 13, should be, by deﬁnition, irrelevant in the limiting case as the slope converges to D(γ). The corresponding result for in Eq. (13). In numerical analysis, the value of K(n, γ) D(γ) is shown in Fig. 2 does aﬀect the numerical result as n is ﬁnite. The Dimen- sion Function D(γ) can be thought of as an alternative ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: )n(nl

label: paragraph

font: default

text: γ

text: )γ(D

text: required

text: to

text: cover

text: the

text: points

text: in

text: a

text: given

text: set,

text: n,

text: changes

text: D(γ)

text: for

text: the

text: uniform

text: Cantor

text: set

text: with

text: the

text: nearest−neighbor

text: method

text: as

text: the

text: size

text: of

text: the

text: partitions

text: changes.

text: The

text: scaling

text: rela-

text: tion

text: can

text: be

text: extracted

text: for

text: a

text: fractal

text: set

text: as

text: the

text: size

text: of

text: the

text: partitions

text: decreases,

text: namely,

label: equation

text: ln

text: D

text: lim

text: ln

text: Due

text: to

text: the

text: simplicity

text: of

text: the

text: method,

text: it

text: is

text: widely

text: used

text: among

text: researchers.

text: However,

text: it

text: has

text: been

text: pointed

text: out

text: by

text: many

text: that

text: this

text: method

text: and,

text: more

text: generally,

text: meth-

text: ods

text: that

text: involve

text: partitions

text: of

text: the

text: same

text: size

text: such

text: as

text: the

text: −4

text: −2

text: Correlation

text: method,

text: do

text: not

text: work

text: well

text: for

text: q

text: A

text: γ

text: heuristic

text: explanation

text: is

text: given

text: below

text: to

text: understand

text: this

text: result.

text: In

text: Eq.

text: we

text: can

text: see

text: that

text: if

text: q

text: the

text: con-

text: FIG.

text: In

text: this

text: graph,

text: the

text: Dimension

text: Function

text: D(γ)

text: for

text: the

text: tribution

text: from

text: relatively

text: large

text: pi

text: is

text: emphasized,

text: and

text: if

text: Uniform

text: Cantor

text: set

text: was

text: computed

text: as

text: the

text: slope

text: of

text: the

text: best-ﬁt

text: q

text: the

text: contribution

text: form

text: relatively

text: sm|al|l

text: pi

text: plays

text: the

text: line

text: to

text: the

text: corresponding

text: data

text: set

text: which

text: is

text: partially

text: plotted

text: dominant

text: role.

text: The

text: larger

text: the

text: value

text: of

text: q

text: the

text: greater

text: in

text: Fig.

text: D(γ)

text: diverges

text: from

text: the

text: analytical

text: result

text: which

text: is

text: the

text: eﬀective

text: discrimination.

text: Therefore,

text: the

text: fact

text: that

text: the

text: log

text: log

text: for

text: negative

text: γ.

text: method

text: does

text: not

text: produce

text: a

text: good

text: result

text: for

text: D

text: with

text: q

text: q

text: means

text: that

text: the

text: sparse

text: regions

text: of

text: the

text: set

text: are

text: not

text: well-

text: represented

text: in

text: the

text: ﬁnite

text: representation

text: of

text: the

text: Cantor

text: set.

text: The

text: approach

text: called

text: the

text: “nearest

text: neighbor

text: method”

text: Since

text: a

text: true

text: fractal

text: possesses

text: an

text: inﬁnite

text: number

text: of

text: points

text: was

text: ﬁrst

text: introduced

text: by

text: Badii

text: and

text: This

text: method

text: or

text: elements,

text: any

text: ﬁnite

text: set

text: may

text: not

text: be

text: large

text: enough

text: to

text: is

text: essentially

text: based

text: on

text: their

text: observation

text: that

text: represent

text: the

text: true

text: Cantor

text: set

text: in

text: relatively

text: sparse

text: regions.

text: In

text: some

text: instances,

text: a

text: ﬁnite

text: representation

text: of

text: a

text: fractal

text: may

text: δ

text: n

text: be

text: thought

text: of

text: as

text: a

text: subset

text: of

text: a

text: corresponding

text: fractal

text: as

text: in

text: the

text: H´enon

text: As

text: the

text: size

text: of

text: the

text: cells

text: diminishes,

text: the

text: where

text: δ

text: denotes

text: the

text: mean

text: distance

text: from

text: each

text: point

text: to

text: its

text: nearest

text: neighbor

text: among

text: n

text: randomly

text: chosen

text: points

text: truncated

text: ﬁnite

text: sample

text: no

text: longer

text: statistically

text: represents

text: from

text: a

text: given

text: test

text: set

text: and,

text: as

text: discussed

text: earlier,

text: The

text: value

text: the

text: sparse

text: regions

text: of

text: a

text: true

text: fractal.

text: Under

text: the

text: same

text: con-

text: D

text: is

text: just

text: the

text: box-counting

text: dimension.

text: By

text: naturally

text: ex-

text: dition,

text: the

text: dense

text: regions

text: are

text: aﬀected

text: less

text: by

text: the

text: ﬁnite

text: size

text: tending

text: the

text: premise,

text: the

text: Dimension

text: Function

text: D(γ)

text: can

text: eﬀect.

text: Since

text: numerical

text: methods

text: always

text: have

text: to

text: deal

text: with

text: be

text: computed

text: by

text: using

text: the

text: moments

text: of

text: order

text: γ

text: of

text: the

text: dis-

text: a

text: ﬁnite

text: sample,

text: diﬀerent

text: methods

text: need

text: to

text: be

text: considered

text: tribution

text: function

text: P(δ,

text: n)

text: generated

text: by

text: an

text: ensemble

text: of

text: n

text: to

text: ﬁnd

text: an

text: accurate

text: result

text: for

text: q

text: randomly

text: chosen

text: B.

text: Nearest

text: Neighbor

text: Method

text: δγ

text: Mγ(n)

text: δγP(δ,

text: n)dδ

text: Kn

text: Dγ(γ)

text: The

text: scaling

text: of

text: γ

text: ln(n)

text: vs.

text: −log(Mγ(n))

text: for

text: the

text: uniform

text: Cantor

text: set

text: γ=−5

text: where

text: K

text: is

text: some

text: function

text: of

text: n

text: and

text: γ

text: which

text: asymp-

text: γ=−4

text: γ=−3

text: totically

text: remains

text: bounded

text: as

text: n

text: becomes

text: large.

text: Here,

text: the

text: γ=−2

text: meaning

text: of

text: γ

text: should

text: be

text: clear;

text: the

text: dense

text: region

text: of

text: a

text: given

text: γ=−1

text: set

text: generates

text: smaller

text: values

text: of

text: δ,

text: the

text: distance

text: to

text: the

text: near-

text: est

text: neighbor,

text: and

text: vice

text: versa.

text: The

text: proof

text: of

text: a

text: more

text: general

text: relation

text: is

text: provided

text: by

text: van

text: de

text: Walter

text: and

text: It

text: follows

text: that

text: the

text: Dimension

text: Function

text: D(γ)

text: can

text: be

text: ob-

text: tained

text: by:

text: γ

text: ln

text: n

text: D(γ)

text: lim

text: −log(Mγ(n))

text: n

text: ln

text: Mγ(n)

text: FIG.

text: For

text: the

text: uniform

text: Cantor

text: Set,

text: γ

text: ln(n)

text: vs.

text: ln(Mγ(n))

text: The

text: function

text: K

text: generally

text: depends

text: on

text: n

text: and

text: γ

text: but

text: K

text: is

text: plotted

text: for

text: each

text: γ

text: as

text: n

text: is

text: increased.

text: According

text: to

text: Eq.

text: should

text: be,

text: by

text: deﬁnition,

text: irrelevant

text: in

text: the

text: limiting

text: case

text: as

text: the

text: slope

text: converges

text: to

text: D(γ).

text: The

text: corresponding

text: result

text: for

text: in

text: Eq.

text: In

text: numerical

text: analysis,

text: the

text: value

text: of

text: K(n,

text: γ)

text: D(γ)

text: is

text: shown

text: in

text: Fig.

text: does

text: aﬀect

text: the

text: numerical

text: result

text: as

text: n

text: is

text: ﬁnite.

text: The

text: Dimen-

text: sion

text: Function

text: D(γ)

text: can

text: be

text: thought

text: of

text: as

text: an

text: alternative

text: ##LTFigure##

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text: )γM(nl / )x(nl γ − 6 the k-neighbor method less sensitive to local anomalies mension. Around the box-counting dimension, the near- which often arises from a ﬁnite sampling process. est neighbor method yields a result closest to the an- IV. NUMERICAL IMPLEMENTATION alytical solutions. However, as γ goes away from it, the k-neighbor method produces more accurate results. When dealing with a fractal set numerically, one needs Therefore, at this point, no single method seems reli- to conﬁne oneself to a ﬁnite sample. For a Cantor-like able enough for an extended domain q of the generalized set, the number of iterations m needs to be ﬁnite. The dimension. However, the combination of the aforemen- hierarchy degree m should be chosen in a way that two tioned methods reveals the essential features of a given points in neighboring intervals of the set are distinguish- set such as whether it is a monofractal or multifractal. able within the precision of a given numerical environ- For a multifractal set, how the dimension changes over ment. In our experiment, m = 30 is typically used and the domain q is a key property. The k-neighbor seems therefore we assume a ﬁnite representation of the Can- to be the best method to start with as it can provide tor set which consists of 230 intervals. Generally, the the estimate of the generalized dimension over an ex- larger the number of reference points n is used, the more tended region, albeit not too accurately. To obtain the accurate the result would be obtained, but n can only dimension to a higher accuracy for a particular q or γ, be increased by correspondingly increasing the amount the box-counting or the nearest neighbor method may of computation time but, as we discuss below, there is be used. For q > 1, the box-counting method should be another limitation on n as well. employed and for q < 1, the nearest neighbor, provided that q is not a very large negative number. Therefore, if When the number of sample points exceeds the number possible, the results obtained from these methods should of broken intervals, the expected probability distribution be compared and examined to see if they are consistent does not produce a desirable result since the distribution within the uncertainty of each method. within an interval is nothing but that of a line interval. A. Nearest Neighbor Method Therefore, the scaling property needs to be obtained for n suﬃciently smaller than 2m but large enough to accu- rately reﬂect a given fractal set. Each of the n points is randomly assigned to a particular one among 2m inter- Convergence of D(γ) vals. The position of the point is then randomly chosen 0.95 within the window of the chosen interval. Therefore, in γ= our model, most of sample points are taken from the 0.9 γγ==23 points which are not in the real Cantor Set. However, in γ=4 0.85 γ=5 principle, we can always set the upper limit to the dis- log2 / log3 tance between the sample points and closest points in a 0.8 true set by taking m suﬃciently large. Choosing a par- 0.75 ticular interval randomly among 230 intervals amounts to randomly generating 30 binary digits. This can be seen 0.7 by assigning 0 to the left interval and 1 to the right inter- 0.65 val on each level of the Cantor set. For the uniform Can- tor set, the probability of generating 0 and 1 is exactly 0.6 6 8 10 12 14 16 18 half. For the generalized Cantor set, the corresponding k weight factors are introduced. The Mersenne Twister Pseudo-Random Number Generator2F3IG. 4. This ﬁgure shows how increasing n = 2k aﬀects the − for c++ was our primary choice for obtaining random value of γ ln n/ ln M . The plot was generated for the uni- γ numbers. The built-in c++ random number generator form Cantor set. The analytical value for D(γ) for all γ is was also used. No idiosyncratic behavior from the par- log 2/ log 3 = 0.630... which corresponds to the horizontal line ticular choice of random number generator was observed. in the plot. Due to the limitation of the size of n, an ensemble aver- age must be employed in order to achieve higher accuracy In the nearest neighbor method, the Dimension Func- rather than increasing n. The number of members of the tion D(γ) was extracted from Eq. 13. In Eq. 13, the − ensemble required to stabilize the result depends on the right hand side reads γ lnn before taking the limit. range of γ. See section VI A for details. To investigate how it applnroMaγc(hne)s to the limit, ln n/ ln M 1 V. RESULTS versus ln n for the uniform Cantor set was plotted in Fig. − 4. The points in the plot indicates how γ ln n/ ln M 1 Generally, with a small amount of computational time, seemingly approaches the theoretical limit of ln 2/ ln 3 = both of the methods in the ﬁxed-mass class give good in- 0.63... as ln(n) increases in the case of uniform Cantor dications of the R´enyi Dimension in the vicinity of the set. However, it can be seen that the convergence rate box-counting dimension (q = 0) on various generalized is rather slow. Given that m is large enough, increasing Cantor Sets. This is a major advantage over the box- n can almost always guarantee a higher accuracy around counting method if one seeks to ﬁnd the box-counting di- the box-counting dimension. However, since the conver- ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: )γM(nl

text: )x(nl

text: γ

text: the

text: k-neighbor

text: method

text: less

text: sensitive

text: to

text: local

text: anomalies

text: mension.

text: Around

text: the

text: box-counting

text: dimension,

text: the

text: near-

text: which

text: often

text: arises

text: from

text: a

text: ﬁnite

text: sampling

text: process.

text: est

text: neighbor

text: method

text: yields

text: a

text: result

text: closest

text: to

text: the

text: an-

text: IV.

label: section

text: NUMERICAL

text: IMPLEMENTATION

text: alytical

text: solutions.

text: However,

text: as

text: γ

text: goes

text: away

text: from

text: it,

text: the

text: k-neighbor

text: method

text: produces

text: more

text: accurate

text: results.

text: When

text: dealing

text: with

text: a

text: fractal

text: set

text: numerically,

text: one

text: needs

text: Therefore,

text: at

text: this

text: point,

text: no

text: single

text: method

text: seems

text: reli-

text: to

text: conﬁne

text: oneself

text: to

text: a

text: ﬁnite

text: sample.

text: For

text: a

text: Cantor-like

text: able

text: enough

text: for

text: an

text: extended

text: domain

text: q

text: of

text: the

text: generalized

text: set,

text: the

text: number

text: of

text: iterations

text: m

text: needs

text: to

text: be

text: ﬁnite.

text: The

text: dimension.

text: However,

text: the

text: combination

text: of

text: the

text: aforemen-

text: hierarchy

text: degree

text: m

text: should

text: be

text: chosen

text: in

text: a

text: way

text: that

text: two

text: tioned

text: methods

text: reveals

text: the

text: essential

text: features

text: of

text: a

text: given

text: points

text: in

text: neighboring

text: intervals

text: of

text: the

text: set

text: are

text: distinguish-

text: set

text: such

text: as

text: whether

text: it

text: is

text: a

text: monofractal

text: or

text: multifractal.

text: able

text: within

text: the

text: precision

text: of

text: a

text: given

text: numerical

text: environ-

text: For

text: a

text: multifractal

text: set,

text: how

text: the

text: dimension

text: changes

text: over

text: ment.

text: In

text: our

text: experiment,

text: m

text: is

text: typically

text: used

text: and

text: the

text: domain

text: q

text: is

text: a

text: key

text: property.

text: The

text: k-neighbor

text: seems

text: therefore

text: we

text: assume

text: a

text: ﬁnite

text: representation

text: of

text: the

text: Can-

text: to

text: be

text: the

text: best

text: method

text: to

text: start

text: with

text: as

text: it

text: can

text: provide

text: tor

text: set

text: which

text: consists

text: of

text: intervals.

text: Generally,

text: the

text: the

text: estimate

text: of

text: the

text: generalized

text: dimension

text: over

text: an

text: ex-

text: larger

text: the

text: number

text: of

text: reference

text: points

text: n

text: is

text: used,

text: the

text: more

text: tended

text: region,

text: albeit

text: not

text: too

text: accurately.

text: To

text: obtain

text: the

text: accurate

text: the

text: result

text: would

text: be

text: obtained,

text: but

text: n

text: can

text: only

text: dimension

text: to

text: a

text: higher

text: accuracy

text: for

text: a

text: particular

text: q

text: or

text: γ,

text: be

text: increased

text: by

text: correspondingly

text: increasing

text: the

text: amount

text: the

text: box-counting

text: or

text: the

text: nearest

text: neighbor

text: method

text: may

text: of

text: computation

text: time

text: but,

text: as

text: we

text: discuss

text: below,

text: there

text: is

text: be

text: used.

text: For

text: q

text: the

text: box-counting

text: method

text: should

text: be

text: another

text: limitation

text: on

text: n

text: as

text: well.

text: employed

text: and

text: for

text: q

text: the

text: nearest

text: neighbor,

text: provided

text: that

text: q

text: is

text: not

text: a

text: very

text: large

text: negative

text: number.

text: Therefore,

text: if

text: When

text: the

text: number

text: of

text: sample

text: points

text: exceeds

text: the

text: number

text: possible,

text: the

text: results

text: obtained

text: from

text: these

text: methods

text: should

text: of

text: broken

text: intervals,

text: the

text: expected

text: probability

text: distribution

text: be

text: compared

text: and

text: examined

text: to

text: see

text: if

text: they

text: are

text: consistent

text: does

text: not

text: produce

text: a

text: desirable

text: result

text: since

text: the

text: distribution

text: within

text: the

text: uncertainty

text: of

text: each

text: method.

text: within

text: an

text: interval

text: is

text: nothing

text: but

text: that

text: of

text: a

text: line

text: interval.

text: A.

text: Nearest

text: Neighbor

text: Method

text: Therefore,

text: the

text: scaling

text: property

text: needs

text: to

text: be

text: obtained

text: for

text: n

text: suﬃciently

text: smaller

text: than

text: but

text: large

text: enough

text: to

text: accu-

text: rately

text: reﬂect

text: a

text: given

text: fractal

text: set.

text: Each

text: of

text: the

text: n

text: points

text: is

text: randomly

text: assigned

text: to

text: a

text: particular

text: one

text: among

text: inter-

text: Convergence

text: of

text: D(γ)

text: vals.

text: The

text: position

text: of

text: the

text: point

text: is

text: then

text: randomly

text: chosen

text: within

text: the

text: window

text: of

text: the

text: chosen

text: interval.

text: Therefore,

text: in

text: γ=

text: our

text: model,

text: most

text: of

text: sample

text: points

text: are

text: taken

text: from

text: the

text: points

text: which

text: are

text: not

text: in

text: the

text: real

text: Cantor

text: Set.

text: However,

text: in

text: principle,

text: we

text: can

text: always

text: set

text: the

text: upper

text: limit

text: to

text: the

text: dis-

text: tance

text: between

text: the

text: sample

text: points

text: and

text: closest

text: points

text: in

text: a

text: true

text: set

text: by

text: taking

text: m

text: suﬃciently

text: large.

text: Choosing

text: a

text: par-

text: ticular

text: interval

text: randomly

text: among

text: intervals

text: amounts

text: to

text: randomly

text: generating

text: binary

text: digits.

text: This

text: can

text: be

text: seen

text: by

text: assigning

text: to

text: the

text: left

text: interval

text: and

text: to

text: the

text: right

text: inter-

text: val

text: on

text: each

text: level

text: of

text: the

text: Cantor

text: set.

text: For

text: the

text: uniform

text: Can-

text: tor

text: set,

text: the

text: probability

text: of

text: generating

text: and

text: is

text: exactly

text: half.

text: For

text: the

text: generalized

text: Cantor

text: set,

text: the

text: corresponding

text: k

text: weight

text: factors

text: are

text: introduced.

text: The

text: Mersenne

text: Twister

text: Pseudo-Random

text: Number

text: This

text: ﬁgure

text: shows

text: how

text: increasing

text: n

text: aﬀects

text: the

text: for

text: c++

text: was

text: our

text: primary

text: choice

text: for

text: obtaining

text: random

text: value

text: of

text: γ

text: ln

text: n/

text: ln

text: M

text: The

text: plot

text: was

text: generated

text: for

text: the

text: uni-

text: γ

text: numbers.

text: The

text: built-in

text: c++

text: random

text: number

text: generator

text: form

text: Cantor

text: set.

text: The

text: analytical

text: value

text: for

text: D(γ)

text: for

text: all

text: γ

text: is

text: was

text: also

text: used.

text: No

text: idiosyncratic

text: behavior

text: from

text: the

text: par-

text: log

text: log

text: which

text: corresponds

text: to

text: the

text: horizontal

text: line

text: ticular

text: choice

text: of

text: random

text: number

text: generator

text: was

text: observed.

text: in

text: the

text: plot.

text: Due

text: to

text: the

text: limitation

text: of

text: the

text: size

text: of

text: n,

text: an

text: ensemble

text: aver-

text: age

text: must

text: be

text: employed

text: in

text: order

text: to

text: achieve

text: higher

text: accuracy

text: In

text: the

text: nearest

text: neighbor

text: method,

text: the

text: Dimension

text: Func-

text: rather

text: than

text: increasing

text: n.

text: The

text: number

text: of

text: members

text: of

text: the

text: tion

text: D(γ)

text: was

text: extracted

text: from

text: Eq.

text: In

text: Eq.

text: the

text: ensemble

text: required

text: to

text: stabilize

text: the

text: result

text: depends

text: on

text: the

text: right

text: hand

text: side

text: reads

text: γ

text: lnn

text: before

text: taking

text: the

text: limit.

text: range

text: of

text: γ.

text: See

text: section

text: VI

text: A

text: for

text: details.

text: To

text: investigate

text: how

text: it

text: applnroMaγc(hne)s

text: to

text: the

text: limit,

text: ln

text: n/

text: ln

text: M

text: V.

text: RESULTS

text: versus

text: ln

text: n

text: for

text: the

text: uniform

text: Cantor

text: set

text: was

text: plotted

text: in

text: Fig.

text: The

text: points

text: in

text: the

text: plot

text: indicates

text: how

text: γ

text: ln

text: n/

text: ln

text: M

text: Generally,

text: with

text: a

text: small

text: amount

text: of

text: computational

text: time,

text: seemingly

text: approaches

text: the

text: theoretical

text: limit

text: of

text: ln

text: ln

text: both

text: of

text: the

text: methods

text: in

text: the

text: ﬁxed-mass

text: class

text: give

text: good

text: in-

text: as

text: ln(n)

text: increases

text: in

text: the

text: case

text: of

text: uniform

text: Cantor

text: dications

text: of

text: the

text: R´enyi

text: Dimension

text: in

text: the

text: vicinity

text: of

text: the

text: set.

text: However,

text: it

text: can

text: be

text: seen

text: that

text: the

text: convergence

text: rate

text: box-counting

text: dimension

text: (q

text: on

text: various

text: generalized

text: is

text: rather

text: slow.

text: Given

text: that

text: m

text: is

text: large

text: enough,

text: increasing

text: Cantor

text: Sets.

text: This

text: is

text: a

text: major

text: advantage

text: over

text: the

text: box-

text: n

text: can

text: almost

text: always

text: guarantee

text: a

text: higher

text: accuracy

text: around

text: counting

text: method

text: if

text: one

text: seeks

text: to

text: ﬁnd

text: the

text: box-counting

text: di-

text: the

text: box-counting

text: dimension.

text: However,

text: since

text: the

text: conver-

text: ##LTFigure##

text: ##LTLine##

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text: )γ(D )γ(D )γ(D )γ(D )γ(D 7 Typical Results of D(γ) for various sets slope of log δγ(k, n) versus log k/n. As shown in Fig. 12 , the obtained δγ(k, n) exhibits a periodic pattern, so all Uniform Cantor Set MBP 0.7 1.2 approaches to obtain the slope seem to inject ambiguity. 0.6 0.5 1 We have used the standard linear regression technique24 0.4 0.8 using sample points equally spaced in the logarithmic 0.3 0.6 0.4 scale of k rather than in the k scale. Another considera- −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 3 4 γ γ tion is that the slope, and therefore, the result for D(γ) depends on the range to which the linear regression is Generalized Cantor Set Unit Segement 0.9 1.2 applied. It turns out that the best range seems to diﬀer 0.8 00..67 0. 81 for diﬀerent γ as shown in Fig. 6. The plot shows how 0.5 0.4 0.6 D(γ) varies when increasing the upper bound of the slope 0.3 −4 −3 −2 −1 0 1 2 3 4 0.4−4 −3 −2 −1 0 1 2 3 4 range when applied to the uniform Cantor set with the γ γ analytical dimension of log 2/ log 3 = 0.63... for all γ. Analytical Nearest Neighbor k−neighbor k−range dependence of D(γ) for the uniform Cantor Set FIG. 5. These plots show the typical results of D(γ) for the 0.72 nearest neighbor method and the k-neighbor method applied 0.7 to four diﬀerent sets and the corresponding analytical values. 0.68 “Unit Segment” here means a interval of unit length and can be thought as the 0th ﬁnite representation of the Cantor set. 0.66 For negative γ, numerical results persistently deviate from 0.64 the analytical results considerably for the nearest neighbor 0.62 method. While the k-neighbor method works relatively well 0.6 for all γ, the outcome may not be as accurate as the nearest 0.58 neighbor method for small positive γ. 0.56 γ=−5 0.54 γγ==− 11 γ= 5 0.52 gence rate is rather slow, determining the limit is not a 500 1000 1500 2000 2500 3000 trivial task. For γ = 1, the number of sample points Maximum k n = 29 = 512 was required to obtain the result within 5% accuracy and n = 217 to obtain the result within 3%. FIG. 6. This plot shows how D(γ) diﬀers when a diﬀerent For quick simulations, we typically used n = 216 and 10 range is used to extract the slope in the k-neighbor method. For the uniform Cantor set, increasing the upper bound of k ensembles. In general, we employed the linear regres- generally seems to produce better results. However, this is sion technique and obtained the limit from the slope of not a general result. the appropriate log-log plot. While the overall qualita- tive features of the Dimension Function such as the non- decreasing property are properly reﬂected on the domain As a result of these ﬁndings, we have used two dif- where γ is p−ositive, the deviations and the ﬂuctuations ferent boundaries for computing the slope, one for posi- around γ = 1 seem sudden and uncontrolled.−The dif- tive γ and the other for negative γ, to produce the ﬁnal ﬁculty of obtaining a sensible result for γ < 1 seems results. Since the inaccuracy inherited from these am- persistent throughout the set we have tested. In Fig. 5, biguities cannot be entirely removed by increasing n as the results for various generalized Cantor Sets are shown; in the nearest neighbor method, it is more diﬃcult for the domain of γ on which the simulated D(γ) agrees well the k-neighbor method to be adjusted to obtain a better with the analytical results is between 0 and 2. For a result before knowing the theoretical values. Neverthe- multifractal, as γ increases, the numerical results start less, aside from these ambiguities in the method, the k- to diverge from the analytical result as well. neighbor works for both positive and negative ranges of q, and therefore, is a good candidate as an initial method to investigate a given set. In the simulation, the ordering B. k-Neighbor Method of the n points from the reference points according to their relative position takes most of the computational Unlike the nearest neighbor method, where the choice time. Since the ordering takes more time as the topolog- of n is often limited by an available ﬁnite sample and ical dimension increases, the method is said to be espe- computational time, the k-neighbor method can utilize a cially suited for one-dimensional sets. Furthermore, un- larger data set from which the slope is extracted to esti- like the nearest neighbor method, the hierarchy degree mate D(γ). As we can see, the ﬁne structure is clearly m can be substantially small. The scaling region expect- observed in a log-log plot which injects arbitrariness in edly diminishes as m decreases. However, the Dimension a slope-ﬁtting process. This point is covered in detail in Function deduced from the best-linear-ﬁt from the ap- section VI. For a ﬁxed value of n, D(γ) or, to be precise, propriate scaling region produces acceptable results. For the corresponding 1/D(γ) in Eq. (17), is taken as the the uniform Cantor Set, when m is as small as 5, we ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: )γ(D

text: )γ(D

text: )γ(D

text: )γ(D

text: )γ(D

text: Typical

text: Results

text: of

text: D(γ)

text: for

text: various

text: sets

text: slope

text: of

text: log

text: δγ(k,

text: n)

text: versus

text: log

text: k/n.

text: As

text: shown

text: in

text: Fig.

text: the

text: obtained

text: δγ(k,

text: n)

text: exhibits

text: a

text: periodic

text: pattern,

text: so

text: all

text: Uniform

text: Cantor

text: Set

text: MBP

text: approaches

text: to

text: obtain

text: the

text: slope

text: seem

text: to

text: inject

text: ambiguity.

text: We

text: have

text: used

text: the

text: standard

text: linear

text: regression

text: using

text: sample

text: points

text: equally

text: spaced

text: in

text: the

text: logarithmic

text: scale

text: of

text: k

text: rather

text: than

text: in

text: the

text: k

text: scale.

text: Another

text: considera-

text: −4

text: −3

text: −2

text: −1

text: −4

text: −3

text: −2

text: −1

text: γ

text: γ

text: tion

text: is

text: that

text: the

text: slope,

text: and

text: therefore,

text: the

text: result

text: for

text: D(γ)

text: depends

text: on

text: the

text: range

text: to

text: which

text: the

text: linear

text: regression

text: is

text: Generalized

text: Cantor

text: Set

text: Unit

text: Segement

text: applied.

text: It

text: turns

text: out

text: that

text: the

text: best

text: range

text: seems

text: to

text: diﬀer

text: for

text: diﬀerent

text: γ

text: as

text: shown

text: in

text: Fig.

text: The

text: plot

text: shows

text: how

text: D(γ)

text: varies

text: when

text: increasing

text: the

text: upper

text: bound

text: of

text: the

text: slope

text: −4

text: −3

text: −2

text: −1

text: −3

text: −2

text: −1

text: range

text: when

text: applied

text: to

text: the

text: uniform

text: Cantor

text: set

text: with

text: the

text: γ

text: γ

text: analytical

text: dimension

text: of

text: log

text: log

text: for

text: all

text: γ.

text: Analytical

text: Nearest

text: Neighbor

text: k−neighbor

text: k−range

text: dependence

text: of

text: D(γ)

text: for

text: the

text: uniform

text: Cantor

text: Set

text: FIG.

text: These

text: plots

text: show

text: the

text: typical

text: results

text: of

text: D(γ)

text: for

text: the

text: nearest

text: neighbor

text: method

text: and

text: the

text: k-neighbor

text: method

text: applied

text: to

text: four

text: diﬀerent

text: sets

text: and

text: the

text: corresponding

text: analytical

text: values.

text: “Unit

text: Segment”

text: here

text: means

text: a

text: interval

text: of

text: unit

text: length

text: and

text: can

text: be

text: thought

text: as

text: the

text: ﬁnite

text: representation

text: of

text: the

text: Cantor

text: set.

text: For

text: negative

text: γ,

text: numerical

text: results

text: persistently

text: deviate

text: from

text: the

text: analytical

text: results

text: considerably

text: for

text: the

text: nearest

text: neighbor

text: method.

text: While

text: the

text: k-neighbor

text: method

text: works

text: relatively

text: well

text: for

text: all

text: γ,

text: the

text: outcome

text: may

text: not

text: be

text: as

text: accurate

text: as

text: the

text: nearest

text: neighbor

text: method

text: for

text: small

text: positive

text: γ.

text: γ=−5

text: γγ==−

text: γ=

text: gence

text: rate

text: is

text: rather

text: slow,

text: determining

text: the

text: limit

text: is

text: not

text: a

text: trivial

text: task.

text: For

text: γ

text: the

text: number

text: of

text: sample

text: points

text: Maximum

text: k

text: n

text: was

text: required

text: to

text: obtain

text: the

text: result

text: within

text: accuracy

text: and

text: n

text: to

text: obtain

text: the

text: result

text: within

text: FIG.

text: This

text: plot

text: shows

text: how

text: D(γ)

text: diﬀers

text: when

text: a

text: diﬀerent

text: For

text: quick

text: simulations,

text: we

text: typically

text: used

text: n

text: and

text: range

text: is

text: used

text: to

text: extract

text: the

text: slope

text: in

text: the

text: k-neighbor

text: method.

text: For

text: the

text: uniform

text: Cantor

text: set,

text: increasing

text: the

text: upper

text: bound

text: of

text: k

text: ensembles.

text: In

text: general,

text: we

text: employed

text: the

text: linear

text: regres-

text: generally

text: seems

text: to

text: produce

text: better

text: results.

text: However,

text: this

text: is

text: sion

text: technique

text: and

text: obtained

text: the

text: limit

text: from

text: the

text: slope

text: of

text: not

text: a

text: general

text: result.

text: the

text: appropriate

text: log-log

text: plot.

text: While

text: the

text: overall

text: qualita-

text: tive

text: features

text: of

text: the

text: Dimension

text: Function

text: such

text: as

text: the

text: non-

text: decreasing

text: property

text: are

text: properly

text: reﬂected

text: on

text: the

text: domain

text: As

text: a

text: result

text: of

text: these

text: ﬁndings,

text: we

text: have

text: used

text: two

text: dif-

text: where

text: γ

text: is

text: p−ositive,

text: the

text: deviations

text: and

text: the

text: ﬂuctuations

text: ferent

text: boundaries

text: for

text: computing

text: the

text: slope,

text: one

text: for

text: posi-

text: around

text: γ

text: seem

text: sudden

text: and

text: uncontrolled.−The

text: dif-

text: tive

text: γ

text: and

text: the

text: other

text: for

text: negative

text: γ,

text: to

text: produce

text: the

text: ﬁnal

text: ﬁculty

text: of

text: obtaining

text: a

text: sensible

text: result

text: for

text: γ

text: seems

text: results.

text: Since

text: the

text: inaccuracy

text: inherited

text: from

text: these

text: am-

text: persistent

text: throughout

text: the

text: set

text: we

text: have

text: tested.

text: In

text: Fig.

text: biguities

text: cannot

text: be

text: entirely

text: removed

text: by

text: increasing

text: n

text: as

text: the

text: results

text: for

text: various

text: generalized

text: Cantor

text: Sets

text: are

text: shown;

text: in

text: the

text: nearest

text: neighbor

text: method,

text: it

text: is

text: more

text: diﬃcult

text: for

text: the

text: domain

text: of

text: γ

text: on

text: which

text: the

text: simulated

text: D(γ)

text: agrees

text: well

text: the

text: k-neighbor

text: method

text: to

text: be

text: adjusted

text: to

text: obtain

text: a

text: better

text: with

text: the

text: analytical

text: results

text: is

text: between

text: and

text: For

text: a

text: result

text: before

text: knowing

text: the

text: theoretical

text: values.

text: Neverthe-

text: multifractal,

text: as

text: γ

text: increases,

text: the

text: numerical

text: results

text: start

text: less,

text: aside

text: from

text: these

text: ambiguities

text: in

text: the

text: method,

text: the

text: k-

text: to

text: diverge

text: from

text: the

text: analytical

text: result

text: as

text: well.

text: neighbor

text: works

text: for

text: both

text: positive

text: and

text: negative

text: ranges

text: of

text: q,

text: and

text: therefore,

text: is

text: a

text: good

text: candidate

text: as

text: an

text: initial

text: method

text: to

text: investigate

text: a

text: given

text: set.

text: In

text: the

text: simulation,

text: the

text: ordering

text: B.

text: k-Neighbor

text: Method

text: of

text: the

text: n

text: points

text: from

text: the

text: reference

text: points

text: according

text: to

text: their

text: relative

text: position

text: takes

text: most

text: of

text: the

text: computational

text: Unlike

text: the

text: nearest

text: neighbor

text: method,

text: where

text: the

text: choice

text: time.

text: Since

text: the

text: ordering

text: takes

text: more

text: time

text: as

text: the

text: topolog-

text: of

text: n

text: is

text: often

text: limited

text: by

text: an

text: available

text: ﬁnite

text: sample

text: and

text: ical

text: dimension

text: increases,

text: the

text: method

text: is

text: said

text: to

text: be

text: espe-

text: computational

text: time,

text: the

text: k-neighbor

text: method

text: can

text: utilize

text: a

text: cially

text: suited

text: for

text: one-dimensional

text: sets.

text: Furthermore,

text: un-

text: larger

text: data

text: set

text: from

text: which

text: the

text: slope

text: is

text: extracted

text: to

text: esti-

text: like

text: the

text: nearest

text: neighbor

text: method,

text: the

text: hierarchy

text: degree

text: mate

text: D(γ).

text: As

text: we

text: can

text: see,

text: the

text: ﬁne

text: structure

text: is

text: clearly

text: m

text: can

text: be

text: substantially

text: small.

text: The

text: scaling

text: region

text: expect-

text: observed

text: in

text: a

text: log-log

text: plot

text: which

text: injects

text: arbitrariness

text: in

text: edly

text: diminishes

text: as

text: m

text: decreases.

text: However,

text: the

text: Dimension

text: a

text: slope-ﬁtting

text: process.

text: This

text: point

text: is

text: covered

text: in

text: detail

text: in

text: Function

text: deduced

text: from

text: the

text: best-linear-ﬁt

text: from

text: the

text: ap-

text: section

text: VI.

text: For

text: a

text: ﬁxed

text: value

text: of

text: n,

text: D(γ)

text: or,

text: to

text: be

text: precise,

text: propriate

text: scaling

text: region

text: produces

text: acceptable

text: results.

text: For

text: the

text: corresponding

text: in

text: Eq.

text: is

text: taken

text: as

text: the

text: the

text: uniform

text: Cantor

text: Set,

text: when

text: m

text: is

text: as

text: small

text: as

text: we

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text: generality of the Green’s function approach with the comparative accessi- bility of Weinstein’s approach to NLS. Additionally, we demonstrate that these methods do generalize to soliton perturbation in higher dimension, be- yond (1+1)D, and how they can be used to determine the evolution of large numbers of parameters for a single soliton. Two-dimensional moving droplet solutions have been computed Piette and Zakrzewski [1998], Hoefer and Sommacal [2012] and studied asymptoti- cally in the weakly nonlinear regime Ivanov et al. [2001]. They can be accel- erated and controlled by an inhomogeneous, external magnetic ﬁeld Hoefer et al. [2012] and exhibit novel interaction properties Maiden et al. [2014]. Droplets can also experience a drift instability in NC-STOs, whose origin is not well-understood Hoefer et al. [2010]. Because droplets are relatively new physical features of nanomagnetic systems, they hold potential for applica- tions such as spintronic information storage and transfer or probing material properties. Moreover, their fundamental physics are not well understood. A more general theory to describe the motion of droplets in realistic physical systems is desirable. Toward this end, we here present an analytical framework for investi- gating the impact of a large class of physical eﬀects on a six parameter propagating droplet soliton. In Section , we derive an approximate solution for the propagating droplet in the close to Zeeman frequency (large mass) and small velocity (order one momentum) regime, which greatly reduces the complexity of the asymptotic theory developed. This allows for explicit ana- lytical results, a powerful feature of soliton perturbation theory considering the droplet’s strongly nonlinear qualities. The main result of this paper, the modulation equations, is presented in Section , utilizing the general soli- ton perturbation theory formulation from Section . The versatility of this framework is subsequently discussed in Sections and through an investiga- tion of a series of perturbations of current physical interest. In particular, we analytically demonstrate a mechanism leading to the NC-STO droplet drift instability and why a droplet can be attracted or repelled by a ferromagnetic boundary. The model under study here is the perturbed Landau-Lifshitz equation for the magnetization, m = [m , m , m ], of a two-dimensional ferromag- x y x netic ﬁlm in nondimensional form ∂m − × R × ∞ → S = m h + (cid:15)p, m : 2 (0, ) 2, eﬀ ∂t (1) ∇ h = 2m + (h + m )ˆz, lim m = ˆz, eﬀ 0 z | |→∞ x where p is a perturbation that preserves the magnetization’s length · ≡ (cid:28) (p m 0) and 0 < (cid:15) 1 is a small parameter encoding the strength of the perturbation. The perturbation can depend explicitly on time so long 4 ##LTLine##

text: generality

text: of

text: the

text: Green’s

text: function

text: approach

text: with

text: the

text: comparative

text: accessi-

text: bility

text: of

text: Weinstein’s

text: approach

text: to

text: NLS.

text: Additionally,

text: we

text: demonstrate

text: that

text: these

text: methods

text: do

text: generalize

text: to

text: soliton

text: perturbation

text: in

text: higher

text: dimension,

text: be-

text: yond

text: and

text: how

text: they

text: can

text: be

text: used

text: to

text: determine

text: the

text: evolution

text: of

text: large

text: numbers

text: of

text: parameters

text: for

text: a

text: single

text: soliton.

text: Two-dimensional

text: moving

text: droplet

text: solutions

text: have

text: been

text: computed

text: Piette

text: and

text: Zakrzewski

text: Hoefer

text: and

text: Sommacal

text: and

text: studied

text: asymptoti-

text: cally

text: in

text: the

text: weakly

text: nonlinear

text: regime

text: Ivanov

text: et

text: al.

text: They

text: can

text: be

text: accel-

text: erated

text: and

text: controlled

text: by

text: an

text: inhomogeneous,

text: external

text: magnetic

text: ﬁeld

text: Hoefer

text: et

text: al.

text: and

text: exhibit

text: novel

text: interaction

text: properties

text: Maiden

text: et

text: al.

text: Droplets

text: can

text: also

text: experience

text: a

text: drift

text: instability

text: in

text: NC-STOs,

text: whose

text: origin

text: is

text: not

text: well-understood

text: Hoefer

text: et

text: al.

text: Because

text: droplets

text: are

text: relatively

text: new

text: physical

text: features

text: of

text: nanomagnetic

text: systems,

text: they

text: hold

text: potential

text: for

text: applica-

text: tions

text: such

text: as

text: spintronic

text: information

text: storage

text: and

text: transfer

text: or

text: probing

text: material

text: properties.

text: Moreover,

text: their

text: fundamental

text: physics

text: are

text: not

text: well

text: understood.

text: A

text: more

text: general

text: theory

text: to

text: describe

text: the

text: motion

text: of

text: droplets

text: in

text: realistic

text: physical

text: systems

text: is

text: desirable.

text: Toward

text: this

text: end,

text: we

text: here

text: present

text: an

text: analytical

text: framework

text: for

text: investi-

text: gating

text: the

text: impact

text: of

text: a

text: large

text: class

text: of

text: physical

text: eﬀects

text: on

text: a

text: six

text: parameter

text: propagating

text: droplet

text: soliton.

text: In

text: Section

text: we

text: derive

text: an

text: approximate

text: solution

text: for

text: the

text: propagating

text: droplet

text: in

text: the

text: close

text: to

text: Zeeman

text: frequency

text: (large

text: mass)

text: and

text: small

text: velocity

text: (order

text: one

text: momentum)

text: regime,

text: which

text: greatly

text: reduces

text: the

text: complexity

text: of

text: the

text: asymptotic

text: theory

text: developed.

text: This

text: allows

text: for

text: explicit

text: ana-

text: lytical

text: results,

text: a

text: powerful

text: feature

text: of

text: soliton

text: perturbation

text: theory

text: considering

text: the

text: droplet’s

text: strongly

text: nonlinear

text: qualities.

text: The

text: main

text: result

text: of

text: this

text: paper,

text: the

text: modulation

text: equations,

text: is

text: presented

text: in

text: Section

text: utilizing

text: the

text: general

text: soli-

text: ton

text: perturbation

text: theory

text: formulation

text: from

text: Section

text: The

text: versatility

text: of

text: this

text: framework

text: is

text: subsequently

text: discussed

text: in

text: Sections

text: and

text: through

text: an

text: investiga-

text: tion

text: of

text: a

text: series

text: of

text: perturbations

text: of

text: current

text: physical

text: interest.

text: In

text: particular,

text: we

text: analytically

text: demonstrate

text: a

text: mechanism

text: leading

text: to

text: the

text: NC-STO

text: droplet

text: drift

text: instability

text: and

text: why

text: a

text: droplet

text: can

text: be

text: attracted

text: or

text: repelled

text: by

text: a

text: ferromagnetic

text: boundary.

text: The

text: model

text: under

text: study

text: here

text: is

text: the

text: perturbed

text: Landau-Lifshitz

text: equation

text: for

text: the

text: magnetization,

text: m

text: [m

text: m

text: m

text: of

text: a

text: two-dimensional

text: ferromag-

text: x

text: y

text: x

text: netic

text: ﬁlm

text: in

text: nondimensional

text: form

text: ∂m

text: R

text: S

text: m

text: h

text: m

text: eﬀ

text: ∂t

text: h

text: (h

text: m

text: )ˆz,

text: lim

text: m

text: ˆz,

text: eﬀ

text: z

text: x

text: where

text: p

text: is

text: a

text: perturbation

text: that

text: preserves

text: the

text: magnetization’s

text: length

text: (p

text: m

text: and

text: is

text: a

text: small

text: parameter

text: encoding

text: the

text: strength

text: of

text: the

text: perturbation.

text: The

text: perturbation

text: can

text: depend

text: explicitly

text: on

text: time

text: so

text: long

text: ##LTLine##


text: Equation (64) is solved by Θ = 0. Substituting the approximate solution (cid:18) 1 (cid:18) (cid:19) (cid:19) (cid:18) (cid:19) for Θ into Eq. (63) 0 − − − 1 ∂Φ − 1 ∆Φ cos(ϕ) 2 tanh ρ 1 sech ρ = 0 (65) 1 ω ∂ρ ω (cid:28) The residual in Eq. (65) is determined by two considerations. If 1 − − − ρ ω 1, sech (ρ 1/ω) dominates and the residual is exponentially small. ∼ − − In the other case, i.e. ρ ω 1, the residual will only be small if ∆Φ − − − O 1 cos(ϕ) 2 tanh (ρ 1/ω) ∂Φ1 is small since sech (ρ 1/ω) is (1). This ∂ρ ∇ suggests that the boundary condition, lim Φ = 1xˆ, may be neglected. ρ 1 2 Assuming Φ is separable of the form Φ (ρ, ϕ) = f(ρ) cos(ϕ), Eq. (65) 1 (cid:18) (cid:18) (cid:19)1 (cid:19) simpliﬁes to the ordinary diﬀerential equation d2f 1 − − 1 df − 1 + 2 tanh ρ f = 1. (66) dρ2 ρ ω dρ ρ2 (cid:0) (cid:1) Numerical solutions of Eq. (66) demonstrate that f becomes quite large, O approximately 1 near ρ = 1 . Factoring this into the analysis, we ω2 ω − change to the coordinate system R = ρ 1 and expand f in the series ω f (ρ) f (ρ) · · · f(ρ) = 0 + 1 + f (ρ) + . (67) ω2 ω 2 ≡ − Let L d2 2 tanh(R) d . Substituting the ansatz in Eq. (67) into dR2 dR Eq. (66) yields, (cid:18) (cid:19) O (cid:18)1 (cid:19) : Lf = 0 (68) ω2 0 O 1 −df : Lf = 0 (69) 1 ω dR O −df df (1) : Lf = 1 + 1 + f + R 0 (70) 2 0 dR dR Eq. (68) admits any constant solution. Take f = A. Substituting this 0 expression for f into Eq. (69), yields Lf = 0. Thus, any constant solution 0 1 is admissible for f as well. Take f = B. Substituting these expressions for 1 1 f and f into Eq. (70), yields Lf = 1+A. However, constant solutions are 0 1 2 − in the kernel of L, so solvability of this equation requires A = 1. Similarly, O − solvability at (ω) requires B = 0. Hence, we take f(ρ) = 1 , which gives ω2 rise to the form of the approximate droplet in Eq. (13). ∇ Since Φ enters into Eq. 11 only in the form of Φ, the asymptotic ap- ∇ proximations here are valid provided that Φ remains small. This condition 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Equation

text: is

text: solved

text: by

text: Θ

text: Substituting

text: the

text: approximate

text: solution

text: for

text: Θ

text: into

text: Eq.

text: ∂Φ

text: ∆Φ

text: cos(ϕ)

text: tanh

text: ρ

text: sech

text: ρ

text: ω

text: ∂ρ

text: ω

text: The

text: residual

text: in

text: Eq.

text: is

text: determined

text: by

text: two

text: considerations.

text: If

text: ρ

text: ω

text: sech

text: (ρ

text: dominates

text: and

text: the

text: residual

text: is

text: exponentially

text: small.

text: In

text: the

text: other

text: case,

text: i.e.

text: ρ

text: ω

text: the

text: residual

text: will

text: only

text: be

text: small

text: if

text: ∆Φ

text: O

text: cos(ϕ)

text: tanh

text: (ρ

text: ∂Φ1

text: is

text: small

text: since

text: sech

text: (ρ

text: is

text: This

text: ∂ρ

text: suggests

text: that

text: the

text: boundary

text: condition,

text: lim

text: Φ

text: may

text: be

text: neglected.

text: ρ

text: Assuming

text: Φ

text: is

text: separable

text: of

text: the

text: form

text: Φ

text: (ρ,

text: ϕ)

text: f(ρ)

text: cos(ϕ),

text: Eq.

text: simpliﬁes

text: to

text: the

text: ordinary

text: diﬀerential

text: equation

text: df

text: tanh

text: ρ

text: f

text: ρ

text: ω

text: dρ

text: ρ2

text: Numerical

text: solutions

text: of

text: Eq.

text: demonstrate

text: that

text: f

text: becomes

text: quite

text: large,

text: O

text: approximately

text: near

text: ρ

text: Factoring

text: this

text: into

text: the

text: analysis,

text: we

text: ω2

text: ω

text: change

text: to

text: the

text: coordinate

text: system

text: R

text: ρ

text: and

text: expand

text: f

text: in

text: the

text: series

text: ω

text: f

text: (ρ)

text: f

text: (ρ)

text: f(ρ)

text: f

text: (ρ)

text: ω2

text: ω

text: Let

text: L

text: tanh(R)

text: d

text: Substituting

text: the

text: ansatz

text: in

text: Eq.

text: into

text: dR

text: Eq.

text: yields,

text: O

text: Lf

text: ω2

text: O

text: −df

text: Lf

text: ω

text: dR

text: O

text: −df

text: df

text: Lf

text: f

text: R

text: dR

text: dR

text: Eq.

text: admits

text: any

text: constant

text: solution.

text: Take

text: f

text: A.

text: Substituting

text: this

text: expression

text: for

text: f

text: into

text: Eq.

text: yields

text: Lf

text: Thus,

text: any

text: constant

text: solution

text: is

text: admissible

text: for

text: f

text: as

text: well.

text: Take

text: f

text: B.

text: Substituting

text: these

text: expressions

text: for

text: f

text: and

text: f

text: into

text: Eq.

text: yields

text: Lf

text: However,

text: constant

text: solutions

text: are

text: in

text: the

text: kernel

text: of

text: L,

text: so

text: solvability

text: of

text: this

text: equation

text: requires

text: A

text: Similarly,

text: O

text: solvability

text: at

text: (ω)

text: requires

text: B

text: Hence,

text: we

text: take

text: f(ρ)

text: which

text: gives

text: ω2

text: rise

text: to

text: the

text: form

text: of

text: the

text: approximate

text: droplet

text: in

text: Eq.

text: Since

text: Φ

text: enters

text: into

text: Eq.

text: only

text: in

text: the

text: form

text: of

text: Φ,

text: the

text: asymptotic

text: ap-

text: proximations

text: here

text: are

text: valid

text: provided

text: that

text: Φ

text: remains

text: small.

text: This

text: condition

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ing to propagating, weakly nonlinear droplets approximated by the NLS (cid:28) Townes soliton Ivanov et al. [2001] (ii) 0 < ω 1 with zero velocity corresponding to stationary, strongly nonlinear droplets approximated by a circular domain wall Kosevich et al. [1986], Ivanov and Stephanovich [1989]. We will focus here on large amplitude propagating solitons where the magnetization is nearly reversed because experiments operate in this regime. Note, however, that the weakly nonlinear regime could also be stud- ied. The deﬁning equation for the droplet can be formulated as a bound- ary value problem by expressing the magnetization in spherical variables m = [sin(Θ) cos(Φ), sin→(Θ) sin(−Φ), cos(Θ→(cid:0))] in the fra(cid:1)me moving −and precess- ing withthe soliton Θ Θ(x ξ), Φ Φ0 + (h0 + ω)t + Φ(x ξ):   − · ∇ ∇ · ∇ sin(Θ)V Θ = sin2 Θ Φ)    sin(Θ)(ω − V · ∇Φ) = −∇2Θ + 1 sin(2Θ)(1 + |∇Φ|2) (11) 2 ∇ − lim Φ = V, lim Θ = 0. | |→∞ 2 | |→∞ x x This problem can be further simpliﬁed by exploiting the invariance of Eq. (11) under rotation of the domain to align the x-axis with the propaga- tion direction. In this coordinate system, V = V xˆ. Adding the assumptions of small frequency and propagation speed, a simple correction to the known, approximate stationary drop(cid:18)let can(cid:18)be foun(cid:19)d(cid:19)(see Appendix ). (cid:0) (cid:1) − − 1 O (cid:18) (cid:19) Θ = cos 1 tanh ρ + ω2, V 2 (12) ω − V O V Φ = Φ + (h + ω)t cos(ϕ) + . (13) 0 0 ω2 ω (cid:112) (cid:16) (cid:17) Above, (ρ, ϕ) are polar variables for the plane, whose origin is centered on − − − the droplet. That is ρ = (x ξ )2 + (y ξ )2 and ϕ = arctan y−ξy . See x y x ξx Fig. 1 for a visualization of this approximate solution. This approximation is valid so long as ≤ | | (cid:28) (cid:28) 0 V ω, 0 < ω 1. (14) As for the stationary case, the propagating droplet can be viewed as a precessing, circular domain wall with a radius that is the inverse of the − frequency. The new term V cos(ϕ)/ω2 reveals the deviation of the propa- gating droplet’s phase from spatial uniformity. While the relations in (14) may, at ﬁrst, seem overly restrictive, we will show that important and prac- tical information about propagating droplets can be obtained in this regime. This approximate solution oﬀers both an error estimate and is amenable to 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ing

text: to

text: propagating,

text: weakly

text: nonlinear

text: droplets

text: approximated

text: by

text: the

text: NLS

text: Townes

text: soliton

text: Ivanov

text: et

text: al.

text: (ii)

text: ω

text: with

text: zero

text: velocity

text: corresponding

text: to

text: stationary,

text: strongly

text: nonlinear

text: droplets

text: approximated

text: by

text: a

text: circular

text: domain

text: wall

text: Kosevich

text: et

text: al.

text: Ivanov

text: and

text: Stephanovich

text: We

text: will

text: focus

text: here

text: on

text: large

text: amplitude

text: propagating

text: solitons

text: where

text: the

text: magnetization

text: is

text: nearly

text: reversed

text: because

text: experiments

text: operate

text: in

text: this

text: regime.

text: Note,

text: however,

text: that

text: the

text: weakly

text: nonlinear

text: regime

text: could

text: also

text: be

text: stud-

text: ied.

text: The

text: deﬁning

text: equation

text: for

text: the

text: droplet

text: can

text: be

text: formulated

text: as

text: a

text: bound-

text: ary

text: value

text: problem

text: by

text: expressing

text: the

text: magnetization

text: in

text: spherical

text: variables

text: m

text: [sin(Θ)

text: cos(Φ),

text: sin→(Θ)

text: sin(−Φ),

text: in

text: the

text: moving

text: −and

text: precess-

text: ing

text: withthe

text: soliton

text: Θ

text: Θ(x

text: ξ),

text: Φ

text: Φ0

text: ω)t

text: Φ(x

text: ξ):

text: sin(Θ)V

text: Θ

text: Θ

text: Φ)

text: sin(Θ)(ω

text: V

text: ∇Φ)

text: −∇2Θ

text: lim

text: Φ

text: V,

text: lim

text: Θ

text: x

text: x

text: This

text: problem

text: can

text: be

text: further

text: simpliﬁed

text: by

text: exploiting

text: the

text: invariance

text: of

text: Eq.

text: under

text: rotation

text: of

text: the

text: domain

text: to

text: align

text: the

text: x-axis

text: with

text: the

text: propaga-

text: tion

text: direction.

text: In

text: this

text: coordinate

text: system,

text: V

text: V

text: xˆ.

text: Adding

text: the

text: assumptions

text: of

text: small

text: frequency

text: and

text: propagation

text: speed,

text: a

text: simple

text: correction

text: to

text: the

text: known,

text: approximate

text: stationary

text: Appendix

text: O

text: Θ

text: cos

text: tanh

text: ρ

text: V

text: ω

text: V

text: O

text: V

text: Φ

text: Φ

text: (h

text: ω)t

text: cos(ϕ)

text: ω2

text: ω

text: Above,

text: (ρ,

text: ϕ)

text: are

text: polar

text: variables

text: for

text: the

text: plane,

text: whose

text: origin

text: is

text: centered

text: on

text: the

text: droplet.

text: That

text: is

text: ρ

text: (x

text: ξ

text: (y

text: ξ

text: and

text: ϕ

text: arctan

text: y−ξy

text: See

text: x

text: y

text: x

text: ξx

text: Fig.

text: for

text: a

text: visualization

text: of

text: this

text: approximate

text: solution.

text: This

text: approximation

text: is

text: valid

text: so

text: long

text: as

text: V

text: ω,

text: ω

text: As

text: for

text: the

text: stationary

text: case,

text: the

text: propagating

text: droplet

text: can

text: be

text: viewed

text: as

text: a

text: precessing,

text: circular

text: domain

text: wall

text: with

text: a

text: radius

text: that

text: is

text: the

text: inverse

text: of

text: the

text: frequency.

text: The

text: new

text: term

text: V

text: reveals

text: the

text: deviation

text: of

text: the

text: propa-

text: gating

text: droplet’s

text: phase

text: from

text: spatial

text: uniformity.

text: While

text: the

text: relations

text: in

text: may,

text: at

text: ﬁrst,

text: seem

text: overly

text: restrictive,

text: we

text: will

text: show

text: that

text: important

text: and

text: prac-

text: tical

text: information

text: about

text: propagating

text: droplets

text: can

text: be

text: obtained

text: in

text: this

text: regime.

text: This

text: approximate

text: solution

text: oﬀers

text: both

text: an

text: error

text: estimate

text: and

text: is

text: amenable

text: to

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: relation in (17) holds and (15) reduces to the eigenvalue Ξ ZE = 0, CE = , (13) relation derived by Srinivasan and Young (2012) for the 2r emergence of jets in a barotropic β-plane. We consider the case of a r(cid:112)ing forcing that injects energy is a ﬁxed point of the S3T system (9) and (12) in the at rate ε at the total wavenumber K : f absence of hyperdiﬀusion (cf. Appendix B). The linear − stability of the homogeneous equilibrium can be addressed Ξˆ(k, l) = 2K δ( k2 + l2 K ), (18) f f by performing an eigenanalysis of the S3T system linearized about this equilibrium. The eigenfunctions in this case have the plane wave form and obtain the eigenvalues σ˜ by numerically solving (15). For small values of the energy input rate, σ˜ < 0 for all r wavenumbers and the homogeneous equilibrium is stable. δZ = Z einx+imyeσt , δC = C (x˜, y˜)einx+imyeσt, At a critical ε˜ the homogeneous ﬂow becomes S3T unsta- nm nm c (14) ble and exponentially growing coherent structures emerge. (cid:0) (cid:1) The critical value, ε˜ , is calculated by ﬁrst determining the − − c where x˜ = x x , x = (x + x )/2, y˜ = y y , energy input rate ε˜ (n˜, m˜ ) that renders wavenumbers (n˜, m˜ ) 1 2 1 2 1 2 t y = (y +y )/2, n and m are the x and y wavenumbers of the neutral σ˜ = 0 , and then by ﬁnding the minimum 1 2 r(n˜,m˜ ) eigenfunction and σ = σ + iσ is the eigenvalue with σ = energy input rate over all wavenumbers: ε˜ = min ε˜ . r i r c (n˜,m˜ ) t Re(σ), σi = Im(σ) being the growth rate and frequency The critical energy input rate ε˜c as a function of β˜ is shown of the mode respectively. The eigenvalue σ satisﬁes the in ﬁgure 1. In addition, the corresponding critical zonos- non-dimens(cid:90)ional(cid:90)equation: trophy parameter R = 0.7(ε˜ β˜2)1/20 which was used in β c previous studies to characterize the emergence and struc- ∞ ∞ ε˜ (cid:104) − × (cid:105) ture of zonal jets in planetary turbulence (Galperin et al. dk˜d˜l(1 N˜2/K˜ 2)Ξˆ(k˜, ˜l) 2010), is shown as a function of β˜ in ﬁgure 2. The absolute 2πr3L2 −∞ −∞ f (cid:16) (cid:17) minimum energy input rate required is ε˜ = 67 and occurs − − − c × (m˜ k˜ n˜˜l) n˜m˜ (k˜+2 ˜l+2 ) + (m˜ 2 n˜2)k˜+˜l+ at β˜min = 3.5, while the minimum zonostrophy parameter iβ˜ k˜K˜s2 − (k˜ + n˜)K˜ 2 + (σ˜ + 2)K˜ 2K˜s2 = raenqduiorcecdufrosrfothreβ˜e→me0rg.eFnocer βo˜f≤coβh˜merienn,tthﬂeowstsriuscRtuβre=s t0h.8a2t = (σ˜ + 1)N˜2 − in˜β˜, (15) ﬁrst become marginally stable are zon∼al j−ets (with→n = 0). The critical input rate increases as ε˜ β˜ 2 for β˜ 0 and c the homogeneous equilibrium is structurally stable for all where L is a characteristic length scale, σ˜ = σ/r excitation amplitudes when β˜ = 0. However, the structural f and (n˜, m˜ ) = L (n, m) are the non-dimensional eigenvalue stability for β˜ = 0 is an artifact of the assumed isotropy of f and wavenumbers respectively, ε˜ = ε/(r3L2) is the non- the excitation and the assumption of a barotropic ﬂow. In f dimensional energy injection rate of the forcing, β˜ = βL /r the presence of even the slightest anisotropy (Bakas and f Ioannou 2011, 2013b), or in the case of a stratiﬁed ﬂow is the non-dimens(cid:90)ional(cid:90)planetary vorticity gradient, (Parker and Krommes 2015), zonal jets are S3T unstable and are expected to emerge even in the absence of β. For ∞ ∞ Ξˆ(k, l) = 1 Ξ(x˜, y˜)e−ikx˜−ily˜dx˜dy˜ , (16) β˜ > β˜min, the marg∼inally stable s→tru∞ctures are non-zonal 2π −∞ −∞ and ε˜ grows as ε˜ β˜1/2 for β˜ . Since the critical c c forcing for the emergence of zonal jets (also shown in ﬁg- ∼ → ∞ is the Fourier transform of the forcing covariance, K˜ 2 = ure 1), increases as ε˜c β˜2 for β˜ , for large values of k˜2+˜l2, K˜ 2 = (k˜+n˜)2+(˜l+m˜ )2, N˜2 = n˜2+m˜ 2, k˜ = k˜+n˜/2 β˜ non-zonal structures ﬁrst emerge and only at signiﬁcantly s + and ˜l = ˜l + m˜ /2 (cf. Appendix B). For a forcing with the higher ε˜ zonal jets are expected to appear. Investigation of + − manidrroforrsn˜ym(cid:54)=m0e,ttrhyeΞˆe(igke,nvla)lu=esΞˆs(akt,ils)fyinthwearveelnautimonbse:r space tthheesreerseuslutsltsfowr iβt˜h(cid:29)ot1hearrfeorincidnegpdenisdterinbtuotifotnhseresvtreuacletdurtehaotf the forcing (Bakas et al. 2015). ∗ σ˜ − = σ˜ , and σ˜ − = σ˜ , (17) The parameter regime of S3T instability is now related ( n˜,m˜ ) (n˜,m˜ ) (n˜, m˜ ) (n˜,m˜ ) to the results of previous studies and to geophysical ﬂows. | | | | Previous studies have identiﬁed a parameter regime which implying that the growth rates depend on n˜ and m˜ . is distinguished by robust, slowly varying zonal jets as As a result, the plane wave δZ = cos(nx + my) and an well as propagating, non-dispersive, non-zonal coherent array of localized vortices δZ = cos(nx) cos(my), have the structures (Galperin et al. 2010). This regime that is termed same growth rate, despite their diﬀerent structure. For zon- as zonostrophic, is in a region in parameter space in which ally symmetric perturbations with n˜ = 0, only the second 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: relation

text: in

text: holds

text: and

text: reduces

text: to

text: the

text: eigenvalue

text: Ξ

text: ZE

text: CE

text: relation

text: derived

text: by

text: Srinivasan

text: and

text: Young

text: for

text: the

text: emergence

text: of

text: jets

text: in

text: a

text: barotropic

text: β-plane.

text: We

text: consider

text: the

text: case

text: of

text: a

text: forcing

text: that

text: injects

text: energy

text: is

text: a

text: ﬁxed

text: point

text: of

text: the

text: system

text: and

text: in

text: the

text: at

text: rate

text: ε

text: at

text: the

text: total

text: wavenumber

text: K

text: f

text: absence

text: of

text: hyperdiﬀusion

text: (cf.

text: Appendix

text: B).

text: The

text: linear

text: stability

text: of

text: the

text: homogeneous

text: equilibrium

text: can

text: be

text: addressed

text: Ξˆ(k,

text: l)

text: δ(

text: K

text: f

text: f

text: by

text: performing

text: an

text: eigenanalysis

text: of

text: the

text: system

text: linearized

text: about

text: this

text: equilibrium.

text: The

text: eigenfunctions

text: in

text: this

text: case

text: have

text: the

text: plane

text: wave

text: form

text: and

text: obtain

text: the

text: eigenvalues

text: σ˜

text: by

text: numerically

text: solving

text: For

text: small

text: values

text: of

text: the

text: energy

text: input

text: rate,

text: σ˜

text: for

text: all

text: r

text: wavenumbers

text: and

text: the

text: homogeneous

text: equilibrium

text: is

text: stable.

text: δZ

text: Z

text: einx+imyeσt

text: δC

text: C

text: (x˜,

text: y˜)einx+imyeσt,

text: At

text: a

text: critical

text: ε˜

text: the

text: homogeneous

text: ﬂow

text: becomes

text: unsta-

text: nm

text: nm

text: c

text: ble

text: and

text: exponentially

text: growing

text: coherent

text: structures

text: emerge.

text: The

text: critical

text: value,

text: ε˜

text: is

text: calculated

text: by

text: ﬁrst

text: determining

text: the

text: c

text: where

text: x˜

text: x

text: x

text: x

text: (x

text: x

text: y˜

text: y

text: y

text: energy

text: input

text: rate

text: ε˜

text: (n˜,

text: m˜

text: that

text: renders

text: wavenumbers

text: (n˜,

text: m˜

text: t

text: y

text: (y

text: +y

text: n

text: and

text: m

text: are

text: the

text: x

text: and

text: y

text: wavenumbers

text: of

text: the

text: neutral

text: σ˜

text: and

text: then

text: by

text: ﬁnding

text: the

text: minimum

text: r(n˜,m˜

text: eigenfunction

text: and

text: σ

text: σ

text: iσ

text: is

text: the

text: eigenvalue

text: with

text: σ

text: energy

text: input

text: rate

text: over

text: all

text: wavenumbers:

text: ε˜

text: min

text: ε˜

text: r

text: i

text: r

text: c

text: (n˜,m˜

text: t

text: Re(σ),

text: σi

text: Im(σ)

text: being

text: the

text: growth

text: rate

text: and

text: frequency

text: The

text: critical

text: energy

text: input

text: rate

text: ε˜c

text: as

text: a

text: function

text: of

text: β˜

text: is

text: shown

text: of

text: the

text: mode

text: respectively.

text: The

text: eigenvalue

text: σ

text: satisﬁes

text: the

text: in

text: ﬁgure

text: In

text: addition,

text: the

text: corresponding

text: critical

text: zonos-

text: trophy

text: parameter

text: R

text: which

text: was

text: used

text: in

text: β

text: c

text: previous

text: studies

text: to

text: characterize

text: the

text: emergence

text: and

text: struc-

text: ε˜

text: ture

text: of

text: zonal

text: jets

text: in

text: planetary

text: turbulence

text: (Galperin

text: et

text: al.

text: ˜l)

text: is

text: shown

text: as

text: a

text: function

text: of

text: β˜

text: in

text: ﬁgure

text: The

text: absolute

text: f

text: minimum

text: energy

text: input

text: rate

text: required

text: is

text: ε˜

text: and

text: occurs

text: c

text: (m˜

text: k˜

text: n˜˜l)

text: n˜m˜

text: (m˜

text: at

text: β˜min

text: while

text: the

text: minimum

text: zonostrophy

text: parameter

text: iβ˜

text: (k˜

text: n˜)K˜

text: (σ˜

text: (σ˜

text: in˜β˜,

text: ﬁrst

text: become

text: marginally

text: stable

text: are

text: zon∼al

text: j−ets

text: (with→n

text: The

text: critical

text: input

text: rate

text: increases

text: as

text: ε˜

text: β˜

text: for

text: β˜

text: and

text: c

text: the

text: homogeneous

text: equilibrium

text: is

text: structurally

text: stable

text: for

text: all

text: where

text: L

text: is

text: a

text: characteristic

text: length

text: scale,

text: σ˜

text: σ/r

text: excitation

text: amplitudes

text: when

text: β˜

text: However,

text: the

text: structural

text: f

text: and

text: (n˜,

text: m˜

text: L

text: (n,

text: m)

text: are

text: the

text: non-dimensional

text: eigenvalue

text: stability

text: for

text: β˜

text: is

text: an

text: artifact

text: of

text: the

text: assumed

text: isotropy

text: of

text: f

text: and

text: wavenumbers

text: respectively,

text: ε˜

text: is

text: the

text: non-

text: the

text: excitation

text: and

text: the

text: assumption

text: of

text: a

text: barotropic

text: ﬂow.

text: In

text: f

text: dimensional

text: energy

text: injection

text: rate

text: of

text: the

text: forcing,

text: β˜

text: βL

text: /r

text: the

text: presence

text: of

text: even

text: the

text: slightest

text: anisotropy

text: (Bakas

text: and

text: f

text: Ioannou

text: or

text: in

text: the

text: case

text: of

text: a

text: stratiﬁed

text: ﬂow

text: is

text: the

text: vorticity

text: gradient,

text: (Parker

text: and

text: Krommes

text: zonal

text: jets

text: are

text: unstable

text: and

text: are

text: expected

text: to

text: emerge

text: even

text: in

text: the

text: absence

text: of

text: β.

text: For

text: Ξˆ(k,

text: l)

text: Ξ(x˜,

text: y˜)e−ikx˜−ily˜dx˜dy˜

text: β˜

text: β˜min,

text: the

text: marg∼inally

text: stable

text: s→tru∞ctures

text: are

text: non-zonal

text: 2π

text: and

text: ε˜

text: grows

text: as

text: ε˜

text: for

text: β˜

text: Since

text: the

text: critical

text: c

text: c

text: forcing

text: for

text: the

text: emergence

text: of

text: zonal

text: jets

text: (also

text: shown

text: in

text: ﬁg-

text: is

text: the

text: Fourier

text: transform

text: of

text: the

text: forcing

text: covariance,

text: K˜

text: ure

text: increases

text: as

text: ε˜c

text: β˜2

text: for

text: β˜

text: for

text: large

text: values

text: of

text: K˜

text: k˜

text: β˜

text: non-zonal

text: structures

text: ﬁrst

text: emerge

text: and

text: only

text: at

text: signiﬁcantly

text: s

text: and

text: ˜l

text: ˜l

text: m˜

text: (cf.

text: Appendix

text: B).

text: For

text: a

text: forcing

text: with

text: the

text: higher

text: ε˜

text: zonal

text: jets

text: are

text: expected

text: to

text: appear.

text: Investigation

text: of

text: space

text: tthheesreerseuslutsltsfowr

text: the

text: forcing

text: (Bakas

text: et

text: al.

text: σ˜

text: σ˜

text: and

text: σ˜

text: σ˜

text: The

text: parameter

text: regime

text: of

text: instability

text: is

text: now

text: related

text: n˜,m˜

text: (n˜,m˜

text: (n˜,

text: m˜

text: (n˜,m˜

text: to

text: the

text: results

text: of

text: previous

text: studies

text: and

text: to

text: geophysical

text: ﬂows.

text: Previous

text: studies

text: have

text: identiﬁed

text: a

text: parameter

text: regime

text: which

text: implying

text: that

text: the

text: growth

text: rates

text: depend

text: on

text: n˜

text: and

text: m˜

text: is

text: distinguished

text: by

text: robust,

text: slowly

text: varying

text: zonal

text: jets

text: as

text: As

text: a

text: result,

text: the

text: plane

text: wave

text: δZ

text: cos(nx

text: my)

text: and

text: an

text: well

text: as

text: propagating,

text: non-dispersive,

text: non-zonal

text: coherent

text: array

text: of

text: localized

text: vortices

text: δZ

text: cos(nx)

text: cos(my),

text: have

text: the

text: structures

text: (Galperin

text: et

text: al.

text: This

text: regime

text: that

text: is

text: termed

text: same

text: growth

text: rate,

text: despite

text: their

text: diﬀerent

text: structure.

text: For

text: zon-

text: as

text: zonostrophic,

text: is

text: in

text: a

text: region

text: in

text: parameter

text: space

text: in

text: which

text: ally

text: symmetric

text: perturbations

text: with

text: n˜

text: only

text: the

text: second

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##