text: (cid:2) (cid:3) − (cid:62) (cid:62) (cid:62) (cid:62) κ(x y) can be written as k(x, y) = E ∼ cos(ω x) cos(ω y) + sin(ω x) sin(ω y) , where ω ∈ RD foll(cid:80)ows s(cid:16)ome distribution Λω (sΛpectral measure of(cid:17) the kernel). Thus, by sam- pling i.i.d. vectors ω , . . . , ω from Λ, we can approximate kernel k by kˆ deﬁned by: 1 N (cid:62) (cid:62) (cid:62) (cid:62) kˆ(x, y) = 1 N cos(ω x) cos(ω y) + sin(ω x) sin(ω y) , so that the original feature · N j=1 j j j j map k( , x), potentially living in an inﬁnite-dimensional space, is approximated by an ex- − plicit 2N dimensional(cid:114)feature vector: (cid:104) (cid:105) 1 (cid:62) (cid:62) (cid:62) (cid:62) T Zˆ(x) = cos(ω x), . . . , cos(ω x), sin(ω x), . . . , sin(ω x) . (5) N 1 N 1 N Thus, the mean map and the mean map kernel can be estimated using these ﬁnite- dimensional representations. In this contribution, we will focus on Gaussian RBF kernels for which the spectral measure Λ is also Gaussian. 3.3 Random features mean map on hyperspectral images Let us now turn our attention to a hyperspectral image h. Around each pixel location x , P i (s) we consider a square patch of size s where we will treat the pixels as a random sample P xi from a distribution speciﬁc to the location x . Instead of calculating the kernel between i i individual data points, we will calculate kernel between these distributions. An empirical mean map kernel is thus given simply by: (cid:88) (cid:88) (cid:104) (cid:105) K (x , x ) = µˆ , µˆ H (6) mm i j Pi Pj 1 = s2 (cid:88)∈P (cid:88)∈P k(h(xl1), h(xl2)) l1 xi l2 xj ≈ 1 (cid:62) Zˆ(h(x )) Zˆ(h(x )), s2 ∈P ∈P l1 l2 l1 xi l2 xj where h(x) denotes the measurement vector at location x and in the last line we employ a random feature approximation of k. It should be noted that there may be outliers in a patch, which can damage the estimation of the mean. Similarly to the work of [23], we proposed to use a weighted mean map, where the weights depend on spatial information. The kernels we obtain, called convolutional kernels, have also been used in [24]. In contrast to [24], however, we will use random feature expansions to explicitly represent the feature space. The convolutional kernel is deﬁned as: (cid:88) (cid:88) (cid:92) (cid:107) (cid:107) (cid:107) (cid:107) KCN(x , x ) = h(x ) h(x ) i j l1 2 l2 2 ∈P ∈P l1 xi l2 xj − (cid:107) − (cid:107) − (cid:107) − (cid:107) e 2β12 xl1 xl2 2e 2σ12 h˜(xl1) h˜(xl2) 2, 4 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: κ(x

text: y)

text: can

text: be

text: written

text: as

text: k(x,

text: y)

text: E

text: cos(ω

text: x)

text: cos(ω

text: y)

text: sin(ω

text: x)

text: sin(ω

text: y)

text: where

text: ω

text: RD

text: distribution

text: Λω

text: (sΛpectral

text: measure

text: the

text: kernel).

text: Thus,

text: by

text: sam-

text: pling

text: i.i.d.

text: vectors

text: ω

text: ω

text: from

text: Λ,

text: we

text: can

text: approximate

text: kernel

text: k

text: by

text: kˆ

text: deﬁned

text: by:

text: N

text: kˆ(x,

text: y)

text: N

text: cos(ω

text: x)

text: cos(ω

text: y)

text: sin(ω

text: x)

text: sin(ω

text: y)

text: so

text: that

text: the

text: original

text: feature

text: N

text: j

text: j

text: j

text: j

text: map

text: k(

text: x),

text: potentially

text: living

text: in

text: an

text: inﬁnite-dimensional

text: space,

text: is

text: approximated

text: by

text: an

text: ex-

text: plicit

text: vector:

text: T

text: Zˆ(x)

text: cos(ω

text: x),

text: cos(ω

text: x),

text: sin(ω

text: x),

text: sin(ω

text: x)

text: N

text: N

text: N

text: Thus,

text: the

text: mean

text: map

text: and

text: the

text: mean

text: map

text: kernel

text: can

text: be

text: estimated

text: using

text: these

text: ﬁnite-

text: dimensional

text: representations.

text: In

text: this

text: contribution,

text: we

text: will

text: focus

text: on

text: Gaussian

text: RBF

text: kernels

text: for

text: which

text: the

text: spectral

text: measure

text: Λ

text: is

text: also

text: Gaussian.

text: Random

text: features

text: mean

text: map

text: on

text: hyperspectral

text: images

text: Let

text: us

text: now

text: turn

text: our

text: attention

text: to

text: a

text: hyperspectral

text: image

text: h.

text: Around

text: each

text: pixel

text: location

text: x

text: P

text: i

text: (s)

text: we

text: consider

text: a

text: square

text: patch

text: of

text: size

text: s

text: where

text: we

text: will

text: treat

text: the

text: pixels

text: as

text: a

text: random

text: sample

text: P

text: xi

text: from

text: a

text: distribution

text: speciﬁc

text: to

text: the

text: location

text: x

text: Instead

text: of

text: calculating

text: the

text: kernel

text: between

text: i

text: i

text: individual

text: data

text: points,

text: we

text: will

text: calculate

text: kernel

text: between

text: these

text: distributions.

text: An

text: empirical

text: mean

text: map

text: kernel

text: is

text: thus

text: given

text: simply

text: by:

text: K

text: (x

text: x

text: µˆ

text: µˆ

text: H

text: mm

text: i

text: j

text: Pi

text: Pj

text: xi

text: xj

text: Zˆ(h(x

text: Zˆ(h(x

text: ∈P

text: ∈P

text: xi

text: xj

text: where

text: h(x)

text: denotes

text: the

text: measurement

text: vector

text: at

text: location

text: x

text: and

text: in

text: the

text: last

text: line

text: we

text: employ

text: a

text: random

text: feature

text: approximation

text: of

text: k.

text: It

text: should

text: be

text: noted

text: that

text: there

text: may

text: be

text: outliers

text: in

text: a

text: patch,

text: which

text: can

text: damage

text: the

text: estimation

text: of

text: the

text: mean.

text: Similarly

text: to

text: the

text: work

text: of

text: we

text: proposed

text: to

text: use

text: a

text: weighted

text: mean

text: map,

text: where

text: the

text: weights

text: depend

text: on

text: spatial

text: information.

text: The

text: kernels

text: we

text: obtain,

text: called

text: convolutional

text: kernels,

text: have

text: also

text: been

text: used

text: in

text: In

text: contrast

text: to

text: however,

text: we

text: will

text: use

text: random

text: feature

text: expansions

text: to

text: explicitly

text: represent

text: the

text: feature

text: space.

text: The

text: convolutional

text: kernel

text: is

text: deﬁned

text: as:

text: KCN(x

text: x

text: h(x

text: h(x

text: i

text: j

text: ∈P

text: ∈P

text: xi

text: xj

text: e

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: T−heorem 5 Given the conditions of the previous theorems. Then with probability at least 1 δ : (cid:2) Rˆµˆ(f)(cid:16)− R(f) ≤ ClCf2(cid:112)E(x)(cid:107)x − µx(cid:107)(cid:17)2E(cid:0)(y)(y2(cid:1))(cid:105) + 2Rn(H,P) + R(cid:114) log (1/δ)/n Eˆ (y)2 R G G 8log(2/δ) 3log(2/δ) +8 n( ) + Σ( ) + n n 5 Experiments We evaluate the classiﬁcation accuracy of the proposed approach using two standard datasets: the AVIRIS Indian Pines, and the ROSIS University of Pavia. The ﬁrst data set × is an image of dimension 145 145 pixels, with D = 224 spectral bands and its geometrical resolution is of 3.7 m. The training set is composed of 80 pixels, and the image is composed × of 16 classes. The dimensions of the second data set are 610 340 pixels, with D = 103 spectral bands and its geometrical resolution is of 1.3 m. The training set is composed of 3921 pixels and a testing set of 42776 pixels, and the image is composed of 9 classes. There is a commonly used testing set for Pavia data, and we report performance on this testing set. In the ﬁrst data set there is no testing set so we generate 20 Monte-Carlo simulations, selecting randomly 5 pixels per class, then aggregate the result of the classiﬁcation. We used the Morphological Proﬁle (MP) feature [10, 9] space which is commonly used in pixel classiﬁation and is described in the supplementary material. We also use the product of the two kernels where one is the MP kernel and the other is the KMM kernels. This kind of techinique has been previously explored in [6, 12, 7]. In contrast to the previous work, which approximate this product of kernels thanks to addition of kernels, we can do the real multiplication since we work with ﬁnite dimension Hilbert spaces. The results of classiﬁcation are reported in Table 1 (Indian Pine), and Table 2 (Pavia). The classiﬁcation algorithm used − is the C-SVM [26] were the parameter C was selected with 5 fold cross-validation on a grid ∈ − C = 2i,with i [ 15, 15]. The results on Table 1 and 2 show us that kernel mean map can perform as well as state of art results on these images. The size of scale s seems to be important, this is linked with the theorem 5, where we see that increasing the size of the scale increase in a way the size of the training set. An explanation of the parameters of evaluation can be found in [8] ( pages 166-167) and also in the supplementary materials. 6 conclusion In this article we developed a new method for pixel classiﬁcation in hyperspectral imaging. The method uses spatial information encoded through distributions of the neighbourhood around each pixel. Even with very simple kernel choices (Gaussian RBF applied directly to 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T−heorem

label: paragraph

text: Given

text: the

text: conditions

text: of

text: the

text: previous

text: theorems.

text: Then

text: with

text: probability

text: at

text: least

text: δ

label: equation

text: R(f)

text: log

text: Eˆ

text: R

text: G

text: G

text: n(

text: Σ(

text: n

text: n

label: section

text: Experiments

text: We

text: evaluate

text: the

text: classiﬁcation

text: accuracy

text: of

text: the

text: proposed

text: approach

text: using

text: two

text: standard

text: datasets:

text: the

text: AVIRIS

text: Indian

text: Pines,

text: and

text: the

text: ROSIS

text: University

text: of

text: Pavia.

text: The

text: ﬁrst

text: data

text: set

text: is

text: an

text: image

text: of

text: dimension

text: pixels,

text: with

text: D

text: spectral

text: bands

text: and

text: its

text: geometrical

text: resolution

text: is

text: of

text: m.

text: The

text: training

text: set

text: is

text: composed

text: of

text: pixels,

text: and

text: the

text: image

text: is

text: composed

text: of

text: classes.

text: The

text: dimensions

text: of

text: the

text: second

text: data

text: set

text: are

text: pixels,

text: with

text: D

text: spectral

text: bands

text: and

text: its

text: geometrical

text: resolution

text: is

text: of

text: m.

text: The

text: training

text: set

text: is

text: composed

text: of

text: pixels

text: and

text: a

text: testing

text: set

text: of

text: pixels,

text: and

text: the

text: image

text: is

text: composed

text: of

text: classes.

text: There

text: is

text: a

text: commonly

text: used

text: testing

text: set

text: for

text: Pavia

text: data,

text: and

text: we

text: report

text: performance

text: on

text: this

text: testing

text: set.

text: In

text: the

text: ﬁrst

text: data

text: set

text: there

text: is

text: no

text: testing

text: set

text: so

text: we

text: generate

text: Monte-Carlo

text: simulations,

text: selecting

text: randomly

text: pixels

text: per

text: class,

text: then

text: aggregate

text: the

text: result

text: of

text: the

text: classiﬁcation.

text: We

text: used

text: the

text: Morphological

text: Proﬁle

text: (MP)

text: feature

text: space

text: which

text: is

text: commonly

text: used

text: in

text: pixel

text: classiﬁation

text: and

text: is

text: described

text: in

text: the

text: supplementary

text: material.

text: We

text: also

text: use

text: the

text: product

text: of

text: the

text: two

text: kernels

text: where

text: one

text: is

text: the

text: MP

text: kernel

text: and

text: the

text: other

text: is

text: the

text: KMM

text: kernels.

text: This

text: kind

text: of

text: techinique

text: has

text: been

text: previously

text: explored

text: in

text: In

text: contrast

text: to

text: the

text: previous

text: work,

text: which

text: approximate

text: this

text: product

text: of

text: kernels

text: thanks

text: to

text: addition

text: of

text: kernels,

text: we

text: can

text: do

text: the

text: real

text: multiplication

text: since

text: we

text: work

text: with

text: ﬁnite

text: dimension

text: Hilbert

text: spaces.

text: The

text: results

text: of

text: classiﬁcation

text: are

text: reported

text: in

text: Table

text: (Indian

text: Pine),

text: and

text: Table

text: (Pavia).

text: The

text: classiﬁcation

text: algorithm

text: used

text: is

text: the

text: C-SVM

text: were

text: the

text: parameter

text: C

text: was

text: selected

text: with

text: fold

text: cross-validation

text: on

text: a

text: grid

text: C

text: i

text: The

text: results

text: on

text: Table

text: and

text: show

text: us

text: that

text: kernel

text: mean

text: map

text: can

text: perform

text: as

text: well

text: as

text: state

text: of

text: art

text: results

text: on

text: these

text: images.

text: The

text: size

text: of

text: scale

text: s

text: seems

text: to

text: be

text: important,

text: this

text: is

text: linked

text: with

text: the

text: theorem

text: where

text: we

text: see

text: that

text: increasing

text: the

text: size

text: of

text: the

text: scale

text: increase

text: in

text: a

text: way

text: the

text: size

text: of

text: the

text: training

text: set.

text: An

text: explanation

text: of

text: the

text: parameters

text: of

text: evaluation

text: can

text: be

text: found

text: in

text: pages

text: and

text: also

text: in

text: the

text: supplementary

text: materials.

text: conclusion

text: In

text: this

text: article

text: we

text: developed

text: a

text: new

text: method

text: for

text: pixel

text: classiﬁcation

text: in

text: hyperspectral

text: imaging.

text: The

text: method

text: uses

text: spatial

text: information

text: encoded

text: through

text: distributions

text: of

text: the

text: neighbourhood

text: around

text: each

text: pixel.

text: Even

text: with

text: very

text: simple

text: kernel

text: choices

text: (Gaussian

text: RBF

text: applied

text: directly

text: to

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: jKorea University, Department of Physics, 145 Anam-ro, Seongbuk-gu,Seoul 02841, Republic of Korea kKyungpook National University, 80 Daehak-ro, Buk-gu, Daegu 41566, Republic of Korea lUniversite de Lyon, Universite Claude Bernard Lyon 1, CNRS-IN2P3, Institut de Physique Nucleaire de Lyon, Villeurbanne, France mBenemerita Universidad Autonoma de Puebla, Puebla, Mexico nUniversidad Iberoamericana, Mexico City, Mexico oNuclear Physics Division Bhabha Atomic Research Centre Mumbai 400 085, INDIA pINFN, Sezione di Napoli, Complesso Univ. Monte S. Angelo, Via Cintia, IT-80126 Napoli, Italy qINFN, Sezione di Pavia, Via Bassi 6, IT-Pavia, Italy rSchool of Physics, Peking University, Beijing 100871, China sUniversity of Seoul, 163 Seoulsiripdae-ro, Dongdaemun-gu, Seoul, Republic of Korea tSungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon-si, Gyeonggi-do, Republic of Korea uBulgarian Academy of Sciences, Inst. for Nucl. Res. and Nucl. Energy, Tzarigradsko shaussee Boulevard 72, BG-1784 Soﬁa, Bulgaria vFaculty of Physics, University of Soﬁa,5, James Bourchier Boulevard, BG-1164 Soﬁa, Bulgaria wTbilisi University, 1 Ilia Chavchavadze Ave, Tbilisi 0179, Georgia xUniversidad de Los Andes, Apartado Aereo 4976, Carrera 1E, no. 18A 10, CO-Bogota, Colombia yDept. of Physics, Wisconsin University, Madison, WI 53706, United States zVilnius University, Vilnius, Lithuania E-mail: muhammad.gul@cern.ch Abstract: In the framework of the High Luminosity LHC upgrade program, the CMS muon group built several diﬀerent RPC prototypes that are now under test at the new CERN Gamma Irradia- tion Facility (GIF++). A dedicated Detector Control System (DCS) has been developed using the WinCC-OA tool to control and monitor these prototype detectors and to store the measured param- eters data. Preliminary eﬃciency studies that set the base performance measurements of CMS RPC for starting aging studies are also presented. Keywords: Resistive-plate chambers, Detector Control System, Radiation damage evaluation meth- ods

text: jKorea

label: author

text: University,

text: Department

text: of

text: Physics,

text: Anam-ro,

text: Seongbuk-gu,Seoul

text: Republic

text: of

text: Korea

text: kKyungpook

text: National

text: University,

text: Daehak-ro,

text: Buk-gu,

text: Daegu

text: Republic

text: of

text: Korea

text: lUniversite

text: de

text: Lyon,

text: Universite

text: Claude

text: Bernard

text: Lyon

text: Institut

text: de

text: Physique

text: Nucleaire

text: de

text: Lyon,

text: Villeurbanne,

text: France

text: mBenemerita

text: Universidad

text: Autonoma

text: de

text: Puebla,

text: Puebla,

text: Mexico

text: nUniversidad

text: Iberoamericana,

text: Mexico

text: City,

text: Mexico

text: oNuclear

text: Physics

text: Division

text: Bhabha

text: Atomic

text: Research

text: Centre

text: Mumbai

text: INDIA

text: pINFN,

text: Sezione

text: di

text: Napoli,

text: Complesso

text: Univ.

text: Monte

text: S.

text: Angelo,

text: Via

text: Cintia,

text: Napoli,

text: Italy

text: qINFN,

text: Sezione

text: di

text: Pavia,

text: Via

text: Bassi

text: IT-Pavia,

text: Italy

text: rSchool

text: of

text: Physics,

text: Peking

text: University,

text: Beijing

text: China

text: sUniversity

text: of

text: Seoul,

text: Seoulsiripdae-ro,

text: Dongdaemun-gu,

text: Seoul,

text: Republic

text: of

text: Korea

text: tSungkyunkwan

text: University,

text: Seobu-ro,

text: Jangan-gu,

text: Suwon-si,

text: Gyeonggi-do,

text: Republic

text: of

text: Korea

text: uBulgarian

text: Academy

text: of

text: Sciences,

text: Inst.

text: for

text: Nucl.

text: Res.

text: and

text: Nucl.

text: Energy,

text: Tzarigradsko

text: shaussee

text: Boulevard

text: Soﬁa,

text: Bulgaria

text: vFaculty

text: of

text: Physics,

text: University

text: of

text: James

text: Bourchier

text: Boulevard,

text: Soﬁa,

text: Bulgaria

text: wTbilisi

text: University,

text: Ilia

text: Chavchavadze

text: Ave,

text: Tbilisi

text: Georgia

text: xUniversidad

text: de

text: Los

text: Andes,

text: Apartado

text: Aereo

text: Carrera

text: no.

text: CO-Bogota,

text: Colombia

text: yDept.

text: of

text: Physics,

text: Wisconsin

text: University,

text: Madison,

text: WI

text: United

text: States

text: zVilnius

text: University,

text: Vilnius,

text: Lithuania

text: E-mail:

text: muhammad.gul@cern.ch

text: Abstract:

text: In

text: the

text: framework

text: of

text: the

text: High

text: Luminosity

text: LHC

text: upgrade

text: program,

text: the

text: CMS

text: muon

text: group

text: built

text: several

text: diﬀerent

text: RPC

text: prototypes

text: that

text: are

text: now

text: under

text: test

text: at

text: the

text: new

text: CERN

text: Gamma

text: Irradia-

text: tion

text: Facility

text: (GIF++).

text: A

text: dedicated

text: Detector

text: Control

text: System

text: (DCS)

text: has

text: been

text: developed

text: using

text: the

text: WinCC-OA

text: tool

text: to

text: control

text: and

text: monitor

text: these

text: prototype

text: detectors

text: and

text: to

text: store

text: the

text: measured

text: param-

text: eters

text: data.

text: Preliminary

text: eﬃciency

text: studies

text: that

text: set

text: the

text: base

text: performance

text: measurements

text: of

text: CMS

text: RPC

text: for

text: starting

text: aging

text: studies

text: are

text: also

text: presented.

text: Keywords:

text: Resistive-plate

text: chambers,

text: Detector

text: Control

text: System,

text: Radiation

text: damage

text: evaluation

text: meth-

text: ods


text: [10] S. C. Moreno et al., CMS RPC preliminary results for aging studies at new CERN GIF++ facility, proceedings from the XIII RPC worksop and related detectors, UGent Belgium, February, 2016. [11] The CMS Collaboration, The Muon Project Technical Design Report, CERN/LHC 97-32, CMS TDR 3, 15 December, 1997. [12] M. Abbrescia et al., Operation, performance and upgrade of the CMS Resistive Plate Chamber system at LHC, Nucl. Instr. and Meth. A 732, 195 (2013). [13] F. Thyssen et at., Performance of the Resistive Plate Chambers in the CMS experimentJINST 7 (2012) C01104. doi:10.1088/1748-0221/7/01/C01104. – 8 –

label: reference

text: S.

text: C.

text: Moreno

text: et

text: al.,

text: CMS

text: RPC

text: preliminary

text: results

text: for

text: aging

text: studies

text: at

text: new

text: CERN

text: GIF++

text: facility,

text: proceedings

text: from

text: the

text: XIII

text: RPC

text: worksop

text: and

text: related

text: detectors,

text: UGent

text: Belgium,

text: February,

text: The

text: CMS

text: Collaboration,

text: The

text: Muon

text: Project

text: Technical

text: Design

text: Report,

text: CERN/LHC

text: CMS

text: TDR

text: December,

text: M.

text: Abbrescia

text: et

text: al.,

text: Operation,

text: performance

text: and

text: upgrade

text: of

text: the

text: CMS

text: Resistive

text: Plate

text: Chamber

text: system

text: at

text: LHC,

text: Nucl.

text: Instr.

text: and

text: Meth.

text: A

text: F.

text: Thyssen

text: et

text: at.,

text: Performance

text: of

text: the

text: Resistive

text: Plate

text: Chambers

text: in

text: the

text: CMS

text: experimentJINST


text: ∗ ∗ • G has an S -tree over P , where P is any uncrossing of P . k k k k Proof. In Theorem 15, let S = S be the set of separations of order < k in G, −→ k and let F := P . Then S is a separable submodular separation system in the k k universe of all separations of G, and the regular F-tangles in G are precisely its k-proﬁles. The assertion now follows from Theorem 15, as before. Theorems 1–3 now follow easily: we just have to translate S-trees into tree- decompositions. Proof of Theorems 1–3. Given a set S of separations of G and an S-tree →− →− →− (T, α), with α( e ) =: (A ( e ), B ( e )) say, we obtain a tree-decomposition (cid:92)α α (T, V ) of G with V = (V ) ∈ by letting α α t t T −−−→ →− →− V := { B ( e ) | e = (s, t) ∈ E(T) }. t α (cid:0) (cid:83) (cid:83) (cid:1) Note that (T, α) can be recovered from this tree-decomposition: given just T and V = (V ) ∈ , we let α map each oriented edge →−e = (t , t ) of T to the t t T 1 2 oriented separation of G it induces: the separation ∈ V , ∈ V where t T1 t t T2 t T is the component of T − e containing t . i i Recall that the set T deﬁned in (2.2) contains P. Hence so does any F ⊇ T . For such F, therefore, every F-tangle −o→f Sk is a k-proﬁle, and hence is regular by Lemma 8 if k > 2. For Fk := F ∩ 2Sk and ∗ TF(k) := { (T, V ) | (T, α) is an S -tree over F } α k k we thus obtain Theorem 1 directly from Theorem 15. Similarly, letting ∗ TB(k) := { (T, V ) | (T, α) is an S -tree over B } α k k ∗ TP(k) := { (T, V ) | (T, α) is an S -tree over P } α k k yields Theorems 2 and 3 as corollaries of Theorems 16 and 17. 5 Width parameters In this section we derive some bounds for the block and proﬁle width of a (cid:84) graph that follow easily from our main results combined with those of [9, 10] and [13]. Given a star σ of separations in a graph, let us call the set { B | (A, B) ∈ σ} ∗ the interior of σ in G. For example, every star in P is of the form k {(A, B), (B ∩ C, A ∪ D), (B ∩ D, A ∪ C)} ⊆ S , k and hence every vertex of its interior lies in at least two of the separators A∩B, (B ∩ C) ∩ (A ∪ D) and (B ∩ D) ∩ (A ∪ C). Since all these separations are in S , ∗ k the interior of any star in P thus has size at most 3(k − 1)/2. k We can apply this observation to obtain the following upper bound on the proﬁle-width pw(G) of a graph G in terms of its tree-width tw(G): Theorem 18. For every graph G, 15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: list

text: G

text: has

text: an

text: S

text: -tree

text: over

text: P

text: where

text: P

text: is

text: any

text: uncrossing

text: of

text: P

text: k

text: k

text: k

text: k

text: Proof.

text: In

text: Theorem

text: let

text: S

text: S

text: be

text: the

text: set

text: of

text: separations

text: of

text: order

text: k

text: in

text: G,

text: k

text: and

text: let

text: F

text: P

text: Then

text: S

text: is

text: a

text: separable

text: submodular

text: separation

text: system

text: in

text: the

text: k

text: k

text: universe

text: of

text: all

text: separations

text: of

text: G,

text: and

text: the

text: regular

text: F-tangles

text: in

text: G

text: are

text: precisely

text: its

text: k-proﬁles.

text: The

text: assertion

text: now

text: follows

text: from

text: Theorem

text: as

text: before.

text: Theorems

text: now

text: follow

text: easily:

text: we

text: just

text: have

text: to

text: translate

text: S-trees

text: into

text: tree-

text: decompositions.

text: Proof

text: of

text: Theorems

text: Given

text: a

text: set

text: S

text: of

text: separations

text: of

text: G

text: and

text: an

text: S-tree

text: (T,

text: α),

text: with

text: α(

text: e

text: (A

text: e

text: B

text: e

text: say,

text: we

text: obtain

text: a

text: tree-decomposition

text: α

text: (T,

text: V

text: of

text: G

text: with

text: V

text: (V

text: by

text: letting

text: α

text: α

text: t

text: t

text: T

text: V

text: B

text: e

text: e

text: (s,

text: t)

text: E(T)

text: t

text: α

text: Note

text: that

text: (T,

text: α)

text: can

text: be

text: recovered

text: from

text: this

text: tree-decomposition:

text: given

text: just

text: T

text: and

text: V

text: (V

text: we

text: let

text: α

text: map

text: each

text: oriented

text: edge

text: →−e

text: (t

text: t

text: of

text: T

text: to

text: the

text: t

text: t

text: T

text: oriented

text: separation

text: of

text: G

text: it

text: induces:

text: the

text: separation

text: V

text: V

text: where

text: t

text: t

text: t

text: t

text: T

text: is

text: the

text: component

text: of

text: T

text: e

text: containing

text: t

text: i

text: i

text: Recall

text: that

text: the

text: set

text: T

text: deﬁned

text: in

text: contains

text: P.

text: Hence

text: so

text: does

text: any

text: F

text: T

text: For

text: such

text: F,

text: therefore,

text: every

text: F-tangle

text: −o→f

text: Sk

text: is

text: a

text: k-proﬁle,

text: and

text: hence

text: is

text: regular

text: by

text: Lemma

text: if

text: k

text: For

text: Fk

text: F

text: and

text: TF(k)

text: (T,

text: V

text: (T,

text: α)

text: is

text: an

text: S

text: -tree

text: over

text: F

text: α

text: k

text: k

text: we

text: thus

text: obtain

text: Theorem

text: directly

text: from

text: Theorem

text: Similarly,

text: letting

text: TB(k)

text: (T,

text: V

text: (T,

text: α)

text: is

text: an

text: S

text: -tree

text: over

text: B

text: α

text: k

text: k

text: TP(k)

text: (T,

text: V

text: (T,

text: α)

text: is

text: an

text: S

text: -tree

text: over

text: P

text: α

text: k

text: k

text: yields

text: Theorems

text: and

text: as

text: corollaries

text: of

text: Theorems

text: and

text: Width

text: parameters

text: In

text: this

text: section

text: we

text: derive

text: some

text: bounds

text: for

text: the

text: block

text: and

text: proﬁle

text: width

text: of

text: a

text: graph

text: that

text: follow

text: easily

text: from

text: our

text: main

text: results

text: combined

text: with

text: those

text: of

text: and

text: Given

text: a

text: star

text: σ

text: of

text: separations

text: in

text: a

text: graph,

text: let

text: us

text: call

text: the

text: set

text: B

text: (A,

text: B)

text: σ}

text: the

text: interior

text: of

text: σ

text: in

text: G.

text: For

text: example,

text: every

text: star

text: in

text: P

text: is

text: of

text: the

text: form

text: k

text: {(A,

text: B),

text: (B

text: C,

text: A

text: D),

text: (B

text: D,

text: A

text: C)}

text: S

text: k

text: and

text: hence

text: every

text: vertex

text: of

text: its

text: interior

text: lies

text: in

text: at

text: least

text: two

text: of

text: the

text: separators

text: A∩B,

text: (B

text: C)

text: (A

text: D)

text: and

text: (B

text: D)

text: (A

text: C).

text: Since

text: all

text: these

text: separations

text: are

text: in

text: S

text: k

text: the

text: interior

text: of

text: any

text: star

text: in

text: P

text: thus

text: has

text: size

text: at

text: most

text: k

text: We

text: can

text: apply

text: this

text: observation

text: to

text: obtain

text: the

text: following

text: upper

text: bound

text: on

text: the

text: proﬁle-width

text: pw(G)

text: of

text: a

text: graph

text: G

text: in

text: terms

text: of

text: its

text: tree-width

text: tw(G):

text: Theorem

text: For

text: every

text: graph

text: G,

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: • G has a tree-decomposition in TP(k). For readers already familiar with proﬁles [7] we remark that, in fact, we shall obtain a more general result than Theorem 3: our Theorem 15 is a duality theo- rem for all regular proﬁles in abitrary submodular abstract separation systems, including the standard ones in graphs and matroids but many others too [11]. Like Theorem 1, Theorems 2 and 3 are ‘structural’ duality theorems in that they identify a structure that a graph G cannot have if it contains a k-block or k-proﬁle, and must have if it does not. Alternatively, we can express the same duality more compactly in terms of graph invariants, as follows. Let β(G) := max { k | G has a k-block } π(G) := max { k | G has a k-proﬁle } be the block number and the proﬁle number of G, respectively, and let bw(G) := min { k | G has a tree-decomposition in TB(k + 1) } pw(G) := min { k | G has a tree-decomposition in TP(k + 1) } be its block-width and proﬁle-width. Theorems 2 and 3 can now be rephrased as Corollary 4. As invariants of ﬁnite graphs, the block and proﬁle numbers agree with the block- and proﬁle-widths: β = bw and π = pw. In Section we introduce just enough about abstract separation systems [4] to state the fundamental duality theorem of [9], on which all our proofs will be based. In Section we give a proof of our main result, a duality theorem for regular proﬁles in submodular abstract separation systems. In Section we apply this to obtain structural duality theorems for k-blocks and k-proﬁles, and deduce Theorems 1–3 as corollaries. In Section we derive some bounds for the above width-parameters in terms of tree-width and branch-width. Any terms or notation left undeﬁned in this paper are explained in [5]. 2 Background Material 2.1 Separation systems A separation of a graph G is a set {A, B} of subsets of V (G) such that A ∪ B = V , and there is no edge of G between A \ B and B \ A. There are two oriented separations associated with a separation, (A, B) and (B, A). Informally we think of (A, B) as pointing towards B and away from A. We can deﬁne a partial ordering on the set of oriented separations of G by (A, B) ≤ (C, D) if and only if A ⊆ C and B ⊇ D. 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: G

text: has

text: a

text: tree-decomposition

text: in

text: TP(k).

text: For

text: readers

text: already

text: familiar

text: with

text: proﬁles

text: we

text: remark

text: that,

text: in

text: fact,

text: we

text: shall

text: obtain

text: a

text: more

text: general

text: result

text: than

text: Theorem

text: our

text: Theorem

text: is

text: a

text: duality

text: theo-

text: rem

text: for

text: all

text: regular

text: proﬁles

text: in

text: abitrary

text: submodular

text: abstract

text: separation

text: systems,

text: including

text: the

text: standard

text: ones

text: in

text: graphs

text: and

text: matroids

text: but

text: many

text: others

text: too

text: Like

text: Theorem

text: Theorems

text: and

text: are

text: ‘structural’

text: duality

text: theorems

text: in

text: that

text: they

text: identify

text: a

text: structure

text: that

text: a

text: graph

text: G

text: cannot

text: have

text: if

text: it

text: contains

text: a

text: k-block

text: or

text: k-proﬁle,

text: and

text: must

text: have

text: if

text: it

text: does

text: not.

text: Alternatively,

text: we

text: can

text: express

text: the

text: same

text: duality

text: more

text: compactly

text: in

text: terms

text: of

text: graph

text: invariants,

text: as

text: follows.

text: Let

text: β(G)

text: max

text: k

text: G

text: has

text: a

text: k-block

text: π(G)

text: max

text: k

text: G

text: has

text: a

text: k-proﬁle

text: be

text: the

text: block

text: number

text: and

text: the

text: proﬁle

text: number

text: of

text: G,

text: respectively,

text: and

text: let

text: bw(G)

text: min

text: k

text: G

text: has

text: a

text: tree-decomposition

text: in

text: TB(k

text: pw(G)

text: min

text: k

text: G

text: has

text: a

text: tree-decomposition

text: in

text: TP(k

text: be

text: its

text: block-width

text: and

text: proﬁle-width.

text: Theorems

text: and

text: can

text: now

text: be

text: rephrased

text: as

text: Corollary

text: As

text: invariants

text: of

text: ﬁnite

text: graphs,

text: the

text: block

text: and

text: proﬁle

text: numbers

text: agree

text: with

text: the

text: block-

text: and

text: proﬁle-widths:

text: β

text: bw

text: and

text: π

text: pw.

text: In

text: Section

text: we

text: introduce

text: just

text: enough

text: about

text: abstract

text: separation

text: systems

text: to

text: state

text: the

text: fundamental

text: duality

text: theorem

text: of

text: on

text: which

text: all

text: our

text: proofs

text: will

text: be

text: based.

text: In

text: Section

text: we

text: give

text: a

text: proof

text: of

text: our

text: main

text: result,

text: a

text: duality

text: theorem

text: for

text: regular

text: proﬁles

text: in

text: submodular

text: abstract

text: separation

text: systems.

text: In

text: Section

text: we

text: apply

text: this

text: to

text: obtain

text: structural

text: duality

text: theorems

text: for

text: k-blocks

text: and

text: k-proﬁles,

text: and

text: deduce

text: Theorems

text: as

text: corollaries.

text: In

text: Section

text: we

text: derive

text: some

text: bounds

text: for

text: the

text: above

text: width-parameters

text: in

text: terms

text: of

text: tree-width

text: and

text: branch-width.

text: Any

text: terms

text: or

text: notation

text: left

text: undeﬁned

text: in

text: this

text: paper

text: are

text: explained

text: in

text: Background

text: Material

text: Separation

text: systems

text: A

text: separation

text: of

text: a

text: graph

text: G

text: is

text: a

text: set

text: {A,

text: B}

text: of

text: subsets

text: of

text: V

text: (G)

text: such

text: that

text: A

text: B

text: V

text: and

text: there

text: is

text: no

text: edge

text: of

text: G

text: between

text: A

text: B

text: and

text: B

text: A.

text: There

text: are

text: two

text: oriented

text: separations

text: associated

text: with

text: a

text: separation,

text: (A,

text: B)

text: and

text: (B,

text: A).

text: Informally

text: we

text: think

text: of

text: (A,

text: B)

text: as

text: pointing

text: towards

text: B

text: and

text: away

text: from

text: A.

text: We

text: can

text: deﬁne

text: a

text: partial

text: ordering

text: on

text: the

text: set

text: of

text: oriented

text: separations

text: of

text: G

text: by

text: (A,

text: B)

text: (C,

text: D)

text: if

text: and

text: only

text: if

text: A

text: C

text: and

text: B

text: D.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Thermodynamic Emergence of a Brownian Motor ∗ Alexander Feigel Racah Inst. of Physics, Hebrew University of Jerusalem, Israel Asaf Rozen Physics Department, Weizmann Inst. of Science, Rehovot Israel (Dated: May 29, 2020) Human-created engines and evolutionarily optimized molecular motors exhibit sophisticated de- sign in order to harvest chemical or thermal energy for generating unidirectional motion. The complexity of these motors makes their random emergence unlikely. Molecules capable of loco- motion, however, seem to be essential to the creation of the ﬁrst self-replicator and initiation of Darwinian evolution, posing a question of the physical mechanism that can facilitate emergence of directed motion in an isotropic environment. Here we show a universal thermodynamic mechanism for spontaneous emergence of motor abilities in a mechanical system. A non-equilibrium system with multiple degrees of freedom develops symmetry breaking that favors rectiﬁcation of environmental thermal ﬂuctuations. The corresponding velocities and its ﬂuctuations are calculated. Homochi- rality of living matter is explained as chirality breaking resulting from the emergence of a motor. Universality of the results provides a general extension of the Onsager relations to the non-linear regime. PACS numbers: Thermal ﬂuctuations of the environment cause random ∂x /∂t = L F , where x are degrees of freedom and i ij j i motion of microscopic particles. This Brownian motion F are the corresponding mechanical or thermodynamic j can be rectiﬁed: for instance, two connected particles forces[23–25]. Thus emergence of rectiﬁcation is anal- (cid:54) exhibit unidirectional motion given that their local en- ogous to emergence of Onsager coeﬃcients L = 0 in 12 vironments possess diﬀerent temperatures[1], see Fig 1. relations: Motion emerges in this non-equilibrium system if, in ad- dition to the temperature diﬀerence, at least one of the ∂x ∆T ∂Q ∆T particles is asymmetric. One can then ask a question: = L F + L , = L F + L , (1) if the particles can be in either symmetric or asymmet- ∂t 11 X 12 T2 ∂t 21 X 22 T2 ric state, will thermodynamic forces favor a transition to the asymmetric state and, therefore, the emergence of where x is translation coordinate, F is the correspond- j motion? Answering this question may shed light on a ing mechanical force, Q is thermal ﬂux and ∆T/T2 is the major biological and evolutionary puzzle, since rectiﬁed corresponding thermodynamic force. This set of equa- Brownian motion is believed to provide a major mecha- tion describes a motor that generates motion along axis nism for directed molecular motion in living cells[2]. x using thermal ﬂux Q between two thermal bathes with Rectiﬁcation of thermal ﬂuctuations was ﬁrst pro- temperature diﬀerence ∆T. posed by M. Smoluchowski[3] and popularized by In general, emergence of Onsager coeﬃcients can be R. Feynman[4] as the Brownian ratchet. The explained by the transition of a system to the correspond- ﬁrst theoretical treatment of the relevant phenomena ing state due to the minimization of energy or a random in semiconductors[5–7] was followed by the seminal event. These explanations, however, are debatable when works[8–18] motivated by an analogy between Brown- applied in the Origin of Life. The simplest modules of ian ratchets and the mechanical motion of proteins in living systems, e.g. molecular motors[26], seem to be too living matter (for review, see[19–21]). Motion is es- complex to emerge by chance. In addition to that, emer- sential for transcribing and translating nucleotide se- gence by chance or by energy minimization predicts equal quences into functional proteins; it also plays key roles probability for either L or R chiralities of bio-molecules. in self-replication and cellular motility. Recent exper- Al living matter, on the contrary, is composed of L chiral iments have indicated the presence of multiple ratchet- aminoacids and R chiral sugars[27, 28]. like processes in the proteins responsible for these cellular Emergence of Onsager coeﬃcients L may occur in the ij functions[22]. case of non-linear Onsager relations, ∂x /∂t = L (x )F , i ij k j where coeﬃcients L depend on the degrees of freedom Models of the Brownian motors ﬁt Onsager formalism ij x . In this case, multiple steady states may exist. Devel- k opment of motion corresponds to convergence to a steady state with ﬁnite Onsager coeﬃcient for the rectiﬁcation (cid:54) ∗ of thermal ﬂuctuations, e.g. L = 0 in (1). Linear On- Electronic address: \markcomm{footercolor}{sasha@phys.huji. 12 ac.il} sager relation possesses many universal properties such ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Thermodynamic

label: title

text: Emergence

text: of

text: a

text: Brownian

text: Motor

text: Alexander

text: Feigel

text: Racah

text: Inst.

text: of

text: Physics,

text: Hebrew

text: University

text: of

text: Jerusalem,

text: Israel

text: Asaf

text: Rozen

text: Physics

text: Department,

text: Weizmann

text: Inst.

text: of

text: Science,

text: Rehovot

text: Israel

text: (Dated:

text: May

label: date

text: Human-created

label: abstract

text: engines

text: and

text: evolutionarily

text: optimized

text: molecular

text: motors

text: exhibit

text: sophisticated

text: de-

text: sign

text: in

text: order

text: to

text: harvest

text: chemical

text: or

text: thermal

text: energy

text: for

text: generating

text: unidirectional

text: motion.

text: The

text: complexity

text: of

text: these

text: motors

text: makes

text: their

text: random

text: emergence

text: unlikely.

text: Molecules

text: capable

text: of

text: loco-

text: motion,

text: however,

text: seem

text: to

text: be

text: essential

text: to

text: the

text: creation

text: of

text: the

text: ﬁrst

text: self-replicator

text: and

text: initiation

text: of

text: Darwinian

text: evolution,

text: posing

text: a

text: question

text: of

text: the

text: physical

text: mechanism

text: that

text: can

text: facilitate

text: emergence

text: of

text: directed

text: motion

text: in

text: an

text: isotropic

text: environment.

text: Here

text: we

text: show

text: a

text: universal

text: thermodynamic

text: mechanism

text: for

text: spontaneous

text: emergence

text: of

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text: 2 Here is calculation of transition rate probability viscous coeﬃcient for the corresponding degree of free- (cid:48) (cid:48) W(V, V ) from velocity V to any other velocity V for dom. a macroscopic body of general 2D convex shape in an To ﬁnd velocity of the body V after collision as a func- (cid:48) (cid:48) ideal gas[36]. The body possesses mass M and can move tion of a gas particle velocity prior the collision (v , v ) x y along axis x only. The gas consists of identical particles let us rewrite energy conservation (17) as: of mass m at density ρ and temperature T. The veloci- ties of gas particles (v , v ) follow Maxwell distribution. x y − − − − In the case body is coupled to the several thermal bathes MV 2 m∆v2 m∆v2 (23) the ﬁnal W is the sum over the thermal bathes. (cid:48) − x (cid:48) y This work highlights universality of W for diﬀerent sys- 2mvx∆vx 2mvy∆vy = 0, tems. It allows further extension of the method to the systems with multiple degrees of freedom. Consider collision of the body with a particle of the It can be rewritten further using (18) and expressing (cid:48) gas. In the frame of the reference of the body (V = 0), ∆v and ∆v as a functions of V : x y conservation of energy before and after the collision is: − − − M V (cid:48) − (cid:48) 1 MV 2 + (17) V m Γ2 (c) + 2vx 2vy tan θ = 0, (24) (cid:48) − (cid:48) (cid:48) − (cid:48) V m(v v )(v + v ) + m(v v )(v + v ) = 0, x x x x y y y y where Γ = sin θ. The ﬁnal formula is: (cid:48) V (cid:20) (cid:21) where indicates velocities before collision. Momen- tum conservation along x axis is: 2m/MΓ2 (c) (cid:48) − (cid:48) 1 V = V v v , (25) − 1 + m/MΓ2 (c) x y tan θ m∆v = MV, (18) V x Transition from the frame of reference of the body to where: the laboratory frame of reference: (cid:20) (cid:21) − (cid:48) ∆v = v v , (19) x x x (cid:48) 2m/MΓ2 (c) (cid:48) − (cid:48) − (cid:48) 1 V = V + V v V v , (26) 1 + m/MΓ2 (c) x y tan θ V In addition, in the case of instantaneous collision, mo- mentum of the gas particle conserves along surface of − (cid:48) requires addition of V to the velocities along x co- body: ordinate: (cid:48) → Taking into account (26), transition probability V (cid:48) (cid:48) V is: v cos θ + v sin θ = v cos θ + v sin θ, (20) (cid:90) (cid:90) x y x y ∞ ∞ where θ is the angle between surface of the body and (cid:48) (cid:48) (cid:48) dW(V V ) = SF(θ)dθ dv dv (27) axi−s x.(cid:48) In addition, it is convenient to express ∆vy = (cid:20) −∞ x (cid:20)−∞ y (cid:21)(cid:21) v v as a function of V : (cid:48) − (cid:48) × (cid:48) − (cid:48) (cid:48) (cid:48) × y y H[(V v )e⊥] (V v )e⊥ρφ(v , v ) x y × − (cid:48) − 2m/MΓ2 (c) (cid:48) − (cid:48) − (cid:48) 1 M V δ V V V v V v , ∆v = . (21) 1 + m/MΓ2 (c) x y tan θ y m tan θ V where H is Heaviside step function and φ is Maxwell It follows from (17) and (20) is: distribution: An important consequence of the conservation laws (cid:32) (cid:33) (18) and (21) is: − (cid:48) (cid:48) (cid:48) (cid:48) m m(v 2 + v 2) M2 V 2 φ(v , v ) = exp x y , (28) ∆v2 + ∆v2 = , (22) x y 2πT 2T x y m2 sin2 θ In this expression we will see that geometric factor in The average is calculated using Hadamard the right part of this expression universally deﬁnes linear Stratonovich transformation. The result is: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Here

text: is

text: calculation

text: of

text: transition

text: rate

text: probability

text: viscous

text: coeﬃcient

text: for

text: the

text: corresponding

text: degree

text: of

text: free-

text: W(V,

text: V

text: from

text: velocity

text: V

text: to

text: any

text: other

text: velocity

text: V

text: for

text: dom.

text: a

text: macroscopic

text: body

text: of

text: general

text: convex

text: shape

text: in

text: an

text: To

text: ﬁnd

text: velocity

text: of

text: the

text: body

text: V

text: after

text: collision

text: as

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text: func-

text: ideal

text: The

text: body

text: possesses

text: mass

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text: and

text: can

text: move

text: tion

text: of

text: a

text: gas

text: particle

text: velocity

text: prior

text: the

text: collision

text: (v

text: v

text: x

text: y

text: along

text: axis

text: x

text: only.

text: The

text: gas

text: consists

text: of

text: identical

text: particles

text: let

text: us

text: rewrite

text: energy

text: conservation

text: as:

text: of

text: mass

text: m

text: at

text: density

text: ρ

text: and

text: temperature

text: T.

text: The

text: veloci-

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text: v

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text: distribution.

text: x

text: y

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text: the

text: case

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text: is

text: coupled

text: to

text: the

text: several

text: thermal

text: bathes

text: MV

text: the

text: ﬁnal

text: W

text: is

text: the

text: sum

text: over

text: the

text: thermal

text: bathes.

text: x

text: y

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text: work

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text: universality

text: of

text: W

text: for

text: diﬀerent

text: sys-

text: tems.

text: It

text: allows

text: further

text: extension

text: of

text: the

text: method

text: to

text: the

text: systems

text: with

text: multiple

text: degrees

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text: freedom.

text: Consider

text: collision

text: of

text: the

text: body

text: with

text: a

text: particle

text: of

text: the

text: It

text: can

text: be

text: rewritten

text: further

text: using

text: and

text: expressing

text: gas.

text: In

text: the

text: frame

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text: reference

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text: the

text: body

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text: The

text: ﬁnal

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text: collision.

text: Momen-

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text: conservation

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text: (c)

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text: Key words: instabilities - protoplanetary discs Drag-driven instability of a dust layer in a magnetized protoplanetary disc Mohsen Shadmehri1, Razieh Oudi2, Gohar Rastegarzade2 1 Department of Physics, Faculty of Sciences, Golestan University, Gorgan 49138-15739, Iran 2 Department of Physics,Semnan University, Semnan 35196-45399, Iran Abstract We study drag-driven instability in a protoplanetary disc consisting of a layer of single-sized dust particles which are coupled to the magnetized gas aerodynamically and the particle-to-gas feedback is included. We ﬁnd a dispersion relation for axisymmet- ric linear disturbances and growth rate of the unstable modes are calculated numerically. While the secular gravitational instability in the absence of particle-to-gas feedback pre- dicts the dust layer is unstable, magnetic ﬁelds signiﬁcantly ampliﬁes the instability if the Toomre parameter for the gas component is ﬁxed. We also show that even a weak magnetic ﬁeld is able to amplify the instability more or less irrespective of the dust-gas coupling. 1 INTRODUCTION While the outer parts of protoplanetary discs are prone to the gravitational instability, the inner parts are stable to the gravitational perturbations (e.g., Raﬁkov 2005). It is known that the onset of the gravita- tional instability in an accretion disc occurs when the Toomre parameters becomes less than a threshold value around unity and the survival of the newly formed fragments is guaranteed when the cooling time-scale is less than a few dynamical time-scale (e.g., Gammie 2001). Although the dissipationless gravitational instability is able to explain some of the observational features of structure formation in the protoplanetary discs (e.g., Matzner & Levin 2005), presence of the dust particles can introduce new physical mechanisms in order dust particles clump together to form larger objects that may eventually growth into planet embryos (e.g., Chiang & Youdin 2010). This is mainly because of the interactions between dust particles and the gas. Drag force is proportional to the relative velocity of dust and gas components. But the effect of this exchange of momentum is much stronger on the dust component simply because mass of gas is much larger than the total mass of dust particles. Dynamics of dust par- ticles in a protoplanetray disc is not necessarily the same as gas component. They rotate slower than the local Keplerian velocity because of the pressure gradient which acts opposite to the direction of the central gravitational force. But an individual particle does not accelerate by the pressure gradient when its internal density is much larger than gas density and thereby dust particles rotate at full Keplerian velocity. Although a few authors had already studied gravitational stability of accretion discs consisting of dust particles and gas (e.g., Coradini et al. 1981; Noh et al. 1991), during recent years speciﬁc types

text: Key

text: words:

text: instabilities

text: protoplanetary

text: discs

text: Drag-driven

text: instability

text: of

text: a

text: dust

text: layer

text: in

text: a

text: magnetized

text: protoplanetary

text: disc

text: Mohsen

text: Razieh

text: Gohar

text: Department

text: of

text: Physics,

text: Faculty

text: of

text: Sciences,

text: Golestan

text: University,

text: Gorgan

text: Iran

text: Department

text: of

text: Physics,Semnan

text: University,

text: Semnan

text: Iran

text: Abstract

text: We

text: study

text: drag-driven

text: instability

text: in

text: a

text: protoplanetary

text: disc

text: consisting

text: of

text: a

text: layer

text: of

text: single-sized

text: dust

text: particles

text: which

text: are

text: coupled

text: to

text: the

text: magnetized

text: gas

text: aerodynamically

text: and

text: the

text: particle-to-gas

text: feedback

text: is

text: included.

text: We

text: ﬁnd

text: a

text: dispersion

text: relation

text: for

text: axisymmet-

text: ric

text: linear

text: disturbances

text: and

text: growth

text: rate

text: of

text: the

text: unstable

text: modes

text: are

text: calculated

text: numerically.

text: While

text: the

text: secular

text: gravitational

text: instability

text: in

text: the

text: absence

text: of

text: particle-to-gas

text: feedback

text: pre-

text: dicts

text: the

text: dust

text: layer

text: is

text: unstable,

text: magnetic

text: ﬁelds

text: signiﬁcantly

text: ampliﬁes

text: the

text: instability

text: if

text: the

text: Toomre

text: parameter

text: for

text: the

text: gas

text: component

text: is

text: ﬁxed.

text: We

text: also

text: show

text: that

text: even

text: a

text: weak

text: magnetic

text: ﬁeld

text: is

text: able

text: to

text: amplify

text: the

text: instability

text: more

text: or

text: less

text: irrespective

text: of

text: the

text: dust-gas

text: coupling.

text: INTRODUCTION

text: While

text: the

text: outer

text: parts

text: of

text: protoplanetary

text: discs

text: are

text: prone

text: to

text: the

text: gravitational

text: instability,

text: the

text: inner

text: parts

text: are

text: stable

text: to

text: the

text: gravitational

text: perturbations

text: (e.g.,

text: Raﬁkov

text: It

text: is

text: known

text: that

text: the

text: onset

text: of

text: the

text: gravita-

text: tional

text: instability

text: in

text: an

text: accretion

text: disc

text: occurs

text: when

text: the

text: Toomre

text: parameters

text: becomes

text: less

text: than

text: a

text: threshold

text: value

text: around

text: unity

text: and

text: the

text: survival

text: of

text: the

text: newly

text: formed

text: fragments

text: is

text: guaranteed

text: when

text: the

text: cooling

text: time-scale

text: is

text: less

text: than

text: a

text: few

text: dynamical

text: time-scale

text: (e.g.,

text: Gammie

text: Although

text: the

text: dissipationless

text: gravitational

text: instability

text: is

text: able

text: to

text: explain

text: some

text: of

text: the

text: observational

text: features

text: of

text: structure

text: formation

text: in

text: the

text: protoplanetary

text: discs

text: (e.g.,

text: Matzner

text: Levin

text: presence

text: of

text: the

text: dust

text: particles

text: can

text: introduce

text: new

text: physical

text: mechanisms

text: in

text: order

text: dust

text: particles

text: clump

text: together

text: to

text: form

text: larger

text: objects

text: that

text: may

text: eventually

text: growth

text: into

text: planet

text: embryos

text: (e.g.,

text: Chiang

text: Youdin

text: This

text: is

text: mainly

text: because

text: of

text: the

text: interactions

text: between

text: dust

text: particles

text: and

text: the

text: gas.

text: Drag

text: force

text: is

text: proportional

text: to

text: the

text: relative

text: velocity

text: of

text: dust

text: and

text: gas

text: components.

text: But

text: the

text: effect

text: of

text: this

text: exchange

text: of

text: momentum

text: is

text: much

text: stronger

text: on

text: the

text: dust

text: component

text: simply

text: because

text: mass

text: of

text: gas

text: is

text: much

text: larger

text: than

text: the

text: total

text: mass

text: of

text: dust

text: particles.

text: Dynamics

text: of

text: dust

text: par-

text: ticles

text: in

text: a

text: protoplanetray

text: disc

text: is

text: not

text: necessarily

text: the

text: same

text: as

text: gas

text: component.

text: They

text: rotate

text: slower

text: than

text: the

text: local

text: Keplerian

text: velocity

text: because

text: of

text: the

text: pressure

text: gradient

text: which

text: acts

text: opposite

text: to

text: the

text: direction

text: of

text: the

text: central

text: gravitational

text: force.

text: But

text: an

text: individual

text: particle

text: does

text: not

text: accelerate

text: by

text: the

text: pressure

text: gradient

text: when

text: its

text: internal

text: density

text: is

text: much

text: larger

text: than

text: gas

text: density

text: and

text: thereby

text: dust

text: particles

text: rotate

text: at

text: full

text: Keplerian

text: velocity.

text: Although

text: a

text: few

text: authors

text: had

text: already

text: studied

text: gravitational

text: stability

text: of

text: accretion

text: discs

text: consisting

text: of

text: dust

text: particles

text: and

text: gas

text: (e.g.,

text: Coradini

text: et

text: al.

text: Noh

text: et

text: al.

text: during

text: recent

text: years

text: speciﬁc

text: types


text: Drag-driven instability 11 Bz0(mG) ∞λ Θ (cid:15) f QM 0 1 1 1 1.99 5 4.35 1.05 0.94 1 2.16 10 2.17 1.21 0.78 1 2.79 15 1.45 1.48 0.52 1 4.9 Table 1 Input parameters for different values of the initial vertical magnetic ﬁeld B . z0 − × − 1 B N − (cid:15) = 1 1.36 10 1( )( z0 )2( H ) 2, (47) µ2 mG 1024cm2 QM = 2.12((cid:15)Θµ1232 )(10−2kmΩs−1AU−1 )(1TK)12 (102N4cHm2 )−1, (48) where µ is the molecular weight, N is the hydrogen column density, T is the gas temperature and H Ω is the angular velocity. 4 ANALYSIS We can now investigate axisymmetric unstable modes for different sets of the input parameters in order to explore possible effects of the magnetic ﬁeld on the drag-driven instability. Nonzero values for m do not affect essentially behavior of the unstable solutions. In Figures 1 and 2, we assume H = × − 143AU, T = 250 K, µ = 2.33 and N = 3.46 1024cm 2 (Lizano et al. 2010). Corresponding to H these input parameters, one can calculate the other parameters based on equations (45)-(48) for different values of the initial vertical magnetic ﬁeld (Table 1). We ﬁrst examine the dependence of the growth rate on the grain size, or dimensionless stopping time. Figure 1 shows unstable growth rate for particles with × − different sizes, ranging from strongly coupled particles with dimensionless stopping time τ = 5 10 3 − − (top panel) and τ = 10 2 (middle panel) to a slightly less coupled case with τ = 10 1 (bottom panel). Each curve is labeled by its corresponding value of B . In this ﬁgure, the standard value of disc z0 − metallicity is adopted, i.e. Z = 0.01, and Toomre parameter is Q = 6 and α = 10 6. Here, different M values of τ are considered. Note that our adopted value of the Toomre parameter is greater than the threshold of the instability which means that the system is stable in the absence of dust particles. For some of the input parameters, we found two unstable roots where one root is much smaller than the other one. We actually displayed both roots, though the larger root which speciﬁes the most unstable root is more interesting. We consider different values of the disc metalicity (i.e., Z = 0.01 and Z = 0.1) in our analysis to explore possible effects of its variations on the instability. Figure 1 shows that the instability occurs in the presence of the magnetic ﬁelds, and as the strength of the magnetic ﬁeld increases the instability grows faster. This feature is understandable by the fact that the critical value of the Toomre parameter for the instability in the absence of the dust particles increase with the magnetic ﬁeld (see Table 1). Since we assume a ﬁxed value for the Toomre parameter, the system becomes closer to the threshold of the instability with increasing the magnetic ﬁeld. Figure 2 is same as Figure 1, but the disc metallicity is larger, i.e. Z = 0.1. Again, the system is unstable in the presence of the dust particles. Figure 1 shows that when the disc contains aerodynamically well-coupled dust particles, even weak ##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: Drag-driven

text: instability

label: table

text: ∞λ

text: Θ

text: f

text: QM

text: Table

text: Input

label: caption

text: parameters

text: for

text: different

text: values

text: of

text: the

text: initial

text: vertical

text: magnetic

text: ﬁeld

text: B

text: B

text: N

text: H

text: µ2

text: mG

text: QM

text: where

text: µ

text: is

text: the

text: molecular

text: weight,

text: N

text: is

text: the

text: hydrogen

text: column

text: density,

text: T

text: is

text: the

text: gas

text: temperature

text: and

text: H

text: Ω

text: is

text: the

text: angular

text: velocity.

text: ANALYSIS

text: We

text: can

text: now

text: investigate

text: axisymmetric

text: unstable

text: modes

text: for

text: different

text: sets

text: of

text: the

text: input

text: parameters

text: in

text: order

text: to

text: explore

text: possible

text: effects

text: of

text: the

text: magnetic

text: ﬁeld

text: on

text: the

text: drag-driven

text: instability.

text: Nonzero

text: values

text: for

text: m

text: do

text: not

text: affect

text: essentially

text: behavior

text: of

text: the

text: unstable

text: solutions.

text: In

text: Figures

text: and

text: we

text: assume

text: H

text: T

text: K,

text: µ

text: and

text: N

text: (Lizano

text: et

text: al.

text: Corresponding

text: to

text: H

text: these

text: input

text: parameters,

text: one

text: can

text: calculate

text: the

text: other

text: parameters

text: based

text: on

text: equations

text: for

text: different

text: values

text: of

text: the

text: initial

text: vertical

text: magnetic

text: ﬁeld

text: (Table

text: We

text: ﬁrst

text: examine

text: the

text: dependence

text: of

text: the

text: growth

text: rate

text: on

text: the

text: grain

text: size,

text: or

text: dimensionless

text: stopping

text: time.

text: Figure

text: shows

text: unstable

text: growth

text: rate

text: for

text: particles

text: with

text: different

text: sizes,

text: ranging

text: from

text: strongly

text: coupled

text: particles

text: with

text: dimensionless

text: stopping

text: time

text: τ

text: (top

text: panel)

text: and

text: τ

text: (middle

text: panel)

text: to

text: a

text: slightly

text: less

text: coupled

text: case

text: with

text: τ

text: (bottom

text: panel).

text: Each

text: curve

text: is

text: labeled

text: by

text: its

text: corresponding

text: value

text: of

text: B

text: In

text: this

text: ﬁgure,

text: the

text: standard

text: value

text: of

text: disc

text: metallicity

text: is

text: adopted,

text: i.e.

text: Z

text: and

text: Toomre

text: parameter

text: is

text: Q

text: and

text: α

text: Here,

text: different

text: M

text: values

text: of

text: τ

text: are

text: considered.

text: Note

text: that

text: our

text: adopted

text: value

text: of

text: the

text: Toomre

text: parameter

text: is

text: greater

text: than

text: the

text: threshold

text: of

text: the

text: instability

text: which

text: means

text: that

text: the

text: system

text: is

text: stable

text: in

text: the

text: absence

text: of

text: dust

text: particles.

text: For

text: some

text: of

text: the

text: input

text: parameters,

text: we

text: found

text: two

text: unstable

text: roots

text: where

text: one

text: root

text: is

text: much

text: smaller

text: than

text: the

text: other

text: one.

text: We

text: actually

text: displayed

text: both

text: roots,

text: though

text: the

text: larger

text: root

text: which

text: speciﬁes

text: the

text: most

text: unstable

text: root

text: is

text: more

text: interesting.

text: We

text: consider

text: different

text: values

text: of

text: the

text: disc

text: metalicity

text: (i.e.,

text: Z

text: and

text: Z

text: in

text: our

text: analysis

text: to

text: explore

text: possible

text: effects

text: of

text: its

text: variations

text: on

text: the

text: instability.

text: Figure

text: shows

text: that

text: the

text: instability

text: occurs

text: in

text: the

text: presence

text: of

text: the

text: magnetic

text: ﬁelds,

text: and

text: as

text: the

text: strength

text: of

text: the

text: magnetic

text: ﬁeld

text: increases

text: the

text: instability

text: grows

text: faster.

text: This

text: feature

text: is

text: understandable

text: by

text: the

text: fact

text: that

text: the

text: critical

text: value

text: of

text: the

text: Toomre

text: parameter

text: for

text: the

text: instability

text: in

text: the

text: absence

text: of

text: the

text: dust

text: particles

text: increase

text: with

text: the

text: magnetic

text: ﬁeld

text: (see

text: Table

text: Since

text: we

text: assume

text: a

text: ﬁxed

text: value

text: for

text: the

text: Toomre

text: parameter,

text: the

text: system

text: becomes

text: closer

text: to

text: the

text: threshold

text: of

text: the

text: instability

text: with

text: increasing

text: the

text: magnetic

text: ﬁeld.

text: Figure

text: is

text: same

text: as

text: Figure

text: but

text: the

text: disc

text: metallicity

text: is

text: larger,

text: i.e.

text: Z

text: Again,

text: the

text: system

text: is

text: unstable

text: in

text: the

text: presence

text: of

text: the

text: dust

text: particles.

text: Figure

text: shows

text: that

text: when

text: the

text: disc

text: contains

text: aerodynamically

text: well-coupled

text: dust

text: particles,

text: even

text: weak

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text: 10 M. Shadmehri, R. Oudi, G. Rastegarzade Σ y [x + ξy2] 1d + i u = 0, (41) ZΣ c 1d 0 s 1 − 2 y − u [x + ]u v + i V 1 = 0, (42) fτ 1d f 1d c 1 fτ s 1 f − v [x + ]v + u 1 = 0, (43) fτ 1d 2 1d fτ (cid:112) V + 2Θ12 c2s ( Σ1 + Σ1d ) = 0. (44) 1 Q (cid:15)Σ y 1 + y 1 + αy M 0 τ Thus, we have eight equations and eight unknowns, i.e. Σ , Σ , v , u , v , u , V , B . Since the 1 1d 1 1 1d 1d 1 1z above linearized equations are valid for the perturbations with a wavelength much smaller than radial (cid:29) (cid:29) distance, i.e. kr 1, then we can consider perturbations which satisfy this inequality: kH = y H/r. For instance, in a thin disc with H/r = 0.1, we consider only perturbations which are larger than this (cid:29) value, i.e. y 0.1. In addition to this constraint of local approximation, the validity of the vertically integrated equations requires ku < Ω (e.g., Wu & Li 1996; Kato et al. 1996). this requirement can be − K (cid:28) written as kH < (αH/r) 1 where α is the disc viscosity. Thus, our analysis is valid for (H/r) y < − (αH/r) 1. If we set α = 0.01 and H/r = 0.1, then the valid range of nondimensional wavenumber (cid:28) becomes 0.1 y < 103. Also, validity of vertical integrated equations implies that the growth rates of the unstable modes are less than angular velocity (Kato et al. 1996). This requirement is justiﬁed by the unstable modes as we will show. Existence of a set of nontrivial solutions for the above linearized equations imply that the deter- minant of the coefﬁcients becomes zero which gives us an algebraic equation involving the input pa- rameters, growth rate and the wavenumber of the perturbations. Using MAPLE software, we found the dispersion relation. But the equation is very lengthy, and so, we do not bring it here. However, our analysis is based on the roots of this equation which can be calculated numerically. Obviously, unstable modes correspond to the roots with positive real part, i.e. Re(x) > 0. We generally found one or two unstable modes for a given set of the input parameters. It is useful to re-write the input parameters as follows (Lizano et al. 2010): N B − λ = 2.71µ( H )( z0 ) 1, (45) 1024cm2 1mG × − B H N − T − Θ = 1 + 1.15 10 2( z0 )2( )( H ) 1( ) 1, (46) 1mG 1AU 1024cm2 1K ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: M.

text: Shadmehri,

text: R.

text: Oudi,

text: G.

text: Rastegarzade

text: Σ

text: y

text: [x

text: i

text: u

text: ZΣ

text: c

text: s

text: y

text: u

text: [x

text: ]u

text: v

text: i

text: V

text: fτ

text: f

text: c

text: fτ

text: s

text: f

text: v

text: [x

text: ]v

text: u

text: fτ

text: fτ

text: V

text: Σ1

text: Q

text: y

text: y

text: αy

text: M

text: τ

text: Thus,

text: we

text: have

text: eight

text: equations

text: and

text: eight

text: unknowns,

text: i.e.

text: Σ

text: Σ

text: v

text: u

text: v

text: u

text: V

text: B

text: Since

text: the

text: above

text: linearized

text: equations

text: are

text: valid

text: for

text: the

text: perturbations

text: with

text: a

text: wavelength

text: much

text: smaller

text: than

text: radial

text: distance,

text: i.e.

text: kr

text: then

text: we

text: can

text: consider

text: perturbations

text: which

text: satisfy

text: this

text: inequality:

text: kH

text: y

text: H/r.

text: For

text: instance,

text: in

text: a

text: thin

text: disc

text: with

text: H/r

text: we

text: consider

text: only

text: perturbations

text: which

text: are

text: larger

text: than

text: this

text: value,

text: i.e.

text: y

text: In

text: addition

text: to

text: this

text: constraint

text: of

text: local

text: approximation,

text: the

text: validity

text: of

text: the

text: vertically

text: integrated

text: equations

text: requires

text: ku

text: Ω

text: (e.g.,

text: Wu

text: Li

text: Kato

text: et

text: al.

text: this

text: requirement

text: can

text: be

text: K

text: written

text: as

text: kH

text: (αH/r)

text: where

text: α

text: is

text: the

text: disc

text: viscosity.

text: Thus,

text: our

text: analysis

text: is

text: valid

text: for

text: (H/r)

text: y

text: (αH/r)

text: If

text: we

text: set

text: α

text: and

text: H/r

text: then

text: the

text: valid

text: range

text: of

text: nondimensional

text: wavenumber

text: becomes

text: y

text: Also,

text: validity

text: of

text: vertical

text: integrated

text: equations

text: implies

text: that

text: the

text: growth

text: rates

text: of

text: the

text: unstable

text: modes

text: are

text: less

text: than

text: angular

text: velocity

text: (Kato

text: et

text: al.

text: This

text: requirement

text: is

text: justiﬁed

text: by

text: the

text: unstable

text: modes

text: as

text: we

text: will

text: show.

text: Existence

text: of

text: a

text: set

text: of

text: nontrivial

text: solutions

text: for

text: the

text: above

text: linearized

text: equations

text: imply

text: that

text: the

text: deter-

text: minant

text: of

text: the

text: coefﬁcients

text: becomes

text: zero

text: which

text: gives

text: us

text: an

text: algebraic

text: equation

text: involving

text: the

text: input

text: pa-

text: rameters,

text: growth

text: rate

text: and

text: the

text: wavenumber

text: of

text: the

text: perturbations.

text: Using

text: MAPLE

text: software,

text: we

text: found

text: the

text: dispersion

text: relation.

text: But

text: the

text: equation

text: is

text: very

text: lengthy,

text: and

text: so,

text: we

text: do

text: not

text: bring

text: it

text: here.

text: However,

text: our

text: analysis

text: is

text: based

text: on

text: the

text: roots

text: of

text: this

text: equation

text: which

text: can

text: be

text: calculated

text: numerically.

text: Obviously,

text: unstable

text: modes

text: correspond

text: to

text: the

text: roots

text: with

text: positive

text: real

text: part,

text: i.e.

text: Re(x)

text: We

text: generally

text: found

text: one

text: or

text: two

text: unstable

text: modes

text: for

text: a

text: given

text: set

text: of

text: the

text: input

text: parameters.

text: It

text: is

text: useful

text: to

text: re-write

text: the

text: input

text: parameters

text: as

text: follows

text: (Lizano

text: et

text: al.

text: N

text: B

text: λ

text: H

text: B

text: H

text: N

text: T

text: Θ

text: H

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: (cid:39) critical value a 0.625. We conclude that since the system of equations (II) has ground 1cr state solution with the lower eigenvalue than ground state solution to the system (I), it is enough to study the properties of the couple of equations (26) deﬁned on two-dimensional space-time. Note that numeric solving of the original eigenvalue equations (13) on a three-dimensional domain does not provide high numeric accuracy, especially in the limit of large values of the size L of the numeric domain along the radial direction. Remind that one should prove the positiveness of the ground state eigenvalues in the limit of inﬁnite space when the eigenvalues become very close to zero. Solving the reduced equations (26) deﬁned on two-dimensional space-time can be performed easily using standard numeric packages with suﬃcient accuracy. The obtained numeric accuracy for the eigenvalues λ(L) in solving the two-dimensional × − equations (26) is 1.0 10 5 which is enough to construct the eigenvalue dependence on the ≤ ≤ size L of the box in the range 6π L 64π. 4. Vacuum stability of the monopole ﬁeld in SU(3) QCD The analysis of quantum stability of the stationary monopole solution performed in the previous section can be generalized straightforward to the case of a pure SU(3) QCD. In general the system of two partial diﬀerential equations (3) admits non-stationary and quasi-stationary solutions. We are interested in stationary monopole solutions which can be ≡ obtained in a full analogy with the case of SU(2) QCD by using a constraint ψ = ψ ψ˜. 1 2 The Schr¨odinger type equation (13) for possible unstable modes corresponding to quan- tum gluon ﬂuctuations contains twenty four partial diﬀerential equations. Since the initial ansatz (2) describes monopole solution corresponding to I- and U-type subgroups SU(2) one expects that factorization property and reduction of the equations on the space of ground state solutions will take place in a similar manner as in the case of SU(2) QCD. Indeed, the numeric analysis of ground state solutions of the full SU(3) Schr¨odinger equation (13) with charge two monopole background ﬁeld ψ˜ implies factorization of the equations in each sector of I, U subgroups SU(2). The obtained numeric solution with the lowest eigenvalue√has a basis which contains six non-vanishing functions Ψ˜ 3, Ψ˜ 2, Ψ˜ 1 = Ψ˜ 6 = Ψ˜ 7 = Ψ˜ 2, Ψ˜ 8 = 3Ψ˜ 3, 1 3 2 2 3 3 1 1 all other functions are vanished identically. An important feature of the numeric solu- tion is that all solution proﬁle functions do not depend on the polar angle θ. One can choose the functions Ψ˜ 3, Ψ˜ 2 as independent ones and verify that twenty four equations in 1 3 the Schr¨odinger type equation (13) reduce to a simple system of two independent partial (cid:16) (cid:17) diﬀerential equations on two-dimensional space-time (r, t) in a consistent manner 2 − 1 (cid:16) − − (cid:17) ∂2Ψ˜ 3 + ∂ Ψ˜ 3 + ∂2Ψ˜ 3 (2ψ˜2 + 1)Ψ˜ 3 2(ψ˜ rψ˜ )Ψ˜ 2 = 0, r 1 r r 1 t 1 r2 1 r 3 (27) 2 − 1 − − − ∂2Ψ˜ 2 + ∂ Ψ˜ 2 + ∂2Ψ˜ 2 (3ψ˜2 2)Ψ˜ 2 4(ψ˜ rψ˜ )Ψ˜ 3 = 0. r 3 r r 3 t 3 2r2 3 r 1 One can show that the structure of the reduced equations is equivalent to the structure of the equations (II)√, (26), obtained by reduction in the case of SU(2) QCD. Simple rescaling of the functio√n ψ˜ = 2ψ representing the monopole background ﬁeld and function renormalization Ψ˜ 2 = 2Ψ2 (Ψ˜ 3 = Ψ3) in the Eqs.(27) lead exactly to the system of equations (II), (26). 3 3 1 1 We have veriﬁed that with increasing the size of the space-time numeric domain in the → ∞ radial direction, L , the corresponding eigenvalue λ(L) vanishes asymptotically from 11/14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: critical

text: value

text: a

text: We

text: conclude

text: that

text: since

text: the

text: system

text: of

text: equations

text: (II)

text: has

text: ground

text: state

text: solution

text: with

text: the

text: lower

text: eigenvalue

text: than

text: ground

text: state

text: solution

text: to

text: the

text: system

text: (I),

text: it

text: is

text: enough

text: to

text: study

text: the

text: properties

text: of

text: the

text: couple

text: of

text: equations

text: deﬁned

text: on

text: two-dimensional

text: space-time.

text: Note

text: that

text: numeric

text: solving

text: of

text: the

text: original

text: eigenvalue

text: equations

text: on

text: a

text: three-dimensional

text: domain

text: does

text: not

text: provide

text: high

text: numeric

text: accuracy,

text: especially

text: in

text: the

text: limit

text: of

text: large

text: values

text: of

text: the

text: size

text: L

text: of

text: the

text: numeric

text: domain

text: along

text: the

text: radial

text: direction.

text: Remind

text: that

text: one

text: should

text: prove

text: the

text: positiveness

text: of

text: the

text: ground

text: state

text: eigenvalues

text: in

text: the

text: limit

text: of

text: inﬁnite

text: space

text: when

text: the

text: eigenvalues

text: become

text: very

text: close

text: to

text: zero.

text: Solving

text: the

text: reduced

text: equations

text: deﬁned

text: on

text: two-dimensional

text: space-time

text: can

text: be

text: performed

text: easily

text: using

text: standard

text: numeric

text: packages

text: with

text: suﬃcient

text: accuracy.

text: The

text: obtained

text: numeric

text: accuracy

text: for

text: the

text: eigenvalues

text: λ(L)

text: in

text: solving

text: the

text: two-dimensional

text: equations

text: is

text: which

text: is

text: enough

text: to

text: construct

text: the

text: eigenvalue

text: dependence

text: on

text: the

text: size

text: L

text: of

text: the

text: box

text: in

text: the

text: range

text: 6π

text: L

text: Vacuum

text: stability

text: of

text: the

text: monopole

text: ﬁeld

text: in

text: QCD

text: The

text: analysis

text: of

text: quantum

text: stability

text: of

text: the

text: stationary

text: monopole

text: solution

text: performed

text: in

text: the

text: previous

text: section

text: can

text: be

text: generalized

text: straightforward

text: to

text: the

text: case

text: of

text: a

text: pure

text: QCD.

text: In

text: general

text: the

text: system

text: of

text: two

text: partial

text: diﬀerential

text: equations

text: admits

text: non-stationary

text: and

text: quasi-stationary

text: solutions.

text: We

text: are

text: interested

text: in

text: stationary

text: monopole

text: solutions

text: which

text: can

text: be

text: obtained

text: in

text: a

text: full

text: analogy

text: with

text: the

text: case

text: of

text: QCD

text: by

text: using

text: a

text: constraint

text: ψ

text: ψ

text: ψ˜.

text: The

text: Schr¨odinger

text: type

text: equation

text: for

text: possible

text: unstable

text: modes

text: corresponding

text: to

text: quan-

text: tum

text: gluon

text: ﬂuctuations

text: contains

text: twenty

text: four

text: partial

text: diﬀerential

text: equations.

text: Since

text: the

text: initial

text: ansatz

text: describes

text: monopole

text: solution

text: corresponding

text: to

text: I-

text: and

text: U-type

text: subgroups

text: one

text: expects

text: that

text: factorization

text: property

text: and

text: reduction

text: of

text: the

text: equations

text: on

text: the

text: space

text: of

text: ground

text: state

text: solutions

text: will

text: take

text: place

text: in

text: a

text: similar

text: manner

text: as

text: in

text: the

text: case

text: of

text: QCD.

text: Indeed,

text: the

text: numeric

text: analysis

text: of

text: ground

text: state

text: solutions

text: of

text: the

text: full

text: Schr¨odinger

text: equation

text: with

text: charge

text: two

text: monopole

text: background

text: ﬁeld

text: ψ˜

text: implies

text: factorization

text: of

text: the

text: equations

text: in

text: each

text: sector

text: of

text: I,

text: U

text: subgroups

text: The

text: obtained

text: numeric

text: solution

text: with

text: the

text: lowest

text: eigenvalue√has

text: a

text: basis

text: which

text: contains

text: six

text: non-vanishing

text: functions

text: Ψ˜

text: Ψ˜

text: Ψ˜

text: Ψ˜

text: Ψ˜

text: Ψ˜

text: Ψ˜

text: 3Ψ˜

text: all

text: other

text: functions

text: are

text: vanished

text: identically.

text: An

text: important

text: feature

text: of

text: the

text: numeric

text: solu-

text: tion

text: is

text: that

text: all

text: solution

text: proﬁle

text: functions

text: do

text: not

text: depend

text: on

text: the

text: polar

text: angle

text: θ.

text: One

text: can

text: choose

text: the

text: functions

text: Ψ˜

text: Ψ˜

text: as

text: independent

text: ones

text: and

text: verify

text: that

text: twenty

text: four

text: equations

text: in

text: the

text: Schr¨odinger

text: type

text: equation

text: reduce

text: to

text: a

text: simple

text: system

text: of

text: two

text: independent

text: partial

text: diﬀerential

text: equations

text: on

text: two-dimensional

text: space-time

text: (r,

text: t)

text: in

text: a

text: consistent

text: manner

text: ∂2Ψ˜

text: Ψ˜

text: ∂2Ψ˜

text: 1)Ψ˜

text: 2(ψ˜

text: rψ˜

text: )Ψ˜

text: r

text: r

text: r

text: t

text: r

text: ∂2Ψ˜

text: Ψ˜

text: ∂2Ψ˜

text: 2)Ψ˜

text: 4(ψ˜

text: rψ˜

text: )Ψ˜

text: r

text: r

text: r

text: t

text: r

text: One

text: can

text: show

text: that

text: the

text: structure

text: of

text: the

text: reduced

text: equations

text: is

text: equivalent

text: to

text: the

text: structure

text: of

text: the

text: equations

text: (II)√,

text: obtained

text: by

text: reduction

text: in

text: the

text: case

text: of

text: QCD.

text: Simple

text: rescaling

text: of

text: the

text: functio√n

text: ψ˜

text: 2ψ

text: representing

text: the

text: monopole

text: background

text: ﬁeld

text: and

text: function

text: renormalization

text: Ψ˜

text: (Ψ˜

text: in

text: the

text: lead

text: exactly

text: to

text: the

text: system

text: of

text: equations

text: (II),

text: We

text: have

text: veriﬁed

text: that

text: with

text: increasing

text: the

text: size

text: of

text: the

text: space-time

text: numeric

text: domain

text: in

text: the

text: radial

text: direction,

text: L

text: the

text: corresponding

text: eigenvalue

text: λ(L)

text: vanishes

text: asymptotically

text: from

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text: Prepared for submission to JHEP On the Study of Non-Relativistic Spin Half-Integer Systems Hadi Sobhani a,1 Hassan Hassanabadia aPhysics Department, Shahrood University of Technology, Shahrood, Iran. E-mail: hadisobhani8637@gmail.com Abstract: In this article, after introducing appropriate equation for non-relativistic spin half-integer system, Lagrange density of such system has been derived. Then time-evolution of this system in presence of a time-dependent interaction has been done. 1Corresponding author. ##LTLine##

text: Prepared

text: for

text: submission

text: to

text: JHEP

text: On

text: the

text: Study

text: of

text: Non-Relativistic

text: Spin

text: Half-Integer

text: Systems

text: Hadi

text: Sobhani

text: Hassan

text: Hassanabadia

text: aPhysics

text: Department,

text: Shahrood

text: University

text: of

text: Technology,

text: Shahrood,

text: Iran.

text: E-mail:

text: Abstract:

text: In

text: this

text: article,

text: after

text: introducing

text: appropriate

text: equation

text: for

text: non-relativistic

text: spin

text: half-integer

text: system,

text: Lagrange

text: density

text: of

text: such

text: system

text: has

text: been

text: derived.

text: Then

text: time-evolution

text: of

text: this

text: system

text: in

text: presence

text: of

text: a

text: time-dependent

text: interaction

text: has

text: been

text: done.

text: author.

text: ##LTLine##

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