text: 9 (cid:18) (cid:19) Using Eq. ((cid:2)S.10) we(cid:3)obta(cid:88)in (cid:2) (cid:3) (cid:2) (cid:3) (cid:104) | (cid:105) (cid:104) | (cid:105) − (cid:104) | (cid:105) (cid:104) | (cid:105) Im (cid:104)ω |ω (cid:105) = (cid:88) Im(cid:2)(cid:104)ω |Υ (cid:105)(cid:3)Re(cid:2)(cid:104)Υ |ψ (cid:105)(cid:3)(cid:18)Re[ Υk ωm ]Im[ Υ(cid:104) k ψ|s ](cid:105) Im(cid:104)[ Υ|k ω(cid:105)m ]Re[ Υk ψs ](cid:19) l m l k k s Re[ Υ ψ ]Im[ Υ ψ ] k s k s k (cid:104) | (cid:105) (cid:104) | (cid:105) (cid:104) | (cid:105) (cid:104) | (cid:105) (cid:104) | (cid:105) (cid:104) | (cid:105) Re[ ω Υ ]Im[ Υ ψ ] + Im[ ω Υ ]Re[ Υ ψ ] = Im ω Υ Re Υ ψ m k (cid:104) k |s (cid:105) (cid:104) m| (cid:105)k k s l k k s Re[ Υ ψ ]Im[ Υ ψ ] k s k s k = 0, (S.13) that vanishes because of Eq. (S.10). Appendix C: Proof of theorem #3. {| (cid:105)(cid:104) |} | (cid:105) | (cid:105) We now consider a more general set of projectors Υ Υ . Some of the states Υ are orthogonal to ψ and (cid:104) | (cid:105) k k k k s we label them with index h ( Υ ψ = 0). To calculate the contribution of these states to the FIM we follow the h s demonstration of Theorem #1. We label the other projectors – those not orthogonal to the probe states – with label (cid:104) | (cid:105) (cid:54) (cid:88) q ( Υ ψ = 0). We have q s (cid:104) | (cid:105)(cid:104) | (cid:105) (cid:104) | (cid:105)(cid:104) | (cid:105) Re[ ∂ ψ Υ Υ ψ ] Re[ ∂ ψ Υ Υ ψ ] [F (θ)] = 4 li(cid:88)m l ϕ h h |(cid:104)ϕ | (cid:105)| m ϕ h h ϕ + l,m δϕ→0 Υh ψϕ 2 h (cid:104) | (cid:105)(cid:104) | (cid:105) (cid:104) | (cid:105)(cid:104) | (cid:105) Re[ ω Υ Υ ψ ] Re[ ω Υ Υ ψ ] (cid:80) (cid:80) +4 l q q |(cid:104)s | (cid:105)| m q q s , (S.1) Υ ψ 2 q s q | (cid:105)(cid:104) | | (cid:105)(cid:104) | | (cid:105) (cid:0) (cid:1) where h Υh Υh + q Υq Υ(cid:88)q = 1 and ωj is given(cid:88)in Eq. (S.2). We have (cid:104) | (cid:105)(cid:104) | (cid:105) 2 |(cid:104) | (cid:105)| − Im[ Ψ Υ Υ ψ ] uTF (θ)u = 4 δϕli(cid:88)m→0 Ψu Υh 2(cid:88)4(cid:0)δϕlim→0 |u(cid:104)Υh(cid:1)h|ψϕ(cid:105)h|2 ϕ h h (cid:104) | (cid:105)(cid:104) | (cid:105) 2 |(cid:104) | (cid:105)| − Im[ Ψ Υ Υ ψ ] +4 Ψ Υ 2 4 (cid:0) |u(cid:104) q| (cid:105)|q s (cid:1) (cid:0) (cid:1) (S.2) u q (cid:88) Υ ψ 2 (cid:88) q s q q (cid:104) | (cid:105)(cid:104) | (cid:105) (cid:104) | (cid:105)(cid:104) | (cid:105) 2 2 − Im[ Ψ Υ Υ ψ ] − Im[ Ψ Υ Υ ψ ] = uTF (θ)u 4 lim |u(cid:104) h| (cid:105)h| ϕ 4 |u(cid:104) q| (cid:105)|q s , (S.3) (cid:80) Q (cid:80) δϕ→0 Υh ψϕ 2 Υq ψs 2 h q (cid:104) | (cid:105)(cid:104) | (cid:105) (cid:104) | (cid:105)(cid:104) | (cid:105) (cid:104) | (cid:105) where Ψ Υ Υ Ψ + 4 Ψ Υ Υ Ψ = 4 Ψ Ψ = uTF (θ)u. The equality F (θ) = F (θ) is h u h h u q u q q u u u Q Q recovered if and only if (cid:104) | (cid:105)(cid:104) | (cid:105) (cid:2) Im[ ∂ ψ Υ Υ(cid:3) ψ(cid:12) ] (cid:12) (cid:2)∀ ∀ (cid:3) lim l |(cid:104)ϕ |h (cid:105)|h (cid:12)ϕ = 0,(cid:12) l, h, (S.4a) → δϕ 0 Υh ψϕ (cid:104) | (cid:105)(cid:104) | (cid:105) − (cid:104) | (cid:105) (cid:104) | (cid:105) ∀ ∀ 2 Im ∂ ψ Υ Υ ψ ψ Υ Im ∂ ψ ψ = 0, l, q. (S.4b) l s q q s s q l s s Appendix D: Example: the multimode Mach-Zehnder interferometers We discuss in details the example provided in the main text. Let us consider a n-mode interferometer, composed − by two cascaded multiport splitters. The parameters to be estimated are a set of d = n 1 optical phases θ = (θ , . . . , θ − ), whose action on the probe state is described by a unitary evolution U(θ). 1 n 1 In the three-mode case, the modes of the interferometer transform according to the unitary U(θ) = − [U(3)] 1U (θ)U(3), where U(3) is a tritter and U (θ) provides a shift of phase of two modes with respect to the ps ps third mode:         √1 1 ei2π/3 ei2π/3 eiθ1 0 0 U(3) = ei2π/3 1 ei2π/3 , and Ups(θ) = 0 eiθ2 0 . (S.1) 3 ei2π/3 ei2π/3 1 0 0 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Using

text: Eq.

text: Im

text: |ω

text: |Υ

text: |ψ

text: Υk

text: ωm

text: ]Im[

text: k

text: ψ|s

text: Υ|k

text: ]Re[

text: Υk

text: ψs

text: l

text: m

text: l

text: k

text: k

text: s

text: Re[

text: Υ

text: ψ

text: ]Im[

text: Υ

text: ψ

text: k

text: s

text: k

text: s

text: k

text: Re[

text: ω

text: Υ

text: ]Im[

text: Υ

text: ψ

text: Im[

text: ω

text: Υ

text: ]Re[

text: Υ

text: ψ

text: Im

text: ω

text: Υ

text: Re

text: Υ

text: ψ

text: m

text: k

text: k

text: |s

text: m|

text: k

text: s

text: l

text: k

text: k

text: s

text: Re[

text: Υ

text: ψ

text: ]Im[

text: Υ

text: ψ

text: k

text: s

text: k

text: s

text: k

text: that

text: vanishes

text: because

text: of

text: Eq.

text: Appendix

text: C:

text: Proof

text: of

text: theorem

text: We

text: now

text: consider

text: a

text: more

text: general

text: set

text: of

text: projectors

text: Υ

text: Υ

text: Some

text: of

text: the

text: states

text: Υ

text: are

text: orthogonal

text: to

text: ψ

text: and

text: k

text: k

text: k

text: k

text: s

text: we

text: label

text: them

text: with

text: index

text: h

text: Υ

text: ψ

text: To

text: calculate

text: the

text: contribution

text: of

text: these

text: states

text: to

text: the

text: FIM

text: we

text: follow

text: the

text: h

text: s

text: demonstration

text: of

text: Theorem

text: We

text: label

text: the

text: other

text: projectors

text: those

text: not

text: orthogonal

text: to

text: the

text: probe

text: states

text: with

text: label

text: q

text: Υ

text: ψ

text: We

text: have

text: q

text: s

text: Re[

text: ψ

text: Υ

text: Υ

text: ψ

text: Re[

text: ψ

text: Υ

text: Υ

text: ψ

text: [F

text: (θ)]

text: l

text: ϕ

text: h

text: h

text: m

text: ϕ

text: h

text: h

text: ϕ

text: l,m

text: δϕ→0

text: Υh

text: ψϕ

text: h

text: Re[

text: ω

text: Υ

text: Υ

text: ψ

text: Re[

text: ω

text: Υ

text: Υ

text: ψ

text: l

text: q

text: q

text: m

text: q

text: q

text: s

text: Υ

text: ψ

text: q

text: s

text: q

text: where

text: h

text: Υh

text: Υh

text: q

text: Υq

text: and

text: ωj

text: is

text: Eq.

text: We

text: have

text: Im[

text: Ψ

text: Υ

text: Υ

text: ψ

text: uTF

text: (θ)u

text: Ψu

text: Υh

text: ϕ

text: h

text: h

text: Im[

text: Ψ

text: Υ

text: Υ

text: ψ

text: Ψ

text: Υ

text: q|

text: s

text: u

text: q

text: Υ

text: ψ

text: q

text: s

text: q

text: q

text: Im[

text: Ψ

text: Υ

text: Υ

text: ψ

text: Im[

text: Ψ

text: Υ

text: Υ

text: ψ

text: uTF

text: (θ)u

text: lim

text: h|

text: ϕ

text: q|

text: s

text: Q

text: δϕ→0

text: Υh

text: ψϕ

text: Υq

text: ψs

text: h

text: q

text: where

text: Ψ

text: Υ

text: Υ

text: Ψ

text: Ψ

text: Υ

text: Υ

text: Ψ

text: Ψ

text: Ψ

text: uTF

text: (θ)u.

text: The

text: equality

text: F

text: (θ)

text: F

text: (θ)

text: is

text: h

text: u

text: h

text: h

text: u

text: q

text: u

text: q

text: q

text: u

text: u

text: u

text: Q

text: Q

text: recovered

text: if

text: and

text: only

text: if

text: Im[

text: ψ

text: Υ

text: lim

text: l

text: |h

text: l,

text: h,

text: δϕ

text: Υh

text: ψϕ

text: Im

text: ψ

text: Υ

text: Υ

text: ψ

text: ψ

text: Υ

text: Im

text: ψ

text: ψ

text: l,

text: q.

text: l

text: s

text: q

text: q

text: s

text: s

text: q

text: l

text: s

text: s

text: Appendix

text: D:

text: Example:

text: the

text: multimode

text: Mach-Zehnder

text: interferometers

text: We

text: discuss

text: in

text: details

text: the

text: example

text: provided

text: in

text: the

text: main

text: text.

text: Let

text: us

text: consider

text: a

text: n-mode

text: interferometer,

text: composed

text: by

text: two

text: cascaded

text: multiport

text: splitters.

text: The

text: parameters

text: to

text: be

text: estimated

text: are

text: a

text: set

text: of

text: d

text: n

text: optical

text: phases

text: θ

text: (θ

text: θ

text: whose

text: action

text: on

text: the

text: probe

text: state

text: is

text: described

text: by

text: a

text: unitary

text: evolution

text: U(θ).

text: n

text: In

text: the

text: three-mode

text: case,

text: the

text: modes

text: of

text: the

text: interferometer

text: transform

text: according

text: to

text: the

text: unitary

text: U(θ)

text: where

text: is

text: a

text: tritter

text: and

text: U

text: (θ)

text: provides

text: a

text: shift

text: of

text: phase

text: of

text: two

text: modes

text: with

text: respect

text: to

text: the

text: ps

text: ps

text: third

text: mode:

text: √1

text: and

text: Ups(θ)

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: etc. Numerical analyses of soil-structure interactions (usually by ﬁnite element methods) are used to predict the performance of the structure including the loads in the structural elements, magnitudes of wall deﬂections and ground deformations at all stages of the construction process. During construction, TERS systems are extensively monitored to ensure the stability of the works and to control/mitigate impacts of excavation-induced ground movements that can aﬀect adjacent structures and facilities. Field monitoring typically includes measurements of compression forces in the strutting system (via strain gauges or load cells), deﬂections (and ﬂexure) of the retaining wall (via inclinomet- ers), surface and subsurface ground movements (optical surveys, inclinometers and extensometers) and pore water pressures (piezometers). In current prac- tice, the monitoring data are primarily used to i) compare against the expected values predicted by prior analyses of soil-structure interaction, to determine if the design speciﬁcation has been exceeded and ii) trigger static threshold based ﬂags/alarms. The high unit costs for installing these instruments (particularly subsurface sensors) leads to sparse spatial coverage at most construction sites. Furthermore, there is little use of the data to interpret the actual ﬁeld perform- ance and hence, to update model predictions and reduce risks associated with the construction process. Wireless sensing oﬀers an attractive alternate approach for monitoring struc- tural performance. By reducing the unit cost of sensor installation, wireless sensor networks can increase the spatial and temporal resolution for measuring forces in the bracing system (compared to wired devices). The goal of this research is to enable robust, reliable wireless collection of strain measurements in an excavation environment and make it available for on- line integration, in real-time, with a user application. This paper demonstrates: • the feasibility of using Wireless Sensor Networks (WSNs) for TERS. An end-to-end proof-of-concept system (ReStructure) has been developed and deployed on an excavation site (alongside a conventional wired system, where forces were monitored by vibrating wire strain gauges, done by a specialist instrumentation contractor) to acquire, transmit and aggregate strain data. Criteria used for evaluating the suitability of the prototype for the application have been: i) the validity and accuracy of the measure- ments against the industry gold-standard systems, ii) wireless communic- ation reliability and data yield, iii) system survivability in ﬁeld, and iv) system and battery life. • The paper provides insights into the feasibility of large scale, low cost in- telligent WSNs monitoring for TERS through analysis of the data archive created through the pilot deployment. Lessons are drawn from the system design, implementation and deployment that add to the existing body of knowledge on deployable WSNs for structural monitoring. In this paper we review the current literature (Section 6) on related wireless sys- tems for structural health monitoring. We then discuss the design speciﬁcations for the current application to Temporary Earth Retaining systems (Section 2), and present details of the prototype design and implementation (Section 3). We 2

text: etc.

label: paragraph

text: Numerical

text: analyses

text: of

text: soil-structure

text: interactions

text: (usually

text: by

text: ﬁnite

text: element

text: methods)

text: are

text: used

text: to

text: predict

text: the

text: performance

text: of

text: the

text: structure

text: including

text: the

text: loads

text: in

text: the

text: structural

text: elements,

text: magnitudes

text: of

text: wall

text: deﬂections

text: and

text: ground

text: deformations

text: at

text: all

text: stages

text: of

text: the

text: construction

text: process.

text: During

text: construction,

text: TERS

text: systems

text: are

text: extensively

text: monitored

text: to

text: ensure

text: the

text: stability

text: of

text: the

text: works

text: and

text: to

text: control/mitigate

text: impacts

text: of

text: excavation-induced

text: ground

text: movements

text: that

text: can

text: aﬀect

text: adjacent

text: structures

text: and

text: facilities.

text: Field

text: monitoring

text: typically

text: includes

text: measurements

text: of

text: compression

text: forces

text: in

text: the

text: strutting

text: system

text: (via

text: strain

text: gauges

text: or

text: load

text: cells),

text: deﬂections

text: (and

text: ﬂexure)

text: of

text: the

text: retaining

text: wall

text: (via

text: inclinomet-

text: ers),

text: surface

text: and

text: subsurface

text: ground

text: movements

text: (optical

text: surveys,

text: inclinometers

text: and

text: extensometers)

text: and

text: pore

text: water

text: pressures

text: (piezometers).

text: In

text: current

text: prac-

text: tice,

text: the

text: monitoring

text: data

text: are

text: primarily

text: used

text: to

text: i)

text: compare

text: against

text: the

text: expected

text: values

text: predicted

text: by

text: prior

text: analyses

text: of

text: soil-structure

text: interaction,

text: to

text: determine

text: if

text: the

text: design

text: speciﬁcation

text: has

text: been

text: exceeded

text: and

text: ii)

text: trigger

text: static

text: threshold

text: based

text: ﬂags/alarms.

text: The

text: high

text: unit

text: costs

text: for

text: installing

text: these

text: instruments

text: (particularly

text: subsurface

text: sensors)

text: leads

text: to

text: sparse

text: spatial

text: coverage

text: at

text: most

text: construction

text: sites.

text: Furthermore,

text: there

text: is

text: little

text: use

text: of

text: the

text: data

text: to

text: interpret

text: the

text: actual

text: ﬁeld

text: perform-

text: ance

text: and

text: hence,

text: to

text: update

text: model

text: predictions

text: and

text: reduce

text: risks

text: associated

text: with

text: the

text: construction

text: process.

text: Wireless

text: sensing

text: oﬀers

text: an

text: attractive

text: alternate

text: approach

text: for

text: monitoring

text: struc-

text: tural

text: performance.

text: By

text: reducing

text: the

text: unit

text: cost

text: of

text: sensor

text: installation,

text: wireless

text: sensor

text: networks

text: can

text: increase

text: the

text: spatial

text: and

text: temporal

text: resolution

text: for

text: measuring

text: forces

text: in

text: the

text: bracing

text: system

text: (compared

text: to

text: wired

text: devices).

text: The

text: goal

text: of

text: this

text: research

text: is

text: to

text: enable

text: robust,

text: reliable

text: wireless

text: collection

text: of

text: strain

text: measurements

text: in

text: an

text: excavation

text: environment

text: and

text: make

text: it

text: available

text: for

text: on-

text: line

text: integration,

text: in

text: real-time,

text: with

text: a

text: user

text: application.

text: This

text: paper

text: demonstrates:

text: the

text: feasibility

text: of

text: using

text: Wireless

text: Sensor

text: Networks

text: (WSNs)

text: for

text: TERS.

text: An

text: end-to-end

text: proof-of-concept

text: system

text: (ReStructure)

text: has

text: been

text: developed

text: and

text: deployed

text: on

text: an

text: excavation

text: site

text: (alongside

text: a

text: conventional

text: wired

text: system,

text: where

text: forces

text: were

text: monitored

text: by

text: vibrating

text: wire

text: strain

text: gauges,

text: done

text: by

text: a

text: specialist

text: instrumentation

text: contractor)

text: to

text: acquire,

text: transmit

text: and

text: aggregate

text: strain

text: data.

text: Criteria

text: used

text: for

text: evaluating

text: the

text: suitability

text: of

text: the

text: prototype

text: for

text: the

text: application

text: have

text: been:

text: i)

text: the

text: validity

text: and

text: accuracy

text: of

text: the

text: measure-

text: ments

text: against

text: the

text: industry

text: gold-standard

text: systems,

text: ii)

text: wireless

text: communic-

text: ation

text: reliability

text: and

text: data

text: yield,

text: iii)

text: system

text: survivability

text: in

text: ﬁeld,

text: and

text: iv)

text: system

text: and

text: battery

text: life.

text: The

text: paper

text: provides

text: insights

text: into

text: the

text: feasibility

text: of

text: large

text: scale,

text: low

text: cost

text: in-

text: telligent

text: WSNs

text: monitoring

text: for

text: TERS

text: through

text: analysis

text: of

text: the

text: data

text: archive

text: created

text: through

text: the

text: pilot

text: deployment.

text: Lessons

text: are

text: drawn

text: from

text: the

text: system

text: design,

text: implementation

text: and

text: deployment

text: that

text: add

text: to

text: the

text: existing

text: body

text: of

text: knowledge

text: on

text: deployable

text: WSNs

text: for

text: structural

text: monitoring.

text: In

text: this

text: paper

text: we

text: review

text: the

text: current

text: literature

text: (Section

text: on

text: related

text: wireless

text: sys-

text: tems

text: for

text: structural

text: health

text: monitoring.

text: We

text: then

text: discuss

text: the

text: design

text: speciﬁcations

text: for

text: the

text: current

text: application

text: to

text: Temporary

text: Earth

text: Retaining

text: systems

text: (Section

text: and

text: present

text: details

text: of

text: the

text: prototype

text: design

text: and

text: implementation

text: (Section

text: We


text: Gateway Depth S1 51 VW-L2 1.34 m VW-L1 55 S2 50 VW-L2 5.15 m VW-L1 52 S3 VW-L2 8.59 m VW-L1 56 Figure 8: Deployment layout of ReStructure wireless bonded foil (WBF) strain gauges (50–56) and commercial vibrating wire (VW) gauges (VW-L1/L2) across three strut levels (S1, S2, S3). Distance from deployment location to gateway is shown in metres. compression during subsequent excavation events. This allowed us to calibrate the WBF strain gauge in an unstressed/zero strain conﬁguration. The WBF strain gauges were spot welded aligned to the length of the strut on the web (vertical section of strut, see Figure 7) and the strain bridge was manually balanced by adjusting the on-board potentiometer while the node is connected to a laptop displaying the current reading. This balancing procedure allows a reference point to be set that corresponds to zero load on the strut. Figure 8 shows a schematic elevation of the cross section of the struts and the relative distance between the nodes and the gateway. Data was collected at 5 minute intervals, and nodes communicated on 802.15.4 channel 26 in an attempt to avoid any WI-FI interference and the maximum number of retries for each packet (sent by Contiki Collect) was set at 16. 4.1 Network and energy performance Table 4 shows the deployment duration and data yield across the deployment. There were two major outages where the node gateway batteries had exhausted: 12 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Gateway

font: default

text: Depth

text: m

text: m

text: m

text: Figure

text: Deployment

label: caption

text: layout

text: of

text: ReStructure

text: wireless

text: bonded

text: foil

text: (WBF)

text: strain

text: gauges

text: and

text: commercial

text: vibrating

text: wire

text: (VW)

text: gauges

text: across

text: three

text: strut

text: levels

text: Distance

text: from

text: deployment

text: location

text: to

text: gateway

text: is

text: shown

text: in

text: metres.

text: compression

text: during

text: subsequent

text: excavation

text: events.

text: This

text: allowed

text: us

text: to

text: calibrate

text: the

text: WBF

text: strain

text: gauge

text: in

text: an

text: unstressed/zero

text: strain

text: conﬁguration.

text: The

text: WBF

text: strain

text: gauges

text: were

text: spot

text: welded

text: aligned

text: to

text: the

text: length

text: of

text: the

text: strut

text: on

text: the

text: web

text: (vertical

text: section

text: of

text: strut,

text: see

text: Figure

text: and

text: the

text: strain

text: bridge

text: was

text: manually

text: balanced

text: by

text: adjusting

text: the

text: on-board

text: potentiometer

text: while

text: the

text: node

text: is

text: connected

text: to

text: a

text: laptop

text: displaying

text: the

text: current

text: reading.

text: This

text: balancing

text: procedure

text: allows

text: a

text: reference

text: point

text: to

text: be

text: set

text: that

text: corresponds

text: to

text: zero

text: load

text: on

text: the

text: strut.

text: Figure

text: shows

text: a

text: schematic

text: elevation

text: of

text: the

text: cross

text: section

text: of

text: the

text: struts

text: and

text: the

text: relative

text: distance

text: between

text: the

text: nodes

text: and

text: the

text: gateway.

text: Data

text: was

text: collected

text: at

text: minute

text: intervals,

text: and

text: nodes

text: communicated

text: on

text: channel

text: in

text: an

text: attempt

text: to

text: avoid

text: any

text: WI-FI

text: interference

text: and

text: the

text: maximum

text: number

text: of

text: retries

text: for

text: each

text: packet

text: (sent

text: by

text: Contiki

text: Collect)

text: was

text: set

text: at

text: Network

text: and

text: energy

text: performance

text: Table

text: shows

text: the

text: deployment

text: duration

text: and

text: data

text: yield

text: across

text: the

text: deployment.

text: There

text: were

text: two

text: major

text: outages

text: where

text: the

text: node

text: gateway

text: batteries

text: had

text: exhausted:

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:90) (cid:90) (cid:90) R2o (cid:90) R1 1 ρodx + ρodx = N , 1 1 2 1 0 Ro 2 R1 R2s 1 (31) ρodx + ρsdx = N ; 2 2 2 2 R2o R1 ρo(R , t) = ρs(R , t). 2 1 2 1 Here N and N are are the numbers of particles in the ﬁrst and second components, respectively, 1 2 − R is the coordinate where the ﬁrst component becomes zero (ρo(R , t) = αo (t) αo(t)R2 = 0), 1 1 1 1,0 1 1 Ro and Rs are the coordinates where the second condensate component becomes zero (ρn(Rn, t) = 2 − 2 2 2 αn (t) αn(t)Rn2 = 0). The point Ro corresponds to zero density of the second component in the 2,0 2 2 2 overlap region, while Rs denotes the coordinate at which the density of the second component becomes 2 (cid:114) (cid:114) (cid:114) zero in the singlet region (see Fig. 4(a)). Consequently, these parameters are deﬁned by the relations αo (t) αo (t) αs (t) R = 1,0 , Ro = 2,0 , Rs = 2,0 . (32) 1 αo(t) 2 αo(t) 2 αs(t) 1 2 2 The coordinates R , Ro and Rs are functions of time. Thus, we have reduced the problem to inte- 1 2 2 grating the ordinary diﬀerential equations (30) with their initial conditions determined by the original component density distributions in the trap. In general, this system must be solved numerically, which is considerably easier than the solution of the Gross-Pitaevskii equations. However, an important case where system (30) is simpliﬁed considerably and admits the derivation of some relations in a closed form is noteworthy. 4.1 The Case of Miscibility Let we have an initial overlap conﬁguration where the ﬁrst component has no singlet region (see Fig. 2(a)-(c) and Fig. 2(h)-(j)), i.e., the components are miscible. System (31) will then be written as (cid:90) (cid:90) R1 (cid:90)1 ρodx = N , 1 2 1 0 R1 R2 1 (33) ρodx + ρsdx = N ; 2 2 2 2 0 R1 ρo(R , t) = ρs(R , t). 2 1 2 1 Here, R is the coordinate where the particle number density of the second component becomes 2 zero (see Fig. 4(b)). In this case, the relations for αn (t) expressed via αn(t) can be found analytically, i,0 i and the solution of this system will b(cid:20)e (cid:21) (cid:112) (cid:34)3(cid:112) 2/3 (cid:35) αo = N αo , 1,0 4 1 1 2/3 3 αs − − αs = 2(cid:20)(N α(cid:21)o N (αo αs)) , (34) 2,0 4αo 2 1 1 2 2 1 3N 2/3 − αo = αs + 1 (αo αs). 2,0 2,0 4αo 2 2 1 The case of the expansion of a condensate with an asymmetric initial proﬁle (see Fig. 2(k)) will be considered separately below. The diﬀerential equations (30) are Newton-type equations that have the total energy of the system (16) the conservation of the number of particles in the singlet region and the overlap region of each 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: ρodx

text: ρodx

text: N

text: Ro

text: ρodx

text: ρsdx

text: N

text: ρo(R

text: t)

text: ρs(R

text: t).

text: Here

text: N

text: and

text: N

text: are

text: are

text: the

text: numbers

text: of

text: particles

text: in

text: the

text: ﬁrst

text: and

text: second

text: components,

text: respectively,

text: R

text: is

text: the

text: coordinate

text: where

text: the

text: ﬁrst

text: component

text: becomes

text: zero

text: (ρo(R

text: t)

text: αo

text: (t)

text: Ro

text: and

text: Rs

text: are

text: the

text: coordinates

text: where

text: the

text: second

text: condensate

text: component

text: becomes

text: zero

text: (ρn(Rn,

text: t)

text: αn

text: (t)

text: The

text: point

text: Ro

text: corresponds

text: to

text: zero

text: density

text: of

text: the

text: second

text: component

text: in

text: the

text: overlap

text: region,

text: while

text: Rs

text: denotes

text: the

text: coordinate

text: at

text: which

text: the

text: density

text: of

text: the

text: second

text: component

text: becomes

text: zero

text: in

text: the

text: singlet

text: region

text: (see

text: Fig.

text: Consequently,

text: these

text: parameters

text: are

text: deﬁned

text: by

text: the

text: relations

text: αo

text: (t)

text: αo

text: (t)

text: αs

text: (t)

text: R

text: Ro

text: Rs

text: αo(t)

text: αo(t)

text: αs(t)

text: The

text: coordinates

text: R

text: Ro

text: and

text: Rs

text: are

text: functions

text: of

text: time.

text: Thus,

text: we

text: have

text: reduced

text: the

text: problem

text: to

text: inte-

text: grating

text: the

text: ordinary

text: diﬀerential

text: equations

text: with

text: their

text: initial

text: conditions

text: determined

text: by

text: the

text: original

text: component

text: density

text: distributions

text: in

text: the

text: trap.

text: In

text: general,

text: this

text: system

text: must

text: be

text: solved

text: numerically,

text: which

text: is

text: considerably

text: easier

text: than

text: the

text: solution

text: of

text: the

text: Gross-Pitaevskii

text: equations.

text: However,

text: an

text: important

text: case

text: where

text: system

text: is

text: simpliﬁed

text: considerably

text: and

text: admits

text: the

text: derivation

text: of

text: some

text: relations

text: in

text: a

text: closed

text: form

text: is

text: noteworthy.

text: The

text: Case

text: of

text: Miscibility

text: Let

text: we

text: have

text: an

text: initial

text: overlap

text: conﬁguration

text: where

text: the

text: ﬁrst

text: component

text: has

text: no

text: singlet

text: region

text: (see

text: Fig.

text: and

text: Fig.

text: i.e.,

text: the

text: components

text: are

text: miscible.

text: System

text: will

text: then

text: be

text: written

text: as

text: ρodx

text: N

text: ρodx

text: ρsdx

text: N

text: ρo(R

text: t)

text: ρs(R

text: t).

text: Here,

text: R

text: is

text: the

text: coordinate

text: where

text: the

text: particle

text: number

text: density

text: of

text: the

text: second

text: component

text: becomes

text: zero

text: (see

text: Fig.

text: In

text: this

text: case,

text: the

text: relations

text: for

text: αn

text: (t)

text: expressed

text: via

text: αn(t)

text: can

text: be

text: found

text: analytically,

text: i

text: and

text: the

text: solution

text: of

text: this

text: system

text: will

text: αo

text: N

text: αo

text: αs

text: αs

text: N

text: (αo

text: αs))

text: αo

text: αs

text: (αo

text: αs).

text: The

text: case

text: of

text: the

text: expansion

text: of

text: a

text: condensate

text: with

text: an

text: asymmetric

text: initial

text: proﬁle

text: (see

text: Fig.

text: will

text: be

text: considered

text: separately

text: below.

text: The

text: diﬀerential

text: equations

text: are

text: Newton-type

text: equations

text: that

text: have

text: the

text: total

text: energy

text: of

text: the

text: system

text: the

text: conservation

text: of

text: the

text: number

text: of

text: particles

text: in

text: the

text: singlet

text: region

text: and

text: the

text: overlap

text: region

text: of

text: each

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: (a) (b) Figure 7: Particle number densities for the symmetric phase with g = 1, g = 1.1, and the same number 11 22 of particles in the components N : N = 100 : 100 (a) and the asymmetric phase with g = 1, g = 1 1 2 11 22 and N : N = 150 : 100 (b) at t = 10. The particle masses and trap frequencies are m = m = 1, 1 2 1 2 ω = ω = 1. 1 2 (cid:90) (cid:90) Rb R2 ρsdx = N , ρsdx = N (56) 1 1 2 2 R1 Rb for the asymmetric one. In principle, these equations allow αs to be expressed via the already i,0 known functions αs through numerically easily solvable algebraic equations (which we do not write i out here, because they are cumbersome). As in the case of g = g , from the system of equations (53) we can ﬁnd the asymptotic solution 11 12 (cid:29) − for ζs corresponding to times t ω 1: i i √ ≈ ζs 2ω1/3t. (57) i i As above, they describe the ﬂow of the condensate cloud by inertia once the potential energy of the condensate compressed in the trap has been converted into the kinetic energy of its ﬂow. The extreme points of the distributions of each condensate component move with maximum velocities: ≈ R (t) ≈ R (t) us b , us 2 ; (58) 1max t 2max t ≈ R (t) ≈ R (t) us 1 , us 2 (59) 1max t 2max t for the symmetric and asymmetric particle number density proﬁles, respectively. The expressions for the particle number densities and velocities at asymptotically large t are ≈ − √m x2 ∼ x ρs(x, t) αs (t) i , us(x, t) . (60) i i,0 4 2g ω t3 i t ii i The functions αs (t) The functions (55) in the symmetric case and (56) in the asymmetric case i,0 for each time t. Knowing αs and αs , we can ﬁnd the forms of the particle number densities and 1,0 2,0 the maximum velocities of each of the condensate components at this time. Thus, the derived formulas give the solution of our problem. It is illustrated in Fig. 7, where the particle number densities are shown for the symmetric and asymmetric phases at a ﬁxed time. 16 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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## ##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## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

text: (b)

text: Figure

text: Particle

text: number

text: densities

text: for

text: the

text: symmetric

text: phase

text: with

text: g

text: g

text: and

text: the

text: same

text: number

text: of

text: particles

text: in

text: the

text: components

text: N

text: N

text: (a)

text: and

text: the

text: asymmetric

text: phase

text: with

text: g

text: g

text: and

text: N

text: N

text: (b)

text: at

text: t

text: The

text: particle

text: masses

text: and

text: trap

text: frequencies

text: are

text: m

text: m

text: ω

text: ω

text: Rb

text: ρsdx

text: N

text: ρsdx

text: N

text: Rb

text: for

text: the

text: asymmetric

text: one.

text: In

text: principle,

text: these

text: equations

text: allow

text: αs

text: to

text: be

text: expressed

text: via

text: the

text: already

text: known

text: functions

text: αs

text: through

text: numerically

text: easily

text: solvable

text: algebraic

text: equations

text: (which

text: we

text: do

text: not

text: write

text: i

text: out

text: here,

text: because

text: they

text: are

text: cumbersome).

text: As

text: in

text: the

text: case

text: of

text: g

text: g

text: from

text: the

text: system

text: of

text: equations

text: we

text: can

text: ﬁnd

text: the

text: asymptotic

text: solution

text: for

text: ζs

text: corresponding

text: to

text: times

text: t

text: ω

text: i

text: i

text: ζs

text: i

text: i

text: As

text: above,

text: they

text: describe

text: the

text: ﬂow

text: of

text: the

text: condensate

text: cloud

text: by

text: inertia

text: once

text: the

text: potential

text: energy

text: of

text: the

text: condensate

text: compressed

text: in

text: the

text: trap

text: has

text: been

text: converted

text: into

text: the

text: kinetic

text: energy

text: of

text: its

text: ﬂow.

text: The

text: extreme

text: points

text: of

text: the

text: distributions

text: of

text: each

text: condensate

text: component

text: move

text: with

text: maximum

text: velocities:

text: R

text: (t)

text: R

text: (t)

text: us

text: b

text: us

text: t

text: t

text: R

text: (t)

text: R

text: (t)

text: us

text: us

text: t

text: t

text: for

text: the

text: symmetric

text: and

text: asymmetric

text: particle

text: number

text: density

text: proﬁles,

text: respectively.

text: The

text: expressions

text: for

text: the

text: particle

text: number

text: densities

text: and

text: velocities

text: at

text: asymptotically

text: large

text: t

text: are

text: √m

text: x

text: ρs(x,

text: t)

text: αs

text: (t)

text: i

text: us(x,

text: t)

text: i

text: ω

text: i

text: t

text: ii

text: i

text: The

text: functions

text: αs

text: (t)

text: The

text: functions

text: in

text: the

text: symmetric

text: case

text: and

text: in

text: the

text: asymmetric

text: case

text: for

text: each

text: time

text: t.

text: Knowing

text: αs

text: and

text: αs

text: we

text: can

text: ﬁnd

text: the

text: forms

text: of

text: the

text: particle

text: number

text: densities

text: and

text: the

text: maximum

text: velocities

text: of

text: each

text: of

text: the

text: condensate

text: components

text: at

text: this

text: time.

text: Thus,

text: the

text: derived

text: formulas

text: give

text: the

text: solution

text: of

text: our

text: problem.

text: It

text: is

text: illustrated

text: in

text: Fig.

text: where

text: the

text: particle

text: number

text: densities

text: are

text: shown

text: for

text: the

text: symmetric

text: and

text: asymmetric

text: phases

text: at

text: a

text: ﬁxed

text: time.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: These are a special case of Eqs. (14) and (15), where we take into account the fact that, in this case, the components are spatially separated, so that the index i = 1 refers to the component to the left of the interface, while the index i = 2 refers to the component to the right of this interface. There is no interaction between the components everywhere except the narrow region near the boundary separating them, but its presence is taken into account by an appropriate boundary condition for the pressures to be equal on both sides of the boundary. In addition, we assume the masses of the atoms in both components to be identical and equal to unity, while the trap potential at the expansion stage to play no role in an obvious way. The solution of such problems is simpliﬁed considerably if we pass from the ordinary physical variables ρ , u to the so-called “Riemann invariants”. For Eqs. (62), i i which coincide in form with the gas-dynamic equations, the Riemann invariants are well known and can be written as (see, e.g., [28]) √ ± r±(i) = u 2 g ρ . (63) i ii i In these variables the hydrodynamic equations (62) take a simple symmetric form: ∂r±(i) ∂r±(i) + v±(i)(r(i), r−(i)) = 0, (64) ∂t + ∂x ± where the ”Riemann velocities v±(i) = u c are expressed via the Riemann invariants by the i i relations 3 1 1 3 v(i) = r(i) + r−(i), v−(i) = r(i) + r−(i). (65) + 4 + 4 4 + 4 If the solution of Eqs. (64) has been found and the Riemann invariants are known, then the physical variables are expressed via them by the formulas − (r(i) r−(i))2 1 ρ = + , u = (r(i) + r−(i)). (66) i 16g i 2 + ii A rarefaction wave is known [27] to belong to the class of “simple waves” characterized by the fact that one of the Riemann invariants has a constant value along the ﬂow. For the case shown in Fig. 9, the rarefaction wave propagates to the left in the ﬁrst condensate. Hence, the following invariant is constant for it: √ √ (1) r = u + 2 g ρ = 2 g ρ , (67) + 1 11 1 11 1,0 where we set its value equal to the value at the boundary with the condensate at rest. It must (cid:112) have the same value at the rarefaction wave boundary with the shelf to the right of it: √ u + 2 g ρ = 2 g ρ . (68) 11 1 11 1,0 The pressures of the components at the boundary between them are equal, which gives the relation g ρ2 = g ρ2. (69) 11 1 22 2 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: These

text: are

text: a

text: special

text: case

text: of

text: Eqs.

text: and

text: where

text: we

text: take

text: into

text: account

text: the

text: fact

text: that,

text: in

text: this

text: case,

text: the

text: components

text: are

text: spatially

text: separated,

text: so

text: that

text: the

text: index

text: i

text: refers

text: to

text: the

text: component

text: to

text: the

text: left

text: of

text: the

text: interface,

text: while

text: the

text: index

text: i

text: refers

text: to

text: the

text: component

text: to

text: the

text: right

text: of

text: this

text: interface.

text: There

text: is

text: no

text: interaction

text: between

text: the

text: components

text: everywhere

text: except

text: the

text: narrow

text: region

text: near

text: the

text: boundary

text: separating

text: them,

text: but

text: its

text: presence

text: is

text: taken

text: into

text: account

text: by

text: an

text: appropriate

text: boundary

text: condition

text: for

text: the

text: pressures

text: to

text: be

text: equal

text: on

text: both

text: sides

text: of

text: the

text: boundary.

text: In

text: addition,

text: we

text: assume

text: the

text: masses

text: of

text: the

text: atoms

text: in

text: both

text: components

text: to

text: be

text: identical

text: and

text: equal

text: to

text: unity,

text: while

text: the

text: trap

text: potential

text: at

text: the

text: expansion

text: stage

text: to

text: play

text: no

text: role

text: in

text: an

text: obvious

text: way.

text: The

text: solution

text: of

text: such

text: problems

text: is

text: simpliﬁed

text: considerably

text: if

text: we

text: pass

text: from

text: the

text: ordinary

text: physical

text: variables

text: ρ

text: u

text: to

text: the

text: so-called

text: “Riemann

text: invariants”.

text: For

text: Eqs.

text: i

text: i

text: which

text: coincide

text: in

text: form

text: with

text: the

text: gas-dynamic

text: equations,

text: the

text: Riemann

text: invariants

text: are

text: well

text: known

text: and

text: can

text: be

text: written

text: as

text: (see,

text: e.g.,

text: r±(i)

text: u

text: g

text: ρ

text: i

text: ii

text: i

text: In

text: these

text: variables

text: the

text: hydrodynamic

text: equations

text: take

text: a

text: simple

text: symmetric

text: form:

text: ∂r±(i)

text: ∂r±(i)

text: v±(i)(r(i),

text: r−(i))

text: ∂t

text: ∂x

text: where

text: the

text: ”Riemann

text: velocities

text: v±(i)

text: u

text: c

text: are

text: expressed

text: via

text: the

text: Riemann

text: invariants

text: by

text: the

text: i

text: i

text: relations

text: v(i)

text: r(i)

text: r−(i),

text: v−(i)

text: r(i)

text: r−(i).

text: If

text: the

text: solution

text: of

text: Eqs.

text: has

text: been

text: found

text: and

text: the

text: Riemann

text: invariants

text: are

text: known,

text: then

text: the

text: physical

text: variables

text: are

text: expressed

text: via

text: them

text: by

text: the

text: formulas

text: (r(i)

text: ρ

text: u

text: (r(i)

text: r−(i)).

text: i

text: i

text: ii

text: A

text: rarefaction

text: wave

text: is

text: known

text: to

text: belong

text: to

text: the

text: class

text: of

text: “simple

text: waves”

text: characterized

text: by

text: the

text: fact

text: that

text: one

text: of

text: the

text: Riemann

text: invariants

text: has

text: a

text: constant

text: value

text: along

text: the

text: ﬂow.

text: For

text: the

text: case

text: shown

text: in

text: Fig.

text: the

text: rarefaction

text: wave

text: propagates

text: to

text: the

text: left

text: in

text: the

text: ﬁrst

text: condensate.

text: Hence,

text: the

text: following

text: invariant

text: is

text: constant

text: for

text: it:

text: r

text: u

text: g

text: ρ

text: g

text: ρ

text: where

text: we

text: set

text: its

text: value

text: equal

text: to

text: the

text: value

text: at

text: the

text: boundary

text: with

text: the

text: condensate

text: at

text: rest.

text: It

text: must

text: have

text: the

text: same

text: value

text: at

text: the

text: rarefaction

text: wave

text: boundary

text: with

text: the

text: shelf

text: to

text: the

text: right

text: of

text: it:

text: u

text: g

text: ρ

text: g

text: ρ

text: The

text: pressures

text: of

text: the

text: components

text: at

text: the

text: boundary

text: between

text: them

text: are

text: equal,

text: which

text: gives

text: the

text: relation

text: g

text: ρ2

text: g

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text: ##LTLine##


text: (a) (b) Figure 3: Comparison of the energies for the symmetric phases (solid lines) and the asymmetric immiscible phase (dashed line) as a function of the interaction constant g . For the symmetric case, the solid curves 11 correspond to two conﬁgurations where the ﬁrst component is inside, while the second one is outside and vice versa. In panel (a) the curves correspond to g = 0.5 and the same number of particles in both 22 components: N : N = 10000 : 10000. In panel (b) we adopted g = 1 and N : N = 10000 : 30000. The 1 2 22 1 2 particle masses and trap frequencies are identical: m = m = 1 and ω = ω = 1. 1 2 1 2 µ m ω2 1 = 1 1 . (24) µ m ω2 2 2 2 From this condition and Eqs. (20) we will derive the following equation for the dash-dotted curve for the region where g2 < g g : 12 11 22 g 1 12 = . (25) − g22 N1 m2ω22 g11 + m2ω22 N1 N2 m1ω12 g12 m1ω12 N2 As we see, it is a hyperbola in the (g /g , g /g ) plane. In the case of immiscible components, 11 12 12 22 comparison of the energies for symmetric conﬁgurations shows (see Fig. 3) that the internal and external components also change places (see Figs. 2(e) and 2(f)) when passing through the hyperbola on which the equality g = g holds. Our numerical calculations show that this hyperbola does not 22 11 depend on the number of particles in the components. This completes the construction of a phase diagram in the Thomas-Fermi approximation. In general terms, In general terms, the constructed diagram gives a correct idea of the pattern of the component distributions in traps, except for the region near the part of the hyperbola that separates the dis- tributions of types Fig. 2(e) and 2(f). The point is that on this curve not only the energies of the symmetric distributions in Fig. 2(e) and 2(f) but also the energy of the asymmetric distribution, where the components are on diﬀerent sides of the trap center (see Fig. 2(k)), are equal to the same value. As a result of such a degeneracy of the energies, which is illustrated in Fig. 3, even a small perturba- tion makes one of the distributions energetically more favorable. As our numerical calculations show, allowance for the dispersion gives an advantage to the asymmetric phase in Fig. 2(k). This diﬀerence is not captured by the Thomas-Fermi approximation and requires a more accurate calculation. The above classiﬁcation of the possible initial states that the condensate has before the trap is switched oﬀ is suﬃcient for our purposes. It should be noted that the distributions found have breaks at the transition points from the overlap regions to the singlet ones. Clearly, the dispersion eﬀects at these points also become signiﬁcant and lead to a smoothing of the curves. In particular, on the solid curves the sharpening in the distribution at the center, where the density of the second component in the Thomas-Fermi approximation is zero, 8 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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## ##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## ##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## ##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: (a)

text: (b)

text: Figure

text: Comparison

text: of

text: the

text: energies

text: for

text: the

text: symmetric

text: phases

text: (solid

text: lines)

text: and

text: the

text: asymmetric

text: immiscible

text: phase

text: (dashed

text: line)

text: as

text: a

text: function

text: of

text: the

text: interaction

text: constant

text: g

text: For

text: the

text: symmetric

text: case,

text: the

text: solid

text: curves

text: correspond

text: to

text: two

text: conﬁgurations

text: where

text: the

text: ﬁrst

text: component

text: is

text: inside,

text: while

text: the

text: second

text: one

text: is

text: outside

text: and

text: vice

text: versa.

text: In

text: panel

text: (a)

text: the

text: curves

text: correspond

text: to

text: g

text: and

text: the

text: same

text: number

text: of

text: particles

text: in

text: both

text: components:

text: N

text: N

text: In

text: panel

text: (b)

text: we

text: adopted

text: g

text: and

text: N

text: N

text: The

text: particle

text: masses

text: and

text: trap

text: frequencies

text: are

text: identical:

text: m

text: m

text: and

text: ω

text: ω

text: µ

text: m

text: ω2

text: µ

text: m

text: ω2

text: From

text: this

text: condition

text: and

text: Eqs.

text: we

text: will

text: derive

text: the

text: following

text: equation

text: for

text: the

text: dash-dotted

text: curve

text: for

text: the

text: region

text: where

text: g

text: g

text: g

text: As

text: we

text: see,

text: it

text: is

text: a

text: hyperbola

text: in

text: the

text: (g

text: /g

text: g

text: /g

text: plane.

text: In

text: the

text: case

text: of

text: immiscible

text: components,

text: comparison

text: of

text: the

text: energies

text: for

text: symmetric

text: conﬁgurations

text: shows

text: (see

text: Fig.

text: that

text: the

text: internal

text: and

text: external

text: components

text: also

text: change

text: places

text: (see

text: Figs.

text: and

text: when

text: passing

text: through

text: the

text: hyperbola

text: on

text: which

text: the

text: equality

text: g

text: g

text: holds.

text: Our

text: numerical

text: calculations

text: show

text: that

text: this

text: hyperbola

text: does

text: not

text: depend

text: on

text: the

text: number

text: of

text: particles

text: in

text: the

text: components.

text: This

text: completes

text: the

text: construction

text: of

text: a

text: phase

text: diagram

text: in

text: the

text: Thomas-Fermi

text: approximation.

text: In

text: general

text: terms,

text: In

text: general

text: terms,

text: the

text: constructed

text: diagram

text: gives

text: a

text: correct

text: idea

text: of

text: the

text: pattern

text: of

text: the

text: component

text: distributions

text: in

text: traps,

text: except

text: for

text: the

text: region

text: near

text: the

text: part

text: of

text: the

text: hyperbola

text: that

text: separates

text: the

text: dis-

text: tributions

text: of

text: types

text: Fig.

text: and

text: The

text: point

text: is

text: that

text: on

text: this

text: curve

text: not

text: only

text: the

text: energies

text: of

text: the

text: symmetric

text: distributions

text: in

text: Fig.

text: and

text: but

text: also

text: the

text: energy

text: of

text: the

text: asymmetric

text: distribution,

text: where

text: the

text: components

text: are

text: on

text: diﬀerent

text: sides

text: of

text: the

text: trap

text: center

text: (see

text: Fig.

text: are

text: equal

text: to

text: the

text: same

text: value.

text: As

text: a

text: result

text: of

text: such

text: a

text: degeneracy

text: of

text: the

text: energies,

text: which

text: is

text: illustrated

text: in

text: Fig.

text: even

text: a

text: small

text: perturba-

text: tion

text: makes

text: one

text: of

text: the

text: distributions

text: energetically

text: more

text: favorable.

text: As

text: our

text: numerical

text: calculations

text: show,

text: allowance

text: for

text: the

text: dispersion

text: gives

text: an

text: advantage

text: to

text: the

text: asymmetric

text: phase

text: in

text: Fig.

text: This

text: diﬀerence

text: is

text: not

text: captured

text: by

text: the

text: Thomas-Fermi

text: approximation

text: and

text: requires

text: a

text: more

text: accurate

text: calculation.

text: The

text: above

text: classiﬁcation

text: of

text: the

text: possible

text: initial

text: states

text: that

text: the

text: condensate

text: has

text: before

text: the

text: trap

text: is

text: switched

text: oﬀ

text: is

text: suﬃcient

text: for

text: our

text: purposes.

text: It

text: should

text: be

text: noted

text: that

text: the

text: distributions

text: found

text: have

text: breaks

text: at

text: the

text: transition

text: points

text: from

text: the

text: overlap

text: regions

text: to

text: the

text: singlet

text: ones.

text: Clearly,

text: the

text: dispersion

text: eﬀects

text: at

text: these

text: points

text: also

text: become

text: signiﬁcant

text: and

text: lead

text: to

text: a

text: smoothing

text: of

text: the

text: curves.

text: In

text: particular,

text: on

text: the

text: solid

text: curves

text: the

text: sharpening

text: in

text: the

text: distribution

text: at

text: the

text: center,

text: where

text: the

text: density

text: of

text: the

text: second

text: component

text: in

text: the

text: Thomas-Fermi

text: approximation

text: is

text: zero,

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text: (a) (b) Figure 4: Two characteristic cases for the particle number densities: (a) the ﬁrst component (solid line) has a singlet region, (b) the components are miscible everywhere. The components are numbered in such a way that the ﬁrst component is always surrounded by the second one. − − α˙ n = αn βn, α˙ n = 3αnβn, (27) i,0 i,0 i i i i − − g − g β˙o = (βo)2 2 ii αo 2 ij αo + ω (t)2, i i m i m j i i i (28) − − g β˙s = (βs)2 2 ii αs + ω (t)2 i i m i i i (the dot denotes a time derivative). Introducing a new variable ζn deﬁned by the relation i m αn = i . (29) i 2g (ζn)3 ii i we simplify considerably these equations: from (27) we ﬁnd that βn = ζ˙n/ζn, and Eqs. (28) will i i i then take the form 1 m g ζo − 1 − ζ¨io = (ζo)2 + mj gij (ζoi)3 ωi(t)2ζio, ζ¨is = (ζs)2 ωi(t)2ζis. (30) i i ii j i This system of six second-order diﬀerential equations deﬁnes the motion of the Bose-Einstein condensate components. The last terms of the equations reﬂect the inﬂuence of the conﬁning potential on the motion, while the other terms arise from the interaction between atoms. In what follows, we will be interested in the condensate dynamics after the trap has been switched oﬀ, i.e., we should set ω (t) = 0 at t > 0. The initial conditions for these equations are determined by the original i conﬁgurations that were described in the previous section. At ﬁxed nonlinear constants they depend on the trap parameters and the number of particles in each component. Having solved the system of equations (30), we can ﬁnd the velocities un(x, t) and coeﬃcients i αn(t). To ﬁnd αn (t), we will use the normalization of the wave functions (20) and the fact that at i i,0 the boundary between the singlet and overlap regions the pressures in them and, consequently, the densities are equal. To be speciﬁc, we number the condensate components in such a way that the ﬁrst component is surrounded by the second one, as shown in Fig. 4 (the ﬁrst and second components are indicated by the solid and dashed lines, respectively), so that we have 10 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (a)

text: (b)

text: Figure

text: Two

text: characteristic

text: cases

text: for

text: the

text: particle

text: number

text: densities:

text: (a)

text: the

text: ﬁrst

text: component

text: (solid

text: line)

text: has

text: a

text: singlet

text: region,

text: (b)

text: the

text: components

text: are

text: miscible

text: everywhere.

text: The

text: components

text: are

text: numbered

text: in

text: such

text: a

text: way

text: that

text: the

text: ﬁrst

text: component

text: is

text: always

text: surrounded

text: by

text: the

text: second

text: one.

text: α˙

text: n

text: αn

text: βn,

text: α˙

text: n

text: i

text: i

text: i

text: i

text: g

text: g

text: β˙o

text: ii

text: αo

text: ij

text: αo

text: ω

text: i

text: i

text: m

text: i

text: m

text: j

text: i

text: i

text: i

text: g

text: β˙s

text: ii

text: αs

text: ω

text: i

text: i

text: m

text: i

text: i

text: i

text: (the

text: dot

text: denotes

text: a

text: time

text: derivative).

text: Introducing

text: a

text: new

text: variable

text: ζn

text: deﬁned

text: by

text: the

text: relation

text: i

text: m

text: αn

text: i

text: i

text: ii

text: i

text: we

text: simplify

text: considerably

text: these

text: equations:

text: from

text: we

text: ﬁnd

text: that

text: βn

text: ζ˙n/ζn,

text: and

text: Eqs.

text: will

text: i

text: i

text: i

text: then

text: take

text: the

text: form

text: m

text: g

text: ζo

text: ζ¨io

text: mj

text: gij

text: ζ¨is

text: i

text: i

text: ii

text: j

text: i

text: This

text: system

text: of

text: six

text: second-order

text: diﬀerential

text: equations

text: deﬁnes

text: the

text: motion

text: of

text: the

text: Bose-Einstein

text: condensate

text: components.

text: The

text: last

text: terms

text: of

text: the

text: equations

text: reﬂect

text: the

text: inﬂuence

text: of

text: the

text: conﬁning

text: potential

text: on

text: the

text: motion,

text: while

text: the

text: other

text: terms

text: arise

text: from

text: the

text: interaction

text: between

text: atoms.

text: In

text: what

text: follows,

text: we

text: will

text: be

text: interested

text: in

text: the

text: condensate

text: dynamics

text: after

text: the

text: trap

text: has

text: been

text: switched

text: oﬀ,

text: i.e.,

text: we

text: should

text: set

text: ω

text: (t)

text: at

text: t

text: The

text: initial

text: conditions

text: for

text: these

text: equations

text: are

text: determined

text: by

text: the

text: original

text: i

text: conﬁgurations

text: that

text: were

text: described

text: in

text: the

text: previous

text: section.

text: At

text: ﬁxed

text: nonlinear

text: constants

text: they

text: depend

text: on

text: the

text: trap

text: parameters

text: and

text: the

text: number

text: of

text: particles

text: in

text: each

text: component.

text: Having

text: solved

text: the

text: system

text: of

text: equations

text: we

text: can

text: ﬁnd

text: the

text: velocities

text: un(x,

text: t)

text: and

text: coeﬃcients

text: i

text: αn(t).

text: To

text: ﬁnd

text: αn

text: (t),

text: we

text: will

text: use

text: the

text: normalization

text: of

text: the

text: wave

text: functions

text: and

text: the

text: fact

text: that

text: at

text: i

text: the

text: boundary

text: between

text: the

text: singlet

text: and

text: overlap

text: regions

text: the

text: pressures

text: in

text: them

text: and,

text: consequently,

text: the

text: densities

text: are

text: equal.

text: To

text: be

text: speciﬁc,

text: we

text: number

text: the

text: condensate

text: components

text: in

text: such

text: a

text: way

text: that

text: the

text: ﬁrst

text: component

text: is

text: surrounded

text: by

text: the

text: second

text: one,

text: as

text: shown

text: in

text: Fig.

text: (the

text: ﬁrst

text: and

text: second

text: components

text: are

text: indicated

text: by

text: the

text: solid

text: and

text: dashed

text: lines,

text: respectively),

text: so

text: that

text: we

text: have

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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text: Figure 1. Let h(x) be a degree d monic polynomial with all other coeﬃcients chosen − independently and randomly from integers in the interval [ K, K]. A result proven by Chela [4] (see Theorem 3.1) implies that the probability that h(x) is reducible divided by the probability the probability that the constant coeﬃcient is zero goes to a limit as K goes to inﬁnity. Above is a plot of data supporting this result when K = 3, 10 and 100, where T is the number of random trials trials to compute the respective data points. The data points are a ratio of the probability of reducibility divided by 1 , which is the probability that the constant coeﬃcient is zero. The 2K+1 ﬁgure indicates that convergence to the limit proven by Chela is relatively fast, ± especially for degree 6 and larger. The 99.9999% conﬁdence intervals are 0.058, ± ± 0.053, and 0.251 for K = 3, 10, and 100, respectively. factors have positive degree, so we factor over the rationals; for example, 2x2 + 4 = 2(x2 + 2) is irreducible over the rationals, because 2 has a multiplicative inverse in the rationals. O’Rourke and Wood [17] study a variety of models, proving that pointwise delocalization of the roots of a random polynomial implies that low-degree factors are very unlikely. Konyagin [12] studied random monic polynomials with 0 or 1 for coeﬃcients and proved a bound on the probability of low-degree factors which he then used to prove a bound on the probability of reducibility. Heuristic 1.1 compliments these results on low-degree factors by suggesting that high-degree factors are even less likely than low-degree factors, and proving results in this direction is an interesting open problem. Kuba [13] studied non-monic polynomials with integer coeﬃcients chosen independently in the − interval [ K, K] (conditioned on the lead coeﬃcient being nonzero), and proved a result implying that the probability of reducibility divided by the probability that the constant term is zero is bounded by a constant. This can be viewed as a step towards proving a version of Chela’s [4] result (and a case of Heuristic 1.1) for non-monic polynomials. In Section , we will test generalizations of Hilbert’s Irreducibility Theorem, studying polynomials with ﬁxed degree in the limit as the support for the coeﬃcients grows, and we will state special cases of Heuristic 1.1 as conjectures. In Hilbert’s Irreducibility Theorem, the degree is ﬁxed and the support of the coeﬃcients is growing, and we believe that Heuristic 1.1 also holds for cases where the range of support of the coeﬃcients is ﬁxed and the degree goes to inﬁnity. For example, let g{ } (x) be a monic 0,1 ,d degree d polynomial with constant coeﬃcient equal to 1 and with every other coeﬃcient equal to 0 or 1 independently with probability 1. In 1993, Odlyzko and Poonen [16] conjectured that the 2 2 ##LTFigure## ##LTLine## ##LTLine##

text: Figure

text: Let

text: h(x)

text: be

text: a

text: degree

text: d

text: monic

text: polynomial

text: with

text: all

text: other

text: coeﬃcients

text: chosen

text: independently

text: and

text: randomly

text: from

text: integers

text: in

text: the

text: interval

text: K,

text: K].

text: A

text: result

text: proven

text: by

text: Chela

text: (see

text: Theorem

text: implies

text: that

text: the

text: probability

text: that

text: h(x)

text: is

text: reducible

text: divided

text: by

text: the

text: probability

text: the

text: probability

text: that

text: the

text: constant

text: coeﬃcient

text: is

text: zero

text: goes

text: to

text: a

text: limit

text: as

text: K

text: goes

text: to

text: inﬁnity.

text: Above

text: is

text: a

text: plot

text: of

text: data

text: supporting

text: this

text: result

text: when

text: K

text: and

text: where

text: T

text: is

text: the

text: number

text: of

text: random

text: trials

text: trials

text: to

text: compute

text: the

text: respective

text: data

text: points.

text: The

text: data

text: points

text: are

text: a

text: ratio

text: of

text: the

text: probability

text: of

text: reducibility

text: divided

text: by

text: which

text: is

text: the

text: probability

text: that

text: the

text: constant

text: coeﬃcient

text: is

text: zero.

text: The

text: ﬁgure

text: indicates

text: that

text: convergence

text: to

text: the

text: limit

text: proven

text: by

text: Chela

text: is

text: relatively

text: fast,

text: especially

text: for

text: degree

text: and

text: larger.

text: The

text: conﬁdence

text: intervals

text: are

text: and

text: for

text: K

text: and

text: respectively.

text: factors

text: have

text: positive

text: degree,

text: so

text: we

text: factor

text: over

text: the

text: rationals;

text: for

text: example,

text: is

text: irreducible

text: over

text: the

text: rationals,

text: because

text: has

text: a

text: multiplicative

text: inverse

text: in

text: the

text: rationals.

text: O’Rourke

text: and

text: Wood

text: study

text: a

text: variety

text: of

text: models,

text: proving

text: that

text: pointwise

text: delocalization

text: of

text: the

text: roots

text: of

text: a

text: random

text: polynomial

text: implies

text: that

text: low-degree

text: factors

text: are

text: very

text: unlikely.

text: Konyagin

text: studied

text: random

text: monic

text: polynomials

text: with

text: or

text: for

text: coeﬃcients

text: and

text: proved

text: a

text: bound

text: on

text: the

text: probability

text: of

text: low-degree

text: factors

text: which

text: he

text: then

text: used

text: to

text: prove

text: a

text: bound

text: on

text: the

text: probability

text: of

text: reducibility.

text: Heuristic

text: compliments

text: these

text: results

text: on

text: low-degree

text: factors

text: by

text: suggesting

text: that

text: high-degree

text: factors

text: are

text: even

text: less

text: likely

text: than

text: low-degree

text: factors,

text: and

text: proving

text: results

text: in

text: this

text: direction

text: is

text: an

text: interesting

text: open

text: problem.

text: Kuba

text: studied

text: non-monic

text: polynomials

text: with

text: integer

text: coeﬃcients

text: chosen

text: independently

text: in

text: the

text: interval

text: K,

text: K]

text: (conditioned

text: on

text: the

text: lead

text: coeﬃcient

text: being

text: nonzero),

text: and

text: proved

text: a

text: result

text: implying

text: that

text: the

text: probability

text: of

text: reducibility

text: divided

text: by

text: the

text: probability

text: that

text: the

text: constant

text: term

text: is

text: zero

text: is

text: bounded

text: by

text: a

text: constant.

text: This

text: can

text: be

text: viewed

text: as

text: a

text: step

text: towards

text: proving

text: a

text: version

text: of

text: Chela’s

text: result

text: (and

text: a

text: case

text: of

text: Heuristic

text: for

text: non-monic

text: polynomials.

text: In

text: Section

text: we

text: will

text: test

text: generalizations

text: of

text: Hilbert’s

text: Irreducibility

text: Theorem,

text: studying

text: polynomials

text: with

text: ﬁxed

text: degree

text: in

text: the

text: limit

text: as

text: the

text: support

text: for

text: the

text: coeﬃcients

text: grows,

text: and

text: we

text: will

text: state

text: special

text: cases

text: of

text: Heuristic

text: as

text: conjectures.

text: In

text: Hilbert’s

text: Irreducibility

text: Theorem,

text: the

text: degree

text: is

text: ﬁxed

text: and

text: the

text: support

text: of

text: the

text: coeﬃcients

text: is

text: growing,

text: and

text: we

text: believe

text: that

text: Heuristic

text: also

text: holds

text: for

text: cases

text: where

text: the

text: range

text: of

text: support

text: of

text: the

text: coeﬃcients

text: is

text: ﬁxed

text: and

text: the

text: degree

text: goes

text: to

text: inﬁnity.

text: For

text: example,

text: let

text: g{

text: (x)

text: be

text: a

text: monic

text: ,d

text: degree

text: d

text: polynomial

text: with

text: constant

text: coeﬃcient

text: equal

text: to

text: and

text: with

text: every

text: other

text: coeﬃcient

text: equal

text: to

text: or

text: independently

text: with

text: probability

text: In

text: Odlyzko

text: and

text: Poonen

text: conjectured

text: that

text: the

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: Figure 3. The plot above shows probability of reducibility for monic polynomials − where each coeﬃcient is +1 or 1 independently with probability 1/2 (note that this ﬁgure also appears in [17, Figure 2]). The random data was calculated using 150,000,000 trials, giving a 99.9999% conﬁdence interval of 0.0002 for every value. The lower bound on the probability of reducibility for odd degrees (dashed) is the asymptotic probability that a degree d polynomial has a linear factor (see Proposi- tion 5.3). If the degree is even, the polynomials cannot have a linear factor, and the reducibility probability appears to be distinctly lower. For even degrees, a Galois theory argument ([17, Subsection 2.2]) proves that the reducibility probability is exactly zero whenever d + 1 is prime and 2 generates the multiplicative group of Z × ( /(d + 1)) , which should happen for inﬁnitely many d by Artin’s Conjecture (see, for example, [15]). − that either 1 or 1 is a double root (and they further extend this result to polynomials with coef- − ﬁcients 1, 0, or 1). This result is related to Section where we conjecture for random polynomials − − with 1 or 1 coeﬃcients that, among reducible polynomials, the probability that x + 1 or x 1 is a factor tends to 1 (see Conjecture 2.1). Very roughly, both Peled, Sen, and Zeitouni’s result [18] and Conjecture 2.1 suggest that the coincidence of having a double root or the coincidence of − factoring are most likely to happen in the simplest way, and that is when 1 or 1 is a root. Random polynomials of the sort considered in Hilbert’s Irreducibility Theorem have been studied in probabilistic Galois theory. For example, van der Waerden [25] proved that if h − (x) is [ K,K],d monic with degree d and all of its coeﬃcients are chosen independently and uniformly from integers − in the interval [ K, K], then the probability that the Galois group for h − (x) is S , the [ K,K],d d symmetric group of d elements, tends to 1 as K tends to inﬁnity. Work proving successively better upper bounds on the probability that h − (x) does not have Galois group S has continued [ K,K],d d with the work of Knobloch in 1955 [10] and 1956 [11], Gallagher in 1973 [6], Zywina in 2010 [26], Dietmann in 2013 [5], and ﬁnally Rivin [19] in 2015, who proved the upper bound logc K , K where c = c is a constant depending on d, which is the ﬁrst upper bound demonstrating that the d probability decreases linearly in K, up to a polylog factor. 4 ##LTFigure## ##LTLine##

text: Figure

text: The

text: plot

text: above

text: shows

text: probability

text: of

text: reducibility

text: for

text: monic

text: polynomials

text: where

text: each

text: coeﬃcient

text: is

text: or

text: independently

text: with

text: probability

text: (note

text: that

text: this

text: ﬁgure

text: also

text: appears

text: in

text: Figure

text: The

text: random

text: data

text: was

text: calculated

text: using

text: trials,

text: giving

text: a

text: conﬁdence

text: interval

text: of

text: for

text: every

text: value.

text: The

text: lower

text: bound

text: on

text: the

text: probability

text: of

text: reducibility

text: for

text: odd

text: degrees

text: (dashed)

text: is

text: the

text: asymptotic

text: probability

text: that

text: a

text: degree

text: d

text: polynomial

text: has

text: a

text: linear

text: factor

text: (see

text: Proposi-

text: tion

text: If

text: the

text: degree

text: is

text: even,

text: the

text: polynomials

text: cannot

text: have

text: a

text: linear

text: factor,

text: and

text: the

text: reducibility

text: probability

text: appears

text: to

text: be

text: distinctly

text: lower.

text: For

text: even

text: degrees,

text: a

text: Galois

text: theory

text: argument

text: Subsection

text: proves

text: that

text: the

text: reducibility

text: probability

text: is

text: exactly

text: zero

text: whenever

text: d

text: is

text: prime

text: and

text: generates

text: the

text: multiplicative

text: group

text: of

text: Z

text: /(d

text: which

text: should

text: happen

text: for

text: inﬁnitely

text: many

text: d

text: by

text: Artin’s

text: Conjecture

text: (see,

text: for

text: example,

text: that

text: either

text: or

text: is

text: a

text: double

text: root

text: (and

text: they

text: further

text: extend

text: this

text: result

text: to

text: polynomials

text: with

text: coef-

text: ﬁcients

text: or

text: This

text: result

text: is

text: related

text: to

text: Section

text: where

text: we

text: conjecture

text: for

text: random

text: polynomials

text: with

text: or

text: coeﬃcients

text: that,

text: among

text: reducible

text: polynomials,

text: the

text: probability

text: that

text: x

text: or

text: x

text: is

text: a

text: factor

text: tends

text: to

text: (see

text: Conjecture

text: Very

text: roughly,

text: both

text: Peled,

text: Sen,

text: and

text: Zeitouni’s

text: result

text: and

text: Conjecture

text: suggest

text: that

text: the

text: coincidence

text: of

text: having

text: a

text: double

text: root

text: or

text: the

text: coincidence

text: of

text: factoring

text: are

text: most

text: likely

text: to

text: happen

text: in

text: the

text: simplest

text: way,

text: and

text: that

text: is

text: when

text: or

text: is

text: a

text: root.

text: Random

text: polynomials

text: of

text: the

text: sort

text: considered

text: in

text: Hilbert’s

text: Irreducibility

text: Theorem

text: have

text: been

text: studied

text: in

text: probabilistic

text: Galois

text: theory.

text: For

text: example,

text: van

text: der

text: Waerden

text: proved

text: that

text: if

text: h

text: (x)

text: is

text: K,K],d

text: monic

text: with

text: degree

text: d

text: and

text: all

text: of

text: its

text: coeﬃcients

text: are

text: chosen

text: independently

text: and

text: uniformly

text: from

text: integers

text: in

text: the

text: interval

text: K,

text: K],

text: then

text: the

text: probability

text: that

text: the

text: Galois

text: group

text: for

text: h

text: (x)

text: is

text: S

text: the

text: K,K],d

text: d

text: symmetric

text: group

text: of

text: d

text: elements,

text: tends

text: to

text: as

text: K

text: tends

text: to

text: inﬁnity.

text: Work

text: proving

text: successively

text: better

text: upper

text: bounds

text: on

text: the

text: probability

text: that

text: h

text: (x)

text: does

text: not

text: have

text: Galois

text: group

text: S

text: has

text: continued

text: K,K],d

text: d

text: with

text: the

text: work

text: of

text: Knobloch

text: in

text: and

text: Gallagher

text: in

text: Zywina

text: in

text: Dietmann

text: in

text: and

text: ﬁnally

text: Rivin

text: in

text: who

text: proved

text: the

text: upper

text: bound

text: logc

text: K

text: K

text: where

text: c

text: c

text: is

text: a

text: constant

text: depending

text: on

text: d,

text: which

text: is

text: the

text: ﬁrst

text: upper

text: bound

text: demonstrating

text: that

text: the

text: d

text: probability

text: decreases

text: linearly

text: in

text: K,

text: up

text: to

text: a

text: polylog

text: factor.

text: ##LTFigure##

text: ##LTLine##