text: 7 FIG. 4: (Color online) (a) Trajectories of the head and tail particles of the two swimmers for several initial positions. Here, − ∆ = R1 R2 at t = 0. The swimmer 1(2) is swimming from left(right) to right(left). For ∆/(cid:96) = 1.2, the head particle H,y H,y 0 of each swimmer touches the phantom particle of the other swimmer, and thus, due to the dynamical rule introduced in the present model, the two swimmers stop moving completely. (This would not happen for asymmetric initial conditions, nor in the presence of thermal ﬂuctuations, or disturbance by the ﬂow caused by other swimmers.) In (b), the velocity of the swimmer 1 along the x axis, V 1 , is plotted against R1 . Initially, when the two swimmers are approaching, they repel each other, H,x H,x because the repulsive HIs due to the incompressibility of the solvent are dominant. Then, when they are passing each other, an activity-induced hydrodynamic attraction dominates; that is, one swimmer is attracted to the streamline induced by the other swimmer, resulting in an acceleration of the swim speed. It is evident from the trajectories that such (conﬁguration- dependent) HIs are stronger for smaller separation distances. (c) Snapshot of two swimmers passing each other for ∆/(cid:96) = 1.6 0 and f = 10. We also show the velocity ﬁeld v(x, y, z = 0), where the axes of the two swimmers lie in the plane. The color act bar represents the value of the ﬂuid velocity scaled by the value of v at f = 10 shown in the inset of Fig. 2(a). 0 act (cid:80) Although large Pe suggests that active eﬀects dominate 0 over thermal ﬂuctuations, it should be noted that in di- FIG. 7: (Color online) The pair distribution function laulltoewsorleuotriioenntraottioantioonfatlhBersowwinmiamninmgodtiiorenctiisones.seAnntioatlhteor gnˆm(cid:48) m) (cid:48)−(rθ,]θδ)(|dReﬁαne−dRasαg(cid:48)(cid:48)m| −m(cid:48)r()r,iθn) t=he4πsNtLe32ard2y stαa(cid:54)=tαe(cid:48)aδt[cfos−1=(nˆ1α4· α m m act mechanism involves the “tumbling” of bacteria at ran- with (a) and without (b) HIs. The purple, green, and yellow dom intervals, causing sudden non-Brownian reorienta- curves are the contours corresponding to 1/6, 3/6, and 5/6 of tion; this is not in our model which therefore describes the maximum value of g (cid:48)(r, θ), respectively. mm only “smooth swimming” bacteria. For these, the cho- sen P´eclet numbers are within the achievable range [59]. Notice also that Pe and v are values for an isolated 0 0 swimmer; in the semidilute regime, due to many-body in the steady state. Both with and without HIs, in (cid:54) interactions, a marked slowdown occurs, which will be an active suspension (f = 0), g (cid:48)(r) increases with act mm discussed below. increasing f , suggesting that the swimmers transiently act The work done by the active force is transformed into form clusters. This clustering is more evident from the the kinetic energy of the solvent and of the swimmers, structure factor given by (cid:90) (cid:88) which is eventually dissipated by viscosity. In the present simulation, the increase of the average kinetic energy den- sity of the solvent due to the active force is at most S(k) = 1 + N dre−ik·rg (cid:48)(r), (20) V mm approximately 1% of the prescribed value given by the (cid:48) m,m =H,T equipartition law, in which the velocity of a thermal ﬂuid (cid:104) (cid:48) (cid:105) − (cid:48) obeys v (r)v (r ) = Tδ δ(r r )/ρ [60], where i and j i j ij which is shown in Fig. 6. However, there is a marked denote Cartesian components of v. diﬀerence in the behavior with and without HIs; while with HIs the increase is most pronounced in g (r), with- TT out HIs it is more apparent in g (r). More signiﬁcantly, HH FIG. 5: (Color online) The radial distribution function (cid:48) the overall clustering tendency is signiﬁcantly enhanced g (cid:48)(r) where (m, m ) = (H, T) with (a) and without (b) mm by the addition of HIs. HIs. These diﬀerent behaviors should directly reﬂect the dif- ference in the clustering mechanism with and without HIs. In order to further understand this diﬀerence, we consider the following pair distribution function, FIG. 6: (Color online) The structure factor S(k) for various values of f . With increasing f , S(k) grows at small k. For fact (cid:54)= 0, Sa(ckt ) for 2ak <∼ 1 can bacetdescribed by the Ornstein- (cid:88) L3 Zernike form, which is represented by the black solid curves g (cid:48)(r, θ) = mm 4πN2r2 in the cases with and without HIs, respectively.The results × − · − | − (cid:48) | − with neither the dynamical rule nor HIs are also shown. δ[cos 1(nˆ nˆ (cid:48)) θ]δ( Rα Rα (cid:48) r), α α m m (cid:54) (cid:48) α=α Each simulation starts from a randomly distributed (21) state, and ﬁnally reaches a steady state with nontrivial correlations. In Fig. 5, we show the radial distribution − · function (cid:88) where cos(cid:48) 1(nˆα nˆα(cid:48)) is the relative angle between α th and α th swimmers, and the angular integral of g (cid:48)(θ, ψ) gives g (cid:48)(r). Figures 7(a) and 7(b) repre- L3 | − (cid:48) | − mm mm gmm(cid:48)(r) = 4πN2r2 (cid:54) (cid:48) δ( Rαm Rαm(cid:48) r) (19) sthenetﬁtrhste pseteaakdoyf-sgtate(cid:48)(dri)st(r1ib<utrio/n2ao<f g2m),mf(cid:48)r(oθm, ψw) haircohuwnde α=α mm ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FIG.

text: (Color

text: online)

text: (a)

text: Trajectories

text: of

text: the

text: head

text: and

text: tail

text: particles

text: of

text: the

text: two

text: swimmers

text: for

text: several

text: initial

text: positions.

text: Here,

text: at

text: t

text: The

text: swimmer

text: is

text: swimming

text: from

text: left(right)

text: to

text: right(left).

text: For

text: the

text: head

text: particle

text: H,y

text: H,y

text: of

text: each

text: swimmer

text: touches

text: the

text: phantom

text: particle

text: of

text: the

text: other

text: swimmer,

text: and

text: thus,

text: due

text: to

text: the

text: dynamical

text: rule

text: introduced

text: in

text: the

text: present

text: model,

text: the

text: two

text: swimmers

text: stop

text: moving

text: completely.

text: (This

text: would

text: not

text: happen

text: for

text: asymmetric

text: initial

text: conditions,

text: nor

text: in

text: the

text: presence

text: of

text: thermal

text: ﬂuctuations,

text: or

text: disturbance

text: by

text: the

text: ﬂow

text: caused

text: by

text: other

text: swimmers.)

text: In

text: (b),

text: the

text: velocity

text: of

text: the

text: swimmer

text: along

text: the

text: x

text: axis,

text: V

text: is

text: plotted

text: against

text: Initially,

text: when

text: the

text: two

text: swimmers

text: are

text: approaching,

text: they

text: repel

text: each

text: other,

text: H,x

text: H,x

text: because

text: the

text: repulsive

text: HIs

text: due

text: to

text: the

text: incompressibility

text: of

text: the

text: solvent

text: are

text: dominant.

text: Then,

text: when

text: they

text: are

text: passing

text: each

text: other,

text: an

text: activity-induced

text: hydrodynamic

text: attraction

text: dominates;

text: that

text: is,

text: one

text: swimmer

text: is

text: attracted

text: to

text: the

text: streamline

text: induced

text: by

text: the

text: other

text: swimmer,

text: resulting

text: in

text: an

text: acceleration

text: of

text: the

text: swim

text: speed.

text: It

text: is

text: evident

text: from

text: the

text: trajectories

text: that

text: such

text: (conﬁguration-

text: dependent)

text: HIs

text: are

text: stronger

text: for

text: smaller

text: separation

text: distances.

text: (c)

text: Snapshot

text: of

text: two

text: swimmers

text: passing

text: each

text: other

text: for

text: and

text: f

text: We

text: also

text: show

text: the

text: velocity

text: ﬁeld

text: v(x,

text: y,

text: z

text: where

text: the

text: axes

text: of

text: the

text: two

text: swimmers

text: lie

text: in

text: the

text: plane.

text: The

text: color

text: act

text: bar

text: represents

text: the

text: value

text: of

text: the

text: ﬂuid

text: velocity

text: scaled

text: by

text: the

text: value

text: of

text: v

text: at

text: f

text: shown

text: in

text: the

text: inset

text: of

text: Fig.

text: act

text: Although

text: large

text: Pe

text: suggests

text: that

text: active

text: eﬀects

text: dominate

text: over

text: thermal

text: ﬂuctuations,

text: it

text: should

text: be

text: noted

text: that

text: in

text: di-

text: FIG.

text: (Color

text: online)

text: The

text: pair

text: distribution

text: function

text: m)

text: α

text: m

text: m

text: act

text: mechanism

text: involves

text: the

text: “tumbling”

text: of

text: bacteria

text: at

text: ran-

text: with

text: (a)

text: and

text: without

text: (b)

text: HIs.

text: The

text: purple,

text: green,

text: and

text: yellow

text: dom

text: intervals,

text: causing

text: sudden

text: non-Brownian

text: reorienta-

text: curves

text: are

text: the

text: contours

text: corresponding

text: to

text: and

text: of

text: tion;

text: this

text: is

text: not

text: in

text: our

text: model

text: which

text: therefore

text: describes

text: the

text: maximum

text: value

text: of

text: g

text: θ),

text: respectively.

text: mm

text: only

text: “smooth

text: swimming”

text: bacteria.

text: For

text: these,

text: the

text: cho-

text: sen

text: P´eclet

text: numbers

text: are

text: within

text: the

text: achievable

text: range

text: Notice

text: also

text: that

text: Pe

text: and

text: v

text: are

text: values

text: for

text: an

text: isolated

text: swimmer;

text: in

text: the

text: semidilute

text: regime,

text: due

text: to

text: many-body

text: in

text: the

text: steady

text: state.

text: Both

text: with

text: and

text: without

text: HIs,

text: in

text: interactions,

text: a

text: marked

text: slowdown

text: occurs,

text: which

text: will

text: be

text: an

text: active

text: suspension

text: (f

text: g

text: increases

text: with

text: act

text: mm

text: discussed

text: below.

text: increasing

text: f

text: suggesting

text: that

text: the

text: swimmers

text: transiently

text: act

text: The

text: work

text: done

text: by

text: the

text: active

text: force

text: is

text: transformed

text: into

text: form

text: clusters.

text: This

text: clustering

text: is

text: more

text: evident

text: from

text: the

text: the

text: kinetic

text: energy

text: of

text: the

text: solvent

text: and

text: of

text: the

text: swimmers,

text: structure

text: factor

text: given

text: by

text: which

text: is

text: eventually

text: dissipated

text: by

text: viscosity.

text: In

text: the

text: present

text: simulation,

text: the

text: increase

text: of

text: the

text: average

text: kinetic

text: energy

text: den-

text: sity

text: of

text: the

text: solvent

text: due

text: to

text: the

text: active

text: force

text: is

text: at

text: most

text: S(k)

text: N

text: dre−ik·rg

text: V

text: mm

text: approximately

text: of

text: the

text: prescribed

text: value

text: given

text: by

text: the

text: m,m

text: =H,T

text: equipartition

text: law,

text: in

text: which

text: the

text: velocity

text: of

text: a

text: thermal

text: ﬂuid

text: obeys

text: v

text: (r)v

text: (r

text: Tδ

text: δ(r

text: r

text: )/ρ

text: where

text: i

text: and

text: j

text: i

text: j

text: ij

text: which

text: is

text: shown

text: in

text: Fig.

text: However,

text: there

text: is

text: a

text: marked

text: denote

text: Cartesian

text: components

text: of

text: v.

text: diﬀerence

text: in

text: the

text: behavior

text: with

text: and

text: without

text: HIs;

text: while

text: with

text: HIs

text: the

text: increase

text: is

text: most

text: pronounced

text: in

text: g

text: (r),

text: with-

text: TT

text: out

text: HIs

text: it

text: is

text: more

text: apparent

text: in

text: g

text: (r).

text: More

text: signiﬁcantly,

text: HH

text: FIG.

text: (Color

text: online)

text: The

text: radial

text: distribution

text: function

text: the

text: overall

text: clustering

text: tendency

text: is

text: signiﬁcantly

text: enhanced

text: g

text: where

text: (m,

text: m

text: (H,

text: T)

text: with

text: (a)

text: and

text: without

text: (b)

text: mm

text: by

text: the

text: addition

text: of

text: HIs.

text: HIs.

text: These

text: diﬀerent

text: behaviors

text: should

text: directly

text: reﬂect

text: the

text: dif-

text: ference

text: in

text: the

text: clustering

text: mechanism

text: with

text: and

text: without

text: HIs.

text: In

text: order

text: to

text: further

text: understand

text: this

text: diﬀerence,

text: we

text: consider

text: the

text: following

text: pair

text: distribution

text: function,

text: FIG.

text: (Color

text: online)

text: The

text: structure

text: factor

text: S(k)

text: for

text: various

text: values

text: of

text: f

text: With

text: increasing

text: f

text: S(k)

text: grows

text: at

text: small

text: k.

text: For

text: fact

text: Sa(ckt

text: for

text: can

text: bacetdescribed

text: by

text: the

text: Ornstein-

text: Zernike

text: form,

text: which

text: is

text: represented

text: by

text: the

text: black

text: solid

text: curves

text: g

text: θ)

text: mm

text: in

text: the

text: cases

text: with

text: and

text: without

text: HIs,

text: respectively.The

text: results

text: with

text: neither

text: the

text: dynamical

text: rule

text: nor

text: HIs

text: are

text: also

text: shown.

text: δ[cos

text: nˆ

text: θ]δ(

text: Rα

text: Rα

text: r),

text: α

text: α

text: m

text: m

text: α=α

text: Each

text: simulation

text: starts

text: from

text: a

text: randomly

text: distributed

text: state,

text: and

text: ﬁnally

text: reaches

text: a

text: steady

text: state

text: with

text: nontrivial

text: correlations.

text: In

text: Fig.

text: we

text: show

text: the

text: radial

text: distribution

text: function

text: where

text: is

text: the

text: relative

text: angle

text: between

text: α

text: th

text: and

text: α

text: th

text: swimmers,

text: and

text: the

text: angular

text: integral

text: of

text: g

text: ψ)

text: gives

text: g

text: Figures

text: and

text: repre-

text: mm

text: mm

text: δ(

text: Rαm

text: r)

text: sthenetﬁtrhste

text: ψw)

text: haircohuwnde

text: α=α

text: mm

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 Ψ(<∼ 0.05), such hydrodynamic torques should enhance shown in Fig. 11(b), but beyond this point, rather than rotational relaxation which would otherwise rely solely diverging, S(0) falls again even though the spinodal con- on Brownian motion. This extra rotation results in the dition is satisﬁed by the observed v (Ψ). This suggests s sharply decreasing trend in τ (Ψ) for small Ψ. (Notice that for the dumbbell system at this density, direct repul- R that this eﬀect depends on both the rodlike shape and sion via excluded volume interactions (which are omitted the front-back asymmetric streamlines around the swim- from the theory) are strong enough to overcome the eﬀec- mer within the present model.) However, with crowding, tive attraction caused by the density-dependent motility. the mean separation between swimmers decreases, which However, we have performed additional exploratory sim- should suppress reorientation while enhancing the hydro- ulations without HIs (in a simulation box of L3 = 1283) dynamic attractions. This presumably causes the rever- using a larger self-propulsion force; these suggest that at ∼ sal of the decreasing trend in τ at around Ψ 0.05. In- Ψ = 0.2 and 0.25 phase separation does occur at high R creasing the volume fraction further (Ψ >∼ 0.15), one ex- enough values of f . act pects activity-induced ﬂow ﬁelds to increasingly be sup- pressed, reducing motility and increasing τ . Nonethe- R A broadly similar scenario holds in the presence of HIs, less, due to the incompressible nature of HIs, neighboring as seen in Fig. 11(a). However the maximum compress- swimmers push and pull one another even without direct ibility is signiﬁcantly greater, and furthermore arises at (cid:39) contact, causing the translational and rotational dynam- signiﬁcantly lower density (Ψ 0.15). The peak coin- ics to become slower than in the system without HIs. cides with the state of most pronounced clustering as observed by direct visualization. The data for v (Ψ) s reported in Fig. 11(a) again suggest that the spinodal − FIG. 11: (Color online) The structure factor with (a) and condition dv /dΨ < v /Ψ of [4, 12, 13] is satisﬁed for s s without (b) HIs for various Ψ at fact = 14. Ψ >∼ 0.06, yet once more we observe a maximum, not a diverg∼ence, of S(0) near the state of maximal clustering These diﬀerences in the volume fraction dependence at Ψ = 0.15. [Not∼e that dvs/dΨ + vs/Ψ now has a mini- of the motility with and without HIs aﬀect the physics mum (< 0) at Ψ = 0.12; above this density, the activity- of clustering. In Fig. 11, we show the structure factor induced attraction itself becomes weakened.] The sub- for various Ψ with and without HIs. Without HIs, the sequent decrease might be attributable to steric repul- clustering tendency increases with volume fraction until sions, but their eﬀects would have to be stronger than in ∼ Ψ 0.25. On the other hand, with HIs, this increase con- the case without HIs to start dominating at this some- ∼ tinues only until Ψ 0.15. These clustering tendencies what lower density. An additional factor could be the were also checked by direct visualization. eﬀect of HIs on reducing τR, which is the mechanism In recent theories for active Brownian particles for avoidance of phase separation discussed by Fielding (ABPs), it was predicted that, if the swim speed de- [22]. We have performed additional exploratory simu- creases steeply enough with density, local densiﬁcation lations with HIs [at the density Ψ = 0.15 in a simula- induces a further accumulation of particles, eventually tion box of L3 = (128)3] using a larger self-propulsion leading to unstable growth of ﬂuctuations and phase sep- force; the structure factor S(k) at small k is found to be aration [4, 12, 13]. This mechanism for activity-induced dramatically enhanced by formation of a space-spanning phase separation was conﬁrmed by subsequent simula- structure of the order of the system size, which suggests tions on self-propelled spherical particles with rotational that bulk phase separation occurs for large enough fact. diﬀusion and no HIs [19]. Although the present dumbbell Notice, however, that the formed structure is not com- swimmers can be categorized as ABPs, we have shown pact and exhibits large ﬂuctuations, which may be due that the physics of mutual slowing down depends on par- to HIs and the relatively small system size of the present ticle shape, so that even without HIs there is no exact simulation. To ﬁrmly assess whether bulk phase separa- relationship to these earlier simulations. However, the tion occurs, we need to perform simulations with larger underlying theoretical analysis has some quite generic system sizes. features [4, 12, 13] so it is useful to discuss the clustering behavior observed here in similar terms. Thus we have shown the observed clustering tendency As shown in Fig. 10, without HIs and for Ψ <∼ 0.2, of our dumbbells to be broadly consistent with a theory v (Ψ) decreases almost linearly with Ψ, while the rota- in which collisional interactions are represented at mean- s tional relaxation time τ (Ψ) decreases only slightly from ﬁeld level by a density-dependent swim speed [4, 12, 13]. R the value set by purely Brownian rotation. Within the However, using the numerically obtained v (Ψ) shown in s framework of the theory [4, 12, 13], the eﬀective com- Fig. 10, this theory predicts bulk phase separation in a pressibility S(0) should increase with Ψ, diverging at a regime where we see only clusters. (This applies both − spinodal set by the condition dv /dΨ < v /Ψ. This with and without HIs.) Although steric repulsions of- s s condition holds for Ψ >∼ 0.15 [evaluated from the ﬁtting fer an obvious candidate mechanism for this discrepancy, of vs(Ψ) in Fig. 10]. In this evaluation, dv∼s/dΨ + vs/Ψ the detailed conditions for bulk phase separation in the is found to have a minimum (< 0) at Ψ = 0.23. The present dumbbell model are not well understood, and a compressibility increase is certainly seen for Ψ <∼ 0.25 as further systematic study will be needed to clarify them.

label: paragraph

text: Ψ(<∼

text: such

text: hydrodynamic

text: torques

text: should

text: enhance

text: shown

text: in

text: Fig.

text: but

text: beyond

text: this

text: point,

text: rather

text: than

text: rotational

text: relaxation

text: which

text: would

text: otherwise

text: rely

text: solely

text: diverging,

text: falls

text: again

text: even

text: though

text: the

text: spinodal

text: con-

text: on

text: Brownian

text: motion.

text: This

text: extra

text: rotation

text: results

text: in

text: the

text: dition

text: is

text: satisﬁed

text: by

text: the

text: observed

text: v

text: (Ψ).

text: This

text: suggests

text: s

text: sharply

text: decreasing

text: trend

text: in

text: τ

text: (Ψ)

text: for

text: small

text: Ψ.

text: (Notice

text: that

text: for

text: the

text: dumbbell

text: system

text: at

text: this

text: density,

text: direct

text: repul-

text: R

text: that

text: this

text: eﬀect

text: depends

text: on

text: both

text: the

text: rodlike

text: shape

text: and

text: sion

text: via

text: excluded

text: volume

text: interactions

text: (which

text: are

text: omitted

text: the

text: front-back

text: asymmetric

text: streamlines

text: around

text: the

text: swim-

text: from

text: the

text: theory)

text: are

text: strong

text: enough

text: to

text: overcome

text: the

text: eﬀec-

text: mer

text: within

text: the

text: present

text: model.)

text: However,

text: with

text: crowding,

text: tive

text: attraction

text: caused

text: by

text: the

text: density-dependent

text: motility.

text: the

text: mean

text: separation

text: between

text: swimmers

text: decreases,

text: which

text: However,

text: we

text: have

text: performed

text: additional

text: exploratory

text: sim-

text: should

text: suppress

text: reorientation

text: while

text: enhancing

text: the

text: hydro-

text: ulations

text: without

text: HIs

text: (in

text: a

text: simulation

text: box

text: of

text: dynamic

text: attractions.

text: This

text: presumably

text: causes

text: the

text: rever-

text: using

text: a

text: larger

text: self-propulsion

text: force;

text: these

text: suggest

text: that

text: at

text: sal

text: of

text: the

text: decreasing

text: trend

text: in

text: τ

text: at

text: around

text: Ψ

text: In-

text: Ψ

text: and

text: phase

text: separation

text: does

text: occur

text: at

text: high

text: R

text: creasing

text: the

text: volume

text: fraction

text: further

text: (Ψ

text: one

text: ex-

text: enough

text: values

text: of

text: f

text: act

text: pects

text: activity-induced

text: ﬂow

text: ﬁelds

text: to

text: increasingly

text: be

text: sup-

text: pressed,

text: reducing

text: motility

text: and

text: increasing

text: τ

text: Nonethe-

text: R

text: A

text: broadly

text: similar

text: scenario

text: holds

text: in

text: the

text: presence

text: of

text: HIs,

text: less,

text: due

text: to

text: the

text: incompressible

text: nature

text: of

text: HIs,

text: neighboring

text: as

text: seen

text: in

text: Fig.

text: However

text: the

text: maximum

text: compress-

text: swimmers

text: push

text: and

text: pull

text: one

text: another

text: even

text: without

text: direct

text: ibility

text: is

text: signiﬁcantly

text: greater,

text: and

text: furthermore

text: arises

text: at

text: contact,

text: causing

text: the

text: translational

text: and

text: rotational

text: dynam-

text: signiﬁcantly

text: lower

text: density

text: (Ψ

text: The

text: peak

text: coin-

text: ics

text: to

text: become

text: slower

text: than

text: in

text: the

text: system

text: without

text: HIs.

text: cides

text: with

text: the

text: state

text: of

text: most

text: pronounced

text: clustering

text: as

text: observed

text: by

text: direct

text: visualization.

text: The

text: data

text: for

text: v

text: (Ψ)

text: s

text: reported

text: in

text: Fig.

text: again

text: suggest

text: that

text: the

text: spinodal

text: FIG.

text: (Color

text: online)

text: The

text: structure

text: factor

text: with

text: (a)

text: and

text: condition

text: dv

text: /dΨ

text: v

text: /Ψ

text: of

text: is

text: satisﬁed

text: for

text: s

text: s

text: without

text: (b)

text: HIs

text: for

text: various

text: Ψ

text: at

text: fact

text: Ψ

text: yet

text: once

text: more

text: we

text: observe

text: a

text: maximum,

text: not

text: a

text: diverg∼ence,

text: of

text: near

text: the

text: state

text: of

text: maximal

text: clustering

text: These

text: diﬀerences

text: in

text: the

text: volume

text: fraction

text: dependence

text: at

text: Ψ

text: [Not∼e

text: that

text: dvs/dΨ

text: vs/Ψ

text: now

text: has

text: a

text: mini-

text: of

text: the

text: motility

text: with

text: and

text: without

text: HIs

text: aﬀect

text: the

text: physics

text: mum

text: at

text: Ψ

text: above

text: this

text: density,

text: the

text: activity-

text: of

text: clustering.

text: In

text: Fig.

text: we

text: show

text: the

text: structure

text: factor

text: induced

text: attraction

text: itself

text: becomes

text: weakened.]

text: The

text: sub-

text: for

text: various

text: Ψ

text: with

text: and

text: without

text: HIs.

text: Without

text: HIs,

text: the

text: sequent

text: decrease

text: might

text: be

text: attributable

text: to

text: steric

text: repul-

text: clustering

text: tendency

text: increases

text: with

text: volume

text: fraction

text: until

text: sions,

text: but

text: their

text: eﬀects

text: would

text: have

text: to

text: be

text: stronger

text: than

text: in

text: Ψ

text: On

text: the

text: other

text: hand,

text: with

text: HIs,

text: this

text: increase

text: con-

text: the

text: case

text: without

text: HIs

text: to

text: start

text: dominating

text: at

text: this

text: some-

text: tinues

text: only

text: until

text: Ψ

text: These

text: clustering

text: tendencies

text: what

text: lower

text: density.

text: An

text: additional

text: factor

text: could

text: be

text: the

text: were

text: also

text: checked

text: by

text: direct

text: visualization.

text: eﬀect

text: of

text: HIs

text: on

text: reducing

text: τR,

text: which

text: is

text: the

text: mechanism

text: In

text: recent

text: theories

text: for

text: active

text: Brownian

text: particles

text: for

text: avoidance

text: of

text: phase

text: separation

text: discussed

text: by

text: Fielding

text: (ABPs),

text: it

text: was

text: predicted

text: that,

text: if

text: the

text: swim

text: speed

text: de-

text: We

text: have

text: performed

text: additional

text: exploratory

text: simu-

text: creases

text: steeply

text: enough

text: with

text: density,

text: local

text: densiﬁcation

text: lations

text: with

text: HIs

text: [at

text: the

text: density

text: Ψ

text: in

text: a

text: simula-

text: induces

text: a

text: further

text: accumulation

text: of

text: particles,

text: eventually

text: tion

text: box

text: of

text: using

text: a

text: larger

text: self-propulsion

text: leading

text: to

text: unstable

text: growth

text: of

text: ﬂuctuations

text: and

text: phase

text: sep-

text: force;

text: the

text: structure

text: factor

text: S(k)

text: at

text: small

text: k

text: is

text: found

text: to

text: be

text: aration

text: This

text: mechanism

text: for

text: activity-induced

text: dramatically

text: enhanced

text: by

text: formation

text: of

text: a

text: space-spanning

text: phase

text: separation

text: was

text: conﬁrmed

text: by

text: subsequent

text: simula-

text: structure

text: of

text: the

text: order

text: of

text: the

text: system

text: size,

text: which

text: suggests

text: tions

text: on

text: self-propelled

text: spherical

text: particles

text: with

text: rotational

text: that

text: bulk

text: phase

text: separation

text: occurs

text: for

text: large

text: enough

text: fact.

text: diﬀusion

text: and

text: no

text: HIs

text: Although

text: the

text: present

text: dumbbell

text: Notice,

text: however,

text: that

text: the

text: formed

text: structure

text: is

text: not

text: com-

text: swimmers

text: can

text: be

text: categorized

text: as

text: ABPs,

text: we

text: have

text: shown

text: pact

text: and

text: exhibits

text: large

text: ﬂuctuations,

text: which

text: may

text: be

text: due

text: that

text: the

text: physics

text: of

text: mutual

text: slowing

text: down

text: depends

text: on

text: par-

text: to

text: HIs

text: and

text: the

text: relatively

text: small

text: system

text: size

text: of

text: the

text: present

text: ticle

text: shape,

text: so

text: that

text: even

text: without

text: HIs

text: there

text: is

text: no

text: exact

text: simulation.

text: To

text: ﬁrmly

text: assess

text: whether

text: bulk

text: phase

text: separa-

text: relationship

text: to

text: these

text: earlier

text: simulations.

text: However,

text: the

text: tion

text: occurs,

text: we

text: need

text: to

text: perform

text: simulations

text: with

text: larger

text: underlying

text: theoretical

text: analysis

text: has

text: some

text: quite

text: generic

text: system

text: sizes.

text: features

text: so

text: it

text: is

text: useful

text: to

text: discuss

text: the

text: clustering

text: behavior

text: observed

text: here

text: in

text: similar

text: terms.

text: Thus

text: we

text: have

text: shown

text: the

text: observed

text: clustering

text: tendency

text: As

text: shown

text: in

text: Fig.

text: without

text: HIs

text: and

text: for

text: Ψ

text: of

text: our

text: dumbbells

text: to

text: be

text: broadly

text: consistent

text: with

text: a

text: theory

text: v

text: (Ψ)

text: decreases

text: almost

text: linearly

text: with

text: Ψ,

text: while

text: the

text: rota-

text: in

text: which

text: collisional

text: interactions

text: are

text: represented

text: at

text: mean-

text: s

text: tional

text: relaxation

text: time

text: τ

text: (Ψ)

text: decreases

text: only

text: slightly

text: from

text: ﬁeld

text: level

text: by

text: a

text: density-dependent

text: swim

text: speed

text: R

text: the

text: value

text: set

text: by

text: purely

text: Brownian

text: rotation.

text: Within

text: the

text: However,

text: using

text: the

text: numerically

text: obtained

text: v

text: (Ψ)

text: shown

text: in

text: s

text: framework

text: of

text: the

text: theory

text: the

text: eﬀective

text: com-

text: Fig.

text: this

text: theory

text: predicts

text: bulk

text: phase

text: separation

text: in

text: a

text: pressibility

text: should

text: increase

text: with

text: Ψ,

text: diverging

text: at

text: a

text: regime

text: where

text: we

text: see

text: only

text: clusters.

text: (This

text: applies

text: both

text: spinodal

text: set

text: by

text: the

text: condition

text: dv

text: /dΨ

text: v

text: /Ψ.

text: This

text: with

text: and

text: without

text: HIs.)

text: Although

text: steric

text: repulsions

text: of-

text: s

text: s

text: condition

text: holds

text: for

text: Ψ

text: [evaluated

text: from

text: the

text: ﬁtting

text: fer

text: an

text: obvious

text: candidate

text: mechanism

text: for

text: this

text: discrepancy,

text: of

text: vs(Ψ)

text: in

text: Fig.

text: In

text: this

text: evaluation,

text: dv∼s/dΨ

text: vs/Ψ

text: the

text: detailed

text: conditions

text: for

text: bulk

text: phase

text: separation

text: in

text: the

text: is

text: found

text: to

text: have

text: a

text: minimum

text: at

text: Ψ

text: The

text: present

text: dumbbell

text: model

text: are

text: not

text: well

text: understood,

text: and

text: a

text: compressibility

text: increase

text: is

text: certainly

text: seen

text: for

text: Ψ

text: as

text: further

text: systematic

text: study

text: will

text: be

text: needed

text: to

text: clarify

text: them.


text: 10 [17] K. Fukushima and Y. Hidaka, Phys. Rev. Lett. 110, (1961). 031601 (2013) [arXiv:1209.1319 [hep-ph]]. [33] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 124, 246 [18] T. Kojo and N. Su, Phys. Lett. B 720, 192 (2013) (1961). [arXiv:1211.7318 [hep-ph]]. [34] S. Klimt, M. F. M. Lutz, U. Vogl and W. Weise, Nucl. [19] F. Bruckmann, G. Endrodi and T. G. Kovacs, JHEP Phys. A 516, 429 (1990). 1304, 112 (2013) [arXiv:1303.3972 [hep-lat]]. [35] U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27, 195 [20] J. Chao, P. Chu and M. Huang, Phys. Rev. D 88, 054009 (1991). (2013) [arXiv:1305.1100 [hep-ph]]. [36] S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992). [21] L. Yu, H. Liu and M. Huang, arXiv:1404.6969 [hep-ph]. [37] T. Hatsuda and T. Kunihiro, Phys. Rept. 247, 221 (1994) [22] Y. Hidaka and A. Yamamoto, Phys. Rev. D 87, 094502 [hep-ph/9401310]. (2013) [arXiv:1209.0007 [hep-ph]]. [38] V. I. Ritus, Annals Phys. 69, 555 (1972). [23] M. N. Chernodub, Phys. Rev. D 86, 107703 (2012) [39] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Nucl. [arXiv:1209.3587 [hep-ph]]. Phys. A 836, 311 (2010) [arXiv:0912.2961 [hep-ph]]. [24] M. N. Chernodub, Phys. Rev. D 89, 018501 (2014) [40] Sh. Fayazbakhsh and N. Sadooghi, Phys. Rev. D 88, no. [arXiv:1309.4071 [hep-ph]]. 6, 065030 (2013) [arXiv:1306.2098 [hep-ph]]. [25] C. Li and Q. Wang, Phys. Lett. B 721, 141 (2013) [41] Y. B. He, J. Hufner, S. P. Klevansky and P. Rehberg, [arXiv:1301.7009 [hep-th]]. Nucl. Phys. A 630, 719 (1998) [nucl-th/9712051]. [26] M. Frasca, JHEP 1311, 099 (2013) [arXiv:1309.3966 [42] P. Rehberg and S. P. Klevansky, Annals Phys. 252, 422 [hep-ph]]. (1996) [hep-ph/9510221]. [27] M. A. Andreichikov, B. O. Kerbikov, V. D. Orlovsky and [43] M. Frasca and M. Ruggieri, Phys. Rev. D 83, 094024 Y. . A. Simonov, Phys. Rev. D 87, no. 9, 094029 (2013) (2011) [arXiv:1103.1194 [hep-ph]]. [arXiv:1304.2533 [hep-ph]]. [44] E. V. Luschevskaya and O. V. Larina, JETP Lett. 98, [28] Kunlun Wang, PhD thesis from Peking University. 652 (2014). [29] V. V. Braguta, P. V. Buividovich, M. N. Chernodub, [45] Julian Schwinger. Proceedings of the National Academy A. Y. .Kotov and M. I. Polikarpov, Phys. Lett. B 718, of Sciences, 37(7):452–455, 1951. ibid. 37(7):455–459, 667 (2012) [arXiv:1104.3767 [hep-lat]]. 1951. [30] N. Callebaut, D. Dudal and H. Verschelde, PoS [46] J. Chao, L. Yu and M. Huang, arXiv:1403.0442 [hep-th]. FACESQCD , 046 (2010) [arXiv:1102.3103 [hep-ph]]. [47] Alan Jeﬀrey and Daniel Zwillinger. Table of integrals, [31] M. Ammon, J. Erdmenger, P. Kerner and M. Strydom, series, and products. Academic Press, 2007. Phys. Lett. B 706, 94 (2011) [arXiv:1106.4551 [hep-th]]. [32] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122, 345

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text: When does Active Learning Work? Lewis P. G. Evans1, Niall M. Adams12, and Christoforos Anagnostopoulos1 1 Department of Mathematics, Imperial College London 2 Heilbronn Institute for Mathematical Research, University of Bristol Abstract. Active Learning (AL) methods seek to improve classiﬁer per- formance when labels are expensive or scarce. We consider two central questions: Where does AL work? How much does it help? To address these questions, a comprehensive experimental simulation study of Ac- tive Learning is presented. We consider a variety of tasks, classiﬁers and other AL factors, to present a broad exploration of AL performance in various settings. A precise way to quantify performance is needed in order to know when AL works. Thus we also present a detailed methodology for tackling the complexities of assessing AL performance in the context of this experimental study. Keywords: classiﬁcation, active learning, experimental evaluation, al- gorithms 1 Introduction Active Learning (AL) is an important sub-ﬁeld of classiﬁcation, where a learning system can intelligently select unlabelled examples for labelling, to im- prove classiﬁer performance. The need for AL is often motivated by practical concerns: labelled data is often scarce or expensive compared to unlabelled data [9]. We consider two central questions: Where does AL work? How much does it help? These questions are as yet unresolved, and answers would enable re- searchers to tackle the subsequent questions of how and why AL works. Several studies have shown that it is surprisingly diﬃcult for AL to out- perform the simple benchmark of random selection ([3, 8]). Further, both AL methods and random selection often show high variability which makes compar- isons diﬃcult. There are many studies showing positive results, for example [9, 5]. Notably there are several studies showing negative results, for example [2, 8]. While valuable, such studies do not permit any overview of where and how much AL works. Moreover, this contradiction suggests there are still things to understand, which is the objective of this paper. We take the view that a broader study should try to understand which fac- tors might be expected to aﬀect AL performance. Such factors include the clas- siﬁcation task and the classiﬁer; see Section 2.3. We present a comprehensive

text: When

label: title

text: does

text: Active

text: Learning

text: Work?

text: Lewis

label: author

text: P.

text: G.

text: Niall

text: M.

text: and

text: Christoforos

text: Department

text: of

text: Mathematics,

text: Imperial

text: College

text: London

text: Heilbronn

text: Institute

text: for

text: Mathematical

text: Research,

text: University

text: of

text: Bristol

text: Abstract.

text: Active

label: abstract

text: Learning

text: (AL)

text: methods

text: seek

text: to

text: improve

text: classiﬁer

text: per-

text: formance

text: when

text: labels

text: are

text: expensive

text: or

text: scarce.

text: We

text: consider

text: two

text: central

text: questions:

text: Where

text: does

text: AL

text: work?

text: How

text: much

text: does

text: it

text: help?

text: To

text: address

text: these

text: questions,

text: a

text: comprehensive

text: experimental

text: simulation

text: study

text: of

text: Ac-

text: tive

text: Learning

text: is

text: presented.

text: We

text: consider

text: a

text: variety

text: of

text: tasks,

text: classiﬁers

text: and

text: other

text: AL

text: factors,

text: to

text: present

text: a

text: broad

text: exploration

text: of

text: AL

text: performance

text: in

text: various

text: settings.

text: A

text: precise

text: way

text: to

text: quantify

text: performance

text: is

text: needed

text: in

text: order

text: to

text: know

text: when

text: AL

text: works.

text: Thus

text: we

text: also

text: present

text: a

text: detailed

text: methodology

text: for

text: tackling

text: the

text: complexities

text: of

text: assessing

text: AL

text: performance

text: in

text: the

text: context

text: of

text: this

text: experimental

text: study.

text: Keywords:

text: classiﬁcation,

text: active

text: learning,

text: experimental

text: evaluation,

text: al-

text: gorithms

text: Introduction

text: Active

text: Learning

text: (AL)

text: is

text: an

text: important

text: sub-ﬁeld

text: of

text: classiﬁcation,

text: where

text: a

text: learning

text: system

text: can

text: intelligently

text: select

text: unlabelled

text: examples

text: for

text: labelling,

text: to

text: im-

text: prove

text: classiﬁer

text: performance.

text: The

text: need

text: for

text: AL

text: is

text: often

text: motivated

text: by

text: practical

text: concerns:

text: labelled

text: data

text: is

text: often

text: scarce

text: or

text: expensive

text: compared

text: to

text: unlabelled

text: data

text: We

text: consider

text: two

text: central

text: questions:

text: Where

text: does

text: AL

text: work?

text: How

text: much

text: does

text: it

text: help?

text: These

text: questions

text: are

text: as

text: yet

text: unresolved,

text: and

text: answers

text: would

text: enable

text: re-

text: searchers

text: to

text: tackle

text: the

text: subsequent

text: questions

text: of

text: how

text: and

text: why

text: AL

text: works.

text: Several

text: studies

text: have

text: shown

text: that

text: it

text: is

text: surprisingly

text: diﬃcult

text: for

text: AL

text: to

text: out-

text: perform

text: the

text: simple

text: benchmark

text: of

text: random

text: selection

text: Further,

text: both

text: AL

text: methods

text: and

text: random

text: selection

text: often

text: show

text: high

text: variability

text: which

text: makes

text: compar-

text: isons

text: diﬃcult.

text: There

text: are

text: many

text: studies

text: showing

text: positive

text: results,

text: for

text: example

text: Notably

text: there

text: are

text: several

text: studies

text: showing

text: negative

text: results,

text: for

text: example

text: While

text: valuable,

text: such

text: studies

text: do

text: not

text: permit

text: any

text: overview

text: of

text: where

text: and

text: how

text: much

text: AL

text: works.

text: Moreover,

text: this

text: contradiction

text: suggests

text: there

text: are

text: still

text: things

text: to

text: understand,

text: which

text: is

text: the

text: objective

text: of

text: this

text: paper.

text: We

text: take

text: the

text: view

text: that

text: a

text: broader

text: study

text: should

text: try

text: to

text: understand

text: which

text: fac-

text: tors

text: might

text: be

text: expected

text: to

text: aﬀect

text: AL

text: performance.

text: Such

text: factors

text: include

text: the

text: clas-

text: siﬁcation

text: task

text: and

text: the

text: classiﬁer;

text: see

text: Section

text: We

text: present

text: a

text: comprehensive


text: 5 perform the AL experimental study. The classiﬁer that sees all the labelled data provides a ceiling benchmark, the score S . The classiﬁer that sees only the all initially labelled data provides a ﬂoor benchmark, the score S . To quantify initial the scope for AL to improve performance, we deﬁne the space for AL as a ratio − of performance scores: (S S )/S . This provides a normalised metric all initial all of the potential for AL to improve performance. A Monte Carlo experiment varying these factors provides the opportunity to statistically analyse the behaviour of AL. To get to this point, both a careful experiment and a reﬁned methodology of performance assessment are required. 3 Experimental Method AL is applied iteratively in these experiments: the amount of labelled data grows progressively, as the AL method spends a budget chunk at each time point. We may choose to spend our overall budget all in one go, or iteratively, in smaller batches. In that sense the experimental setup resembles that of the AL challenge described in [5]. We use this iteration over budget because it is realistic for practical AL applications, and because it explores the behaviour of AL as the number of labelled examples grows. Experiments consider AL methods SE and QBC. To motivate our experimental method, we present the summary plots of the relative performances of AL and RS over time, see Figures 2a and 2b. The experimental setup is as follows. Firstly, sample a pair of datasets [D , train D ] from the classiﬁcation task. To simulate label scarcity, split the training test dataset into initally labelled data D and an unlabelled pool D . initial pool The output of one experiment can be described in a single plot, for example Figure 2a. That ﬁgure shows the trajectory of performance scores obtained from progressive labelling, as follows. At each time point the AL method chooses a small set of examples for labelling, which is added to the existing dataset of labelled data. This selection happens repeatedly, creating a trajectory of selected datasets from the unlabelled pool. Each time point gives a performance score, for example error rate, though the framework extends to any performance metric. This gives the overall result of a trajectory of scores over time, denoted S : an i empirical learning curve. Here i denotes the time point as we iterately increase ∈ the amount of labelled data, with i [0, 100]. Given several instances of RS, we form an empirical boxplot, called a sam- pling interval. Figure 2a shows the trajectory of scores for the AL method, and the vertical boxplots show the sampling intervals for the scores for RS. Once this iterative process is done, we obtain a set of scores over the whole budget range, denoted S . Those scores are used to calculate various performance i comparisons, speciﬁcally to see whether AL outperformed RS, see Section 1. The AL method now has a score trajectory S : a set of scores over the whole i budget range. All trajectories begin at the ﬂoor benchmark score S and initial terminate at the ceiling benchmark score S . From the score trajectory S a all i

text: perform

text: the

text: AL

text: experimental

text: study.

text: The

text: classiﬁer

text: that

text: sees

text: all

text: the

text: labelled

text: data

text: provides

text: a

text: ceiling

text: benchmark,

text: the

text: score

text: S

text: The

text: classiﬁer

text: that

text: sees

text: only

text: the

text: all

text: initially

text: labelled

text: data

text: provides

text: a

text: ﬂoor

text: benchmark,

text: the

text: score

text: S

text: To

text: quantify

text: initial

text: the

text: scope

text: for

text: AL

text: to

text: improve

text: performance,

text: we

text: deﬁne

text: the

text: space

text: for

text: AL

text: as

text: a

text: ratio

text: of

text: performance

text: scores:

text: (S

text: S

text: )/S

text: This

text: provides

text: a

text: normalised

text: metric

text: all

text: initial

text: all

text: of

text: the

text: potential

text: for

text: AL

text: to

text: improve

text: performance.

text: A

text: Monte

text: Carlo

text: experiment

text: varying

text: these

text: factors

text: provides

text: the

text: opportunity

text: to

text: statistically

text: analyse

text: the

text: behaviour

text: of

text: AL.

text: To

text: get

text: to

text: this

text: point,

text: both

text: a

text: careful

text: experiment

text: and

text: a

text: reﬁned

text: methodology

text: of

text: performance

text: assessment

text: are

text: required.

text: Experimental

text: Method

text: AL

text: is

text: applied

text: iteratively

text: in

text: these

text: experiments:

text: the

text: amount

text: of

text: labelled

text: data

text: grows

text: progressively,

text: as

text: the

text: AL

text: method

text: spends

text: a

text: budget

text: chunk

text: at

text: each

text: time

text: point.

text: We

text: may

text: choose

text: to

text: spend

text: our

text: overall

text: budget

text: all

text: in

text: one

text: go,

text: or

text: iteratively,

text: in

text: smaller

text: batches.

text: In

text: that

text: sense

text: the

text: experimental

text: setup

text: resembles

text: that

text: of

text: the

text: AL

text: challenge

text: described

text: in

text: We

text: use

text: this

text: iteration

text: over

text: budget

text: because

text: it

text: is

text: realistic

text: for

text: practical

text: AL

text: applications,

text: and

text: because

text: it

text: explores

text: the

text: behaviour

text: of

text: AL

text: as

text: the

text: number

text: of

text: labelled

text: examples

text: grows.

text: Experiments

text: consider

text: AL

text: methods

text: SE

text: and

text: QBC.

text: To

text: motivate

text: our

text: experimental

text: method,

text: we

text: present

text: the

text: summary

text: plots

text: of

text: the

text: relative

text: performances

text: of

text: AL

text: and

text: RS

text: over

text: time,

text: see

text: Figures

text: and

text: The

text: experimental

text: setup

text: is

text: as

text: follows.

text: Firstly,

text: sample

text: a

text: pair

text: of

text: datasets

text: [D

text: train

text: D

text: from

text: the

text: classiﬁcation

text: task.

text: To

text: simulate

text: label

text: scarcity,

text: split

text: the

text: training

text: test

text: dataset

text: into

text: initally

text: labelled

text: data

text: D

text: and

text: an

text: unlabelled

text: pool

text: D

text: initial

text: pool

text: The

text: output

text: of

text: one

text: experiment

text: can

text: be

text: described

text: in

text: a

text: single

text: plot,

text: for

text: example

text: Figure

text: That

text: ﬁgure

text: shows

text: the

text: trajectory

text: of

text: performance

text: scores

text: obtained

text: from

text: progressive

text: labelling,

text: as

text: follows.

text: At

text: each

text: time

text: point

text: the

text: AL

text: method

text: chooses

text: a

text: small

text: set

text: of

text: examples

text: for

text: labelling,

text: which

text: is

text: added

text: to

text: the

text: existing

text: dataset

text: of

text: labelled

text: data.

text: This

text: selection

text: happens

text: repeatedly,

text: creating

text: a

text: trajectory

text: of

text: selected

text: datasets

text: from

text: the

text: unlabelled

text: pool.

text: Each

text: time

text: point

text: gives

text: a

text: performance

text: score,

text: for

text: example

text: error

text: rate,

text: though

text: the

text: framework

text: extends

text: to

text: any

text: performance

text: metric.

text: This

text: gives

text: the

text: overall

text: result

text: of

text: a

text: trajectory

text: of

text: scores

text: over

text: time,

text: denoted

text: S

text: an

text: i

text: empirical

text: learning

text: curve.

text: Here

text: i

text: denotes

text: the

text: time

text: point

text: as

text: we

text: iterately

text: increase

text: the

text: amount

text: of

text: labelled

text: data,

text: with

text: i

text: Given

text: several

text: instances

text: of

text: RS,

text: we

text: form

text: an

text: empirical

text: boxplot,

text: called

text: a

text: sam-

text: pling

text: interval.

text: Figure

text: shows

text: the

text: trajectory

text: of

text: scores

text: for

text: the

text: AL

text: method,

text: and

text: the

text: vertical

text: boxplots

text: show

text: the

text: sampling

text: intervals

text: for

text: the

text: scores

text: for

text: RS.

text: Once

text: this

text: iterative

text: process

text: is

text: done,

text: we

text: obtain

text: a

text: set

text: of

text: scores

text: over

text: the

text: whole

text: budget

text: range,

text: denoted

text: S

text: Those

text: scores

text: are

text: used

text: to

text: calculate

text: various

text: performance

text: i

text: comparisons,

text: speciﬁcally

text: to

text: see

text: whether

text: AL

text: outperformed

text: RS,

text: see

text: Section

text: The

text: AL

text: method

text: now

text: has

text: a

text: score

text: trajectory

text: S

text: a

text: set

text: of

text: scores

text: over

text: the

text: whole

text: i

text: budget

text: range.

text: All

text: trajectories

text: begin

text: at

text: the

text: ﬂoor

text: benchmark

text: score

text: S

text: and

text: initial

text: terminate

text: at

text: the

text: ceiling

text: benchmark

text: score

text: S

text: From

text: the

text: score

text: trajectory

text: S

text: a

text: all

text: i


text: UNIT BALLS OF CONSTANT VOLUME: WHICH ONE HAS OPTIMAL REPRESENTATION?3 (cid:32) (cid:33) ∈ (cid:88) (cid:88) (cid:88) g Hom , let g = (g ) be its vector of coeﬃcients, i.e., d α (cid:55)→ (cid:80) · · · x g(x) = g xα = g xα1 xαn , α = d, α α 1 n i α α i (cid:107) (cid:107) | | | | ∈ with standard (cid:96) -norm g = g = g . With any g Hom is associated 1 ⊂ R 1 α α d its sublevel set G n deﬁned by: (1.3) G := { x ∈ Rn : g(cid:80)(x) ≤ 1 }, g ∈ Hom . d (cid:55)→ ∗ ∗ In particular, with x g ((cid:88)x) := n xd, the sublevel set G is nothing less i=1 i than the standard d-unit ball { n ≤ } { (cid:107) (cid:107) ≤ } B = x : xd 1 =: x : x d 1 , d i d i=1 ∈ whose Lebesgue volume vol (B ) is denoted ρ . (When g Hom is convex then (cid:55)→ (cid:107) (cid:107) d d d x g(x) deﬁnes a norm x := g(x)1/d with G as associated unit ball.) g (a) In a ﬁrst contribution we prove that the optimization problem: { (cid:107) (cid:107) ≤ ∈ } (1.4) P : inf g : vol (G) ρ ; g Hom , 1 1 d d g (cid:0) (cid:1) ∗ (cid:55)→ (cid:107) (cid:107) has a unique optimal solution g which is the L -norm polynomial x x d. ∗ 1 d d Observe that g has the minimal number n of coeﬃcients over potentially s(d) := − 1 ∈ n 1+d coeﬃcients. (Indeed for a polynomial g Hom with m < n non zero d d coeﬃcients, its sublevel set G cannot have ﬁnite Lebesgue volume.) Therefore the ∗ L -norm polynomial g associated with the unit ball B is the “sparsest” solution d ∈ 1 ≤ d ∗ among all g Hom such that vol (G) vol (B ). In particular, g not only solves d d 1 problem P but also solves the non convex optimization problem 1 { (cid:107) (cid:107) ≤ ∈ } P : inf g : vol (G) ρ ; g Hom , 0 0 d d g of which P is a “convex relaxation”. But this is also equivalent to state that among 1 all homogeneous polynomials of degree d with constant (cid:96) -norm, the L -unit ball ≤ 1 d is the one with minimum volume vol (B ) vol(G). d (b) In a second contribution we consider the (cid:96) -norm version of (1.4): 2 { (cid:107) (cid:107) ≤ ∈ } (1.5) P : inf g : vol (G) ρ ; g Hom , 2 2 d d g (cid:88) (cid:55)→ (cid:107) (cid:107) with weighted Euclidean norm g g deﬁned by: 2 (cid:107) (cid:107) ∈ (d)! g 2 := c g2, g Hom , where c := · · · . 2 | | α α d α α1! αn! α =d (cid:0) (cid:1) ∗ We then show that P also has a unique optimal solution g , but in contrast to ∗2 ∗ 2 the optimal solution g of problem P , g is not sparse at all! This is because one − 1 1 2 ∗ | | can show that all n 1+d coeﬃcients of the form g with β = d/2 are non-zero. ∗ d 2β In addition, g is a particular sum of squares (SOS) polynomial as it is a sum of 2 ∗ d-powers of linear forms. (Notice that g is also a (very particular and simple) sum 1 of d-powers of linear forms.) In particular, when d = 4 the optimal solution of P is 2 ##LTLine##

text: UNIT

text: BALLS

text: OF

text: CONSTANT

text: VOLUME:

text: WHICH

text: ONE

text: HAS

text: OPTIMAL

text: g

text: Hom

text: let

text: g

text: (g

text: be

text: its

text: vector

text: of

text: coeﬃcients,

text: i.e.,

text: d

text: α

text: x

text: g(x)

text: g

text: xα

text: g

text: xαn

text: α

text: d,

text: α

text: α

text: n

text: i

text: α

text: α

text: i

text: with

text: standard

text: -norm

text: g

text: g

text: g

text: With

text: any

text: g

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text: is

text: associated

text: R

text: α

text: α

text: d

text: its

text: sublevel

text: set

text: G

text: n

text: deﬁned

text: by:

label: equation

text: G

text: x

text: Rn

text: g

text: Hom

text: d

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text: particular,

text: with

text: x

text: g

text: n

text: xd,

text: the

text: sublevel

text: set

text: G

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text: nothing

text: less

text: i

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text: standard

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text: n

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text: g(x)

text: deﬁnes

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text: g

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text: In

text: a

text: ﬁrst

text: contribution

text: we

text: prove

text: that

text: the

text: optimization

text: problem:

text: P

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text: g

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text: (G)

text: ρ

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text: has

text: a

text: unique

text: optimal

text: solution

text: g

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text: is

text: the

text: L

text: -norm

text: polynomial

text: x

text: x

text: d.

text: d

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text: Observe

text: that

text: g

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text: the

text: minimal

text: number

text: n

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text: coeﬃcients

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text: s(d)

text: n

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text: (Indeed

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text: ﬁnite

text: Lebesgue

text: volume.)

text: Therefore

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text: L

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text: “convex

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text: vol(G).

text: d

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text: -norm

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text: unique

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text: contrast

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text: ∗2

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text: optimal

text: solution

text: g

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text: problem

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text: g

text: is

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text: This

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text: of

text: d-powers

text: of

text: linear

text: forms.)

text: In

text: particular,

text: when

text: d

text: the

text: optimal

text: solution

text: of

text: P

text: is

text: ##LTLine##


text: UNIT BALLS OF CONSTANT VOLUME: WHICH ONE HAS OPTIMAL REPRESENTATION?21 [11] J.B. Lasserre. A quick proof for the volume of n-balls. Amer. Math. Monthly 108, pp. 768– 769, 2001. [12] J.B. Lasserre and T. Netzer. SOS approximations of nonnegative polynomials via simple high degree perturbations Math. Z. 256, pp. 99–112, 2007. [13] J.B. Lasserre and E.S. Zeron. Solving a class of multivariable integration problems via Laplace Lechniques, Appl. Math. (Warsaw) 28, pp. 391–405, 2001. [14] B. Recht, M. Fazel, P. A. Parrilo. Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization, SIAM Review 52, pp. 471–501, 2010. [15] B. Reznick. Sums of even powers of real linear forms, Mem. Amer. Math. Soc. 96, No. 463, 1992. LAAS-CNRS and Institute of Mathematics, University of Toulouse, LAAS, 7 avenue du Colonel Roche, 31077 Toulouse Ce´dex 4, France, Tel: +33561336415 Email address: lasserre@laas.fr

text: UNIT

text: BALLS

text: OF

text: CONSTANT

text: VOLUME:

text: WHICH

text: ONE

text: HAS

text: OPTIMAL

text: J.B.

text: Lasserre.

text: A

text: quick

text: proof

text: for

text: the

text: volume

text: of

text: n-balls.

text: Amer.

text: Math.

text: Monthly

text: pp.

text: J.B.

text: Lasserre

text: and

text: T.

text: Netzer.

text: SOS

text: approximations

text: of

text: nonnegative

text: polynomials

text: via

text: simple

text: high

text: degree

text: perturbations

text: Math.

text: Z.

text: pp.

text: J.B.

text: Lasserre

text: and

text: E.S.

text: Zeron.

text: Solving

text: a

text: class

text: of

text: multivariable

text: integration

text: problems

text: via

text: Laplace

text: Lechniques,

text: Appl.

text: Math.

text: (Warsaw)

text: pp.

text: B.

text: Recht,

text: M.

text: Fazel,

text: P.

text: A.

text: Parrilo.

text: Guaranteed

text: Minimum-Rank

text: Solutions

text: of

text: Linear

text: Matrix

text: Equations

text: via

text: Nuclear

text: Norm

text: Minimization,

text: SIAM

text: Review

text: pp.

text: B.

text: Reznick.

text: Sums

text: of

text: even

text: powers

text: of

text: real

text: linear

text: forms,

text: Mem.

text: Amer.

text: Math.

text: Soc.

text: No.

text: LAAS-CNRS

text: and

text: Institute

text: of

text: Mathematics,

text: University

text: of

text: Toulouse,

text: LAAS,

text: avenue

text: du

text: Colonel

text: Roche,

text: Toulouse

text: Ce´dex

text: France,

text: Tel:

text: Email

text: address:

text: lasserre@laas.fr


text: (cid:80) 4 (cid:80) JEAN B. LASSERRE (cid:80) { ≤ } the Euclidean ball x : x2 1 which has the equivalent quartic representation { ≤ } i i (cid:55)→ x : ( n x2)2 1 and the SOS quartic polynomial x ( n x2)2 solves P . i=1 i i=1 i 2 (c) We also consider the SOS (sum of squares) version of P , that is one now 1 searches for a degree-d SOS homogeneous polynomial g (x) = v (x)Qv (x), (cid:23) | | Q d/2 d/2 Q 0, (where v (x) = (xα), α = d/2). That is, one characterizes the unique d/2 optimal solution of the optimization problem: { ≤ (cid:23) } (1.6) P : inf trace (Q) : vol (G ) ρ ; Q 0 . 3 (cid:23) Q d Q 0 In this matrix context, trace (Q) is the nuclear norm of Q and so solving P ∗ 3 aims at ﬁnding an optimal solution Q with small rank, which translates into an homogeneous polynomial g ∗ which is a sum of a few squares. We also proves that ∗ Q g associated with the L -unit ball cannot be an optimal solution of P (and indeed 1 d 3 g ∗ being a sum of a few squares does not necessarily implies that it has a small Q number of coeﬃcients). (d) Finally we also show that results in (a) and (b) extend to the case of other val- ues of d (including p = 1 and rationals) in which case one now deals with positively (cid:80) homogeneous “generalized polynomials” (instead of homogeneous polynomials) and one has to deﬁne an appropriate ﬁnite-dimensional analogue analogue of Hom . { | | ≤ } d This includes the interesting case of the L -unit ball x : x 1 and when 1 i i p < 1, balls which are not associated with norms. 2. Notation, definitions and preliminary results R Let [x] denote the ring or real polynomials in the variables x = (x , . . . , x ), R 1 n and let [x] be the vector space of real polynomials of degree at most d. Similarly, ⊂ Rd let Σ[x] [x] denote the convex cone of real polynomials that are sums of squares ⊂ (SOS) of polynomials, and Σ[x] Σ[x] its subcone of SOS polynomials of degree S d × at most d. Denote by m the space of m m real symmetric matrices. For a ∈ S (cid:23) (cid:31) given matrix A m, the notation A 0 (resp. A 0) means that A is positive semideﬁnite (psd) (resp. positive deﬁnite (pd)), i.e., all its eigenvalues are real and nonnegative (resp. positive). ∈ R ∈ R ∈ R A polynomial p [x] is homogenous if p(λx) = λdp(x) for all x n, λ . R → R d ∈ R A function f : n is positively homogeneous of degree d if f(λx) = λdf(x) (cid:54) ∈ R (cid:55)→ | | for all 0 = x n, λ > 0. For instance x x is not homogeneous but is positively (cid:80) homogeneous of degree 1. ⊂ R (cid:0) (cid:1) Let Hom [x] be the vector space of homogeneous polynomials of even d N d { } degree d, and let n := (α , . . . , α ) : α = d . For an homogeneous poly- nomial g ∈ R[x] , andd with s1(d) := nn−1+d )i, liet g = (g ) ∈ Rs(d) be its vector of d (cid:88) d (cid:88) α coeﬃcients, i.e.,   (cid:55)→ · · · x g(x) := g xα = g xα1 xαn . α α 1 n ∈N ∈N α n α n ⊂ R d d { ≤ } Denote by G n its associated sublevel set G := x : g(x) 1 . ⊂ Let P[x] Hom be the set of homogeneous polynomials of degree d whose d d associated level set G has ﬁnite Lebesgue volume. It is a convex cone; see [10,

text: JEAN

text: B.

text: LASSERRE

text: the

text: Euclidean

text: ball

text: x

text: which

text: has

text: the

text: equivalent

text: quartic

text: representation

text: i

text: i

text: x

text: n

text: and

text: the

text: SOS

text: quartic

text: polynomial

text: x

text: n

text: solves

text: P

text: i

text: i

text: (c)

text: We

text: also

text: consider

text: the

text: SOS

text: (sum

text: of

text: squares)

text: version

text: of

text: P

text: that

text: is

text: one

text: now

text: searches

text: for

text: a

text: degree-d

text: SOS

text: homogeneous

text: polynomial

text: g

text: (x)

text: v

text: (x)Qv

text: (x),

text: Q

text: Q

text: (where

text: v

text: (x)

text: (xα),

text: α

text: That

text: is,

text: one

text: characterizes

text: the

text: unique

text: optimal

text: solution

text: of

text: the

text: optimization

text: problem:

text: P

text: inf

text: trace

text: (Q)

text: vol

text: (G

text: ρ

text: Q

text: Q

text: d

text: Q

text: In

text: this

text: matrix

text: context,

text: trace

text: (Q)

text: is

text: the

text: nuclear

text: norm

text: of

text: Q

text: and

text: so

text: solving

text: P

text: aims

text: at

text: ﬁnding

text: an

text: optimal

text: solution

text: Q

text: with

text: small

text: rank,

text: which

text: translates

text: into

text: an

text: homogeneous

text: polynomial

text: g

text: which

text: is

text: a

text: sum

text: of

text: a

text: few

text: squares.

text: We

text: also

text: proves

text: that

text: Q

text: g

text: associated

text: with

text: the

text: L

text: -unit

text: ball

text: cannot

text: be

text: an

text: optimal

text: solution

text: of

text: P

text: (and

text: indeed

text: d

text: g

text: being

text: a

text: sum

text: of

text: a

text: few

text: squares

text: does

text: not

text: necessarily

text: implies

text: that

text: it

text: has

text: a

text: small

text: Q

text: number

text: of

text: coeﬃcients).

text: (d)

text: Finally

text: we

text: also

text: show

text: that

text: results

text: in

text: (a)

text: and

text: (b)

text: extend

text: to

text: the

text: case

text: of

text: other

text: val-

text: ues

text: of

text: d

text: (including

text: p

text: and

text: rationals)

text: in

text: which

text: case

text: one

text: now

text: deals

text: with

text: positively

text: homogeneous

text: “generalized

text: polynomials”

text: (instead

text: of

text: homogeneous

text: polynomials)

text: and

text: one

text: has

text: to

text: deﬁne

text: an

text: appropriate

text: ﬁnite-dimensional

text: analogue

text: analogue

text: of

text: Hom

text: d

text: This

text: includes

text: the

text: interesting

text: case

text: of

text: the

text: L

text: -unit

text: ball

text: x

text: x

text: and

text: when

text: i

text: i

text: p

text: balls

text: which

text: are

text: not

text: associated

text: with

text: norms.

text: Notation,

text: definitions

text: and

text: preliminary

text: results

text: R

text: Let

text: [x]

text: denote

text: the

text: ring

text: or

text: real

text: polynomials

text: in

text: the

text: variables

text: x

text: (x

text: x

text: R

text: n

text: and

text: let

text: [x]

text: be

text: the

text: vector

text: space

text: of

text: real

text: polynomials

text: of

text: degree

text: at

text: most

text: d.

text: Similarly,

text: Rd

text: let

text: Σ[x]

text: [x]

text: denote

text: the

text: convex

text: cone

text: of

text: real

text: polynomials

text: that

text: are

text: sums

text: of

text: squares

text: (SOS)

text: of

text: polynomials,

text: and

text: Σ[x]

text: Σ[x]

text: its

text: subcone

text: of

text: SOS

text: polynomials

text: of

text: degree

text: S

text: d

text: at

text: most

text: d.

text: Denote

text: by

text: m

text: the

text: space

text: of

text: m

text: m

text: real

text: symmetric

text: matrices.

text: For

text: a

text: S

text: given

text: matrix

text: A

text: m,

text: the

text: notation

text: A

text: (resp.

text: A

text: means

text: that

text: A

text: is

text: positive

text: semideﬁnite

text: (psd)

text: (resp.

text: positive

text: deﬁnite

text: (pd)),

text: i.e.,

text: all

text: its

text: eigenvalues

text: are

text: real

text: and

text: nonnegative

text: (resp.

text: positive).

text: R

text: R

text: R

text: A

text: polynomial

text: p

text: [x]

text: is

text: homogenous

text: if

text: p(λx)

text: λdp(x)

text: for

text: all

text: x

text: n,

text: λ

text: R

text: R

text: d

text: R

text: A

text: function

text: f

text: n

text: is

text: positively

text: homogeneous

text: of

text: degree

text: d

text: if

text: f(λx)

text: λdf(x)

text: R

text: for

text: all

text: x

text: n,

text: λ

text: For

text: instance

text: x

text: x

text: is

text: not

text: homogeneous

text: but

text: is

text: positively

text: homogeneous

text: of

text: degree

text: R

text: Let

text: Hom

text: [x]

text: be

text: the

text: vector

text: space

text: of

text: homogeneous

text: polynomials

text: of

text: even

text: d

text: N

text: d

text: degree

text: d,

text: and

text: let

text: n

text: (α

text: α

text: α

text: d

text: For

text: an

text: homogeneous

text: poly-

text: nomial

text: g

text: R[x]

text: andd

text: with

text: )i,

text: liet

text: g

text: (g

text: Rs(d)

text: be

text: its

text: vector

text: of

text: d

text: d

text: α

text: coeﬃcients,

text: i.e.,

text: x

text: g(x)

text: g

text: xα

text: g

text: xαn

text: α

text: α

text: n

text: ∈N

text: ∈N

text: α

text: n

text: α

text: n

text: R

text: d

text: d

text: Denote

text: by

text: G

text: n

text: its

text: associated

text: sublevel

text: set

text: G

text: x

text: g(x)

text: Let

text: P[x]

text: Hom

text: be

text: the

text: set

text: of

text: homogeneous

text: polynomials

text: of

text: degree

text: d

text: whose

text: d

text: d

text: associated

text: level

text: set

text: G

text: has

text: ﬁnite

text: Lebesgue

text: volume.

text: It

text: is

text: a

text: convex

text: cone;

text: see


text: 2 present our conclusions and perspective for solid state (cid:88) emulation of the avian compass. II. SPIN DYNAMICS OF THE RADICAL PAIR − i 8 † − 1 † † ρ˙ = (cid:126)[H, ρ] + k PiρPi 2(Pi Piρ + ρPi Pi) (2) In order to study the spin transitions involved in the i=1 RP dynamics, we choose a representative RP system in × − which an unpaired spin on each of two radicals, and a Here k (= 5 105s 1) is the singlet and triplet radical single nucleus on one of them is responsible for an hyper- recombination rate. We note that this method is equiva- ﬁne interaction therein. This model may be directly ex- lent to the Haberkorn approach to modeling radical pair trapolated to multinuclear systems [27, 30]. The nucleus dynamics [19, 25, 27] but is more amenable for discern- preferentially interacts with the spin on the same radical ing the spin transitions involved in the compass action. ⊗ | (cid:105) ⊗ (cid:104) | and both the spins interact with the geomagnetic Zeeman The system starts in the state ρ(0) = 1I ( s s ). 2 ﬁeld. Therefore, the RP Hamiltonian looks like [27]: The ensuing spin evolution involves intersystem cross- ing between singlet and triplet states. It is accompanied by a spin dependent recombination process in which sin- · · · H = γB (Sˆ + Sˆ ) + Iˆ A Sˆ (1) glet and triplet radical pairs recombine through diﬀer- 1 2 2 ent channels; it is this Zeeman ﬁeld dependent diﬀeren- tial spin yield which is used by the avian neural system Sˆ and Sˆ are electron spin operators, and Iˆ is the to sense the geomagnetic ﬁeld [12, 27, 30], although the 1 2 ∈ nuclear spin operator given as: Iˆ, Sˆ , Sˆ 1(σ , σ , σ ), neurological processes involved are not yet fully under- 1 2 2 x y z γ = µ g is gyromagnetic ratio, µ is Bohr magneton and stood [28]. Our aim in this manuscript is to explain: 0 0 g is electron g-factor (= 2). We consider the illustrative one, the details of the spin transitions responsible for case in which the hyperﬁne tensor A = diag(0, 0, a) [30]. the magnetosensitive yield, and two, the role of coher- The external ﬁeld (geomagnetic ﬁeld) is characterized by ent evolution of electron pair spins and decoherence due B = B (sinθcosφ, sinθsinφ, cosθ); B (= 47µT) is the 0 0 local geomagnetic ﬁeld at Frankfurt [21] and θ is the magnetic ﬁeld orientation. The axial symmetry of hy- S Yie ld T0 Yield T+ Yield T- Yield perﬁne tensor allows us to take φ = 0 [27]. The RP 1.00 spin dynamics starts at the instant of radical pair gen- a/ B=0.01 a/ B=0.20 a/ B=0.70 eration, which is taken to be t = 0. The initial state s 0.75 d o|f (cid:105)the radical pair is usually taken to be singlet state el ( s ) with the nuclear spin completely depolarized ow- Yi 0.50 n ing to its interaction with the neighboring soft matter pi S 0.25 environment [13, 27, 31]. We adopt the quantum mas- ter equation approach to simulate the dynamics of the 0.00 RP system, similar to Gauger et al. [27] but modiﬁed 1.00 to highlight the exact spin transitions. Intrinsically, the a/ B=1.00 a/ B=2.00 a/ B=10.00 Hilbert space is eight dimensional. The spin-dependent s 0.75 d relaxation process happens through two channels – the el singlet channel whereby radical pairs in the |s(cid:105) state re- n Yi 0.50 combine and triplet channel whereby radical pairs in the pi | (cid:105) | (cid:105) | (cid:105) S 0.25 three triplet states t , t− and t recombine – which 0 + are included as ‘shelving’ states in the Hilbert space. In 0.00 order to distinguish the spin transitions, we resolve the 0 /8 /4 3 /8 /2 0 /8 /4 3 /8 /2 0 /8 /4 3 /8 /2 triplet channel into its three constituent channels corre- | (cid:105) | (cid:105) | (cid:105) sponding to t , t and t− . This is accommodated 0 + FIG. 1. (Color online) Singlet (S) and triplet (T , T , T−) by augmenting the eight dimensional Hilbert space with 0 + | (cid:105) | (cid:105) | (cid:105) yields vs. geomagnetic ﬁeld orientation for various hyperﬁne four additional shelving states denoted as S , T , T , | (cid:105) 0 + coupling strengths. These ﬁgures illustrates how various spin and T− . | (cid:105) | (cid:105) | (cid:105) | (cid:105) transition pathways between s , t , t , t− make the ﬁ- 0 + This method is quite versatile for the calculation of the nal spin yields angle dependent. Simultaneous Zeeman and yield corresponding to various spin states. The recombi- anisotropic hyperﬁne plays collaborative role in inducing these nation of radical pair into singlet and triplet channels is transitions which ultimately lead to magnetosensitive product modeled through decay operators in the master equation yield.(The various transitions induced by anisotropic hyper- | (cid:105) (cid:104) ↑| | (cid:105) (cid:104) ↓| | (cid:105) (cid:104) ↑| (ME) as: P = S s, , P = S s, , P = T t , , ﬁne and Zeeman interactions are shown in the state transition | (cid:105) (cid:104)1 ↓| | 2 (cid:105) (cid:104) ↑| 3 | 0(cid:105) (cid:104) 0 ↓| P = T t , , P = T t , , P = T t , , diagram, Fig. 2.) Additionally, we observe a conspicuous peak 4 | 0 (cid:105) (cid:104) 0 ↑| 5 + | +(cid:105) (cid:104) ↓|6 + + P = T− t−, and P = T− t−, . The Lindblad appearing in S yield for a range of hyperﬁne constants.This 7 8 peak corresponds to the dip in T and T− yields. We recog- master equation describing the evolution of RP spin sys- + nize that this peak appears due to nuclear decoherence. tem is given by [27]: ##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##

text: present

text: our

text: conclusions

text: and

text: perspective

text: for

text: solid

text: state

text: emulation

text: of

text: the

text: avian

text: compass.

text: II.

label: section

text: SPIN

text: DYNAMICS

text: OF

text: THE

text: RADICAL

text: PAIR

text: i

text: ρ˙

text: ρ]

text: k

text: PiρPi

text: Piρ

text: ρPi

text: Pi)

text: In

text: order

text: to

text: study

text: the

text: spin

text: transitions

text: involved

text: in

text: the

text: RP

text: dynamics,

text: we

text: choose

text: a

text: representative

text: RP

text: system

text: in

text: which

text: an

text: unpaired

text: spin

text: on

text: each

text: of

text: two

text: radicals,

text: and

text: a

text: Here

text: k

text: is

text: the

text: singlet

text: and

text: triplet

text: radical

text: single

text: nucleus

text: on

text: one

text: of

text: them

text: is

text: responsible

text: for

text: an

text: hyper-

text: recombination

text: rate.

text: We

text: note

text: that

text: this

text: method

text: is

text: equiva-

text: ﬁne

text: interaction

text: therein.

text: This

text: model

text: may

text: be

text: directly

text: ex-

text: lent

text: to

text: the

text: Haberkorn

text: approach

text: to

text: modeling

text: radical

text: pair

text: trapolated

text: to

text: multinuclear

text: systems

text: The

text: nucleus

text: dynamics

text: but

text: is

text: more

text: amenable

text: for

text: discern-

text: preferentially

text: interacts

text: with

text: the

text: spin

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text: the

text: same

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text: ing

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text: various

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text: tem

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text: given

text: by

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