text: Sabine Hossenfelder Disentangling the Black Hole Vacuum Whatever happens in the near horizon region, if unitarity is to be maintained, it must induce a suitable Bogoliubov trafo from the usual out-state into a new out-state Φ˜ = CΦ + DΦ¯ . (18) out out out Our aim is now to identify exactly what properties this transformation must have. 1. The operation we are looking for should change the correlations in the radiation I received at +, but not the spectrum. If it would alter the spectrum, one would run into the same problem as pointed out by AMPS, namely that, when tracing the I modes from + to the vicinity of the horizon, they will signiﬁcantly deviate from ‘empty’ space (on scales below the curvature radius) and the local observer will notice. The reason for this problem goes back to the divergence of the stress-energy, which is the relevant observable. As has been shown in [12, 13], the renormalized vacuum expectation value of the stress-energy tensor in the near horizon region is ﬁnite I for the local observer if and only if the spectrum at + is the normal thermal spectrum with temperature determined by the black hole’s mass. Any deviations from this thermal spectrum, regardless how small, will be blueshifted and generate drama for an observer who has to pass through. The stress-energy is determined by the spectral distribution of the modes and the absence of drama requires that this distribution should not be altered. This means that the operation we are looking for should not create additional particles, ie it should not mix positive and negative − frequencies of the outgoing radiation, and thus D = 0, and C 1 = C¯ T. The transformation (18) will then necessarily mix positive and negative frequencies of the ingoing state. (cid:101) 2. The (cid:101)transformation (18), when combined with the normal transformation (10), should no longer result in a mixing of dn-modes and out-modes. With A := CA and B := CB we denote the new transformation from the in-states to the out-states as (cid:101) (cid:101) Φ = A¯ TΦ˜ − BTΦ¯˜ . (19) in out out The second requirement the(cid:101)n takes t(cid:101)he(cid:101)form that − ¯ − V := BA 1 = diag (20) should not have off-diagonal entries. Since we do not produce additional particles (cid:18) (cid:19) and the spectrum outside should not change, this means (cid:101) 0 1 V := V . (21) 1 0 5 ##LTLine##

text: Sabine

text: Hossenfelder

text: Disentangling

text: the

text: Black

text: Hole

text: Vacuum

text: Whatever

text: happens

text: in

text: the

text: near

text: horizon

text: region,

text: if

text: unitarity

text: is

text: to

text: be

text: maintained,

text: it

text: must

text: induce

text: a

text: suitable

text: Bogoliubov

text: trafo

text: from

text: the

text: usual

text: out-state

text: into

text: a

text: new

text: out-state

text: Φ˜

text: CΦ

text: DΦ¯

text: out

text: out

text: out

text: Our

text: aim

text: is

text: now

text: to

text: identify

text: exactly

text: what

text: properties

text: this

text: transformation

text: must

text: have.

text: The

text: operation

text: we

text: are

text: looking

text: for

text: should

text: change

text: the

text: correlations

text: in

text: the

text: radiation

text: I

text: received

text: at

text: but

text: not

text: the

text: spectrum.

text: If

text: it

text: would

text: alter

text: the

text: spectrum,

text: one

text: would

text: run

text: into

text: the

text: same

text: problem

text: as

text: pointed

text: out

text: by

text: AMPS,

text: namely

text: that,

text: when

text: tracing

text: the

text: I

text: modes

text: from

text: to

text: the

text: vicinity

text: of

text: the

text: horizon,

text: they

text: will

text: signiﬁcantly

text: deviate

text: from

text: ‘empty’

text: space

text: (on

text: scales

text: below

text: the

text: curvature

text: radius)

text: and

text: the

text: local

text: observer

text: will

text: notice.

text: The

text: reason

text: for

text: this

text: problem

text: goes

text: back

text: to

text: the

text: divergence

text: of

text: the

text: stress-energy,

text: which

text: is

text: the

text: relevant

text: observable.

text: As

text: has

text: been

text: shown

text: in

text: the

text: renormalized

text: vacuum

text: expectation

text: value

text: of

text: the

text: stress-energy

text: tensor

text: in

text: the

text: near

text: horizon

text: region

text: is

text: ﬁnite

text: I

text: for

text: the

text: local

text: observer

text: if

text: and

text: only

text: if

text: the

text: spectrum

text: at

text: is

text: the

text: normal

text: thermal

text: spectrum

text: with

text: temperature

text: determined

text: by

text: the

text: black

text: hole’s

text: mass.

text: Any

text: deviations

text: from

text: this

text: thermal

text: spectrum,

text: regardless

text: how

text: small,

text: will

text: be

text: blueshifted

text: and

text: generate

text: drama

text: for

text: an

text: observer

text: who

text: has

text: to

text: pass

text: through.

text: The

text: stress-energy

text: is

text: determined

text: by

text: the

text: spectral

text: distribution

text: of

text: the

text: modes

text: and

text: the

text: absence

text: of

text: drama

text: requires

text: that

text: this

text: distribution

text: should

text: not

text: be

text: altered.

text: This

text: means

text: that

text: the

text: operation

text: we

text: are

text: looking

text: for

text: should

text: not

text: create

text: additional

text: particles,

text: ie

text: it

text: should

text: not

text: mix

text: positive

text: and

text: negative

text: frequencies

text: of

text: the

text: outgoing

text: radiation,

text: and

text: thus

text: D

text: and

text: C

text: C¯

text: T.

text: The

text: transformation

text: will

text: then

text: necessarily

text: mix

text: positive

text: and

text: negative

text: frequencies

text: of

text: the

text: ingoing

text: state.

text: The

text: when

text: combined

text: with

text: the

text: normal

text: transformation

text: should

text: no

text: longer

text: result

text: in

text: a

text: mixing

text: of

text: dn-modes

text: and

text: out-modes.

text: With

text: A

text: CA

text: and

text: B

text: CB

text: we

text: denote

text: the

text: new

text: transformation

text: from

text: the

text: in-states

text: to

text: the

text: out-states

text: as

text: Φ

text: A¯

text: TΦ˜

text: BTΦ¯˜

text: in

text: out

text: out

text: The

text: second

text: requirement

text: takes

text: that

text: V

text: BA

text: diag

text: should

text: not

text: have

text: off-diagonal

text: entries.

text: Since

text: we

text: do

text: not

text: produce

text: additional

text: particles

text: and

text: the

text: spectrum

text: outside

text: should

text: not

text: change,

text: this

text: means

text: V

text: V

text: ##LTLine##


text: Sabine Hossenfelder Disentangling the Black Hole Vacuum lost lost out out (u,v) S R v > 0 v > 0 v = v v = v 0 0 (U,V) v = 0 v = 0 v < 0 v < 0 Figure 1: Causal Diagram of black hole formation. The wavy line demarks the singular- ity, the fat black line the horizon, and the thin black line marked with S is the stretched horizon. The grey shaded region is the inside of the collapsing matter; its surface is at R(u, v) = 0. The (red and green) diagonal lines depict modes. The solid parts of these modes depict the usual ray-tracing, the dashed parts depict the new contribution from the boundary condition at S. Left: The outgoing radiation contains information both from ≤ ≥ the region v 0 and v 0. Right: The same setting but traced backwards. The matrix Y is only there to adjust the phase factor. p is the total momentum of the Φ- modes incoming to the boundary and p˜ is the total momentum of the Φ˜-modes outgoing from the boundary in spherical coordinates. We will now go on to demonstrate that this boundary condition fulﬁlls our require- ments. Recall that we are suppressing an index ω labelling the modes. The above ansatz for C means that we take the boundary condition to not depend on the frequency. To begin with, it is CC¯T = 1, so we can interpret this as the result of a scattering process that comes about by some unknown physics located at S = 0, and will not ruin unitarity. The volume factor is there to take into account a suitable normalization procedure for plane waves as usual. In the following we will look at a state (the vacuum state) that preserves the four momentum at the boundary and we will take this factor to 7 ##LTFigure## ##LTFigure## ##LTLine##

text: Sabine

label: paragraph

text: Hossenfelder

text: Disentangling

text: the

text: Black

text: Hole

text: Vacuum

text: lost

font: default

text: lost

text: out

text: out

text: (u,v)

text: S

text: R

text: v

text: v

text: v

text: v

text: v

text: v

text: (U,V)

text: v

text: v

text: v

text: v

text: Figure

text: Causal

label: caption

text: Diagram

text: of

text: black

text: hole

text: formation.

text: The

text: wavy

text: line

text: demarks

text: the

text: singular-

text: ity,

text: the

text: fat

text: black

text: line

text: the

text: horizon,

text: and

text: the

text: thin

text: black

text: line

text: marked

text: with

text: S

text: is

text: the

text: stretched

text: horizon.

text: The

text: grey

text: shaded

text: region

text: is

text: the

text: inside

text: of

text: the

text: collapsing

text: matter;

text: its

text: surface

text: is

text: at

text: R(u,

text: v)

text: The

text: (red

text: and

text: green)

text: diagonal

text: lines

text: depict

text: modes.

text: The

text: solid

text: parts

text: of

text: these

text: modes

text: depict

text: the

text: usual

text: ray-tracing,

text: the

text: dashed

text: parts

text: depict

text: the

text: new

text: contribution

text: from

text: the

text: boundary

text: condition

text: at

text: S.

text: Left:

text: The

text: outgoing

text: radiation

text: contains

text: information

text: both

text: from

text: the

text: region

text: v

text: and

text: v

text: Right:

text: The

text: same

text: setting

text: but

text: traced

text: backwards.

text: The

text: matrix

text: Y

text: is

text: only

text: there

text: to

text: adjust

text: the

text: phase

text: factor.

text: p

text: is

text: the

text: total

text: momentum

text: of

text: the

text: Φ-

text: modes

text: incoming

text: to

text: the

text: boundary

text: and

text: p˜

text: is

text: the

text: total

text: momentum

text: of

text: the

text: Φ˜-modes

text: outgoing

text: from

text: the

text: boundary

text: in

text: spherical

text: coordinates.

text: We

text: will

text: now

text: go

text: on

text: to

text: demonstrate

text: that

text: this

text: boundary

text: condition

text: fulﬁlls

text: our

text: require-

text: ments.

text: Recall

text: that

text: we

text: are

text: suppressing

text: an

text: index

text: ω

text: labelling

text: the

text: modes.

text: The

text: above

text: ansatz

text: for

text: C

text: means

text: that

text: we

text: take

text: the

text: boundary

text: condition

text: to

text: not

text: depend

text: on

text: the

text: frequency.

text: To

text: begin

text: with,

text: it

text: is

text: CC¯T

text: so

text: we

text: can

text: interpret

text: this

text: as

text: the

text: result

text: of

text: a

text: scattering

text: process

text: that

text: comes

text: about

text: by

text: some

text: unknown

text: physics

text: located

text: at

text: S

text: and

text: will

text: not

text: ruin

text: unitarity.

text: The

text: volume

text: factor

text: is

text: there

text: to

text: take

text: into

text: account

text: a

text: suitable

text: normalization

text: procedure

text: for

text: plane

text: waves

text: as

text: usual.

text: In

text: the

text: following

text: we

text: will

text: look

text: at

text: a

text: state

text: (the

text: vacuum

text: state)

text: that

text: preserves

text: the

text: four

text: momentum

text: at

text: the

text: boundary

text: and

text: we

text: will

text: take

text: this

text: factor

text: to

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: Sabine Hossenfelder Disentangling the Black Hole Vacuum be one, though we will come back to the requirement of momentum conservation in the discussion . Before we come to the discussion though, note that here and for the rest of this and the next section we are not talking about what happens to infalling information, I− we are only considering the case in which the ﬁeld is in the vacuum state at . If we make a plane-wave ansatz for Φ, the reﬂecting condition (23) will convert − I ΦT = (exp( iωu), 0) coming from + to away √ √ − − − Φ˜T = (ie iωu/ 2, ie iω ln(v/κ)/ 2) (28) moving towards S = 0. As anticipated, the phase shift from y is cancelled by Y. This means the boundary condition converts both out-modes and dn-modes into combinations of out- and dn-modes at the same frequency. Note that the location of S is essential for this to work. Needless to say, this particular scattering process would not preserve momentum because part of the wave is reﬂected. Alas, in the vacuum state there will be another contribution from φ that provides the missing momentum. The requirement of dn momentum conservation in the chosen base means that the mixture of modes dependent on u and v has to stay the same at the boundary. Since C mixes them in equal parts, this means that the state ingoing to the boundary must also have had contributions in ∼ equal parts, ie Φ˜T (φ , φ ), the relevant point being that the prefactors in both out dn components are identical. For the annihilation and creation operators, the boundary condition performs the fol- lowing mixing † † a˜ = a C¯ T , a˜ = a CT . (29) out out out out The physical picture is the following. We take the in-modes and move them for- I ward to + or to the inside of the horizon respectively. On the way, we have to pass S = 0 and have to take into account the boundary condition. By this, the solutions of the wave-equation change. If we start with deﬁning the in-vacuum as the physically real vacuum, then the out-state that corresponds to the in-vacuum is now a different state. It differs from the ‘normal’ out-vacuum that contains thermal radiation (without the bound- ary condition) by the operation induced by C. The early ingoing modes do not encounter a boundary condition. This seems natural, because in the early phase the matter distribu- tion is nothing like a black hole. The dominant contribution for the Hawking-radiation comes from the near-horizon region which is why the approximation around these rapidly oscillating modes is excellent. Therefore it does not really matter what happens with the early ingoing modes and we will not further speci(cid:18)fy this re(cid:19)gion. In the new basis (cid:101)we(cid:101)have ¯ T 1 0 BB = C¯ BB¯TCT = c2w2 , (30) 0 1 (cid:18) (cid:19) ie the spectrum remains unchanged as expected(cid:101). And further | (cid:105) 1 † † | (cid:105) 0 = eiW exp a V(a )T 0 , (31) in 2 out out out 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Sabine

text: Hossenfelder

text: Disentangling

text: the

text: Black

text: Hole

text: Vacuum

text: be

text: one,

text: though

text: we

text: will

text: come

text: back

text: to

text: the

text: requirement

text: of

text: momentum

text: conservation

text: in

text: the

text: discussion

text: Before

text: we

text: come

text: to

text: the

text: discussion

text: though,

text: note

text: that

text: here

text: and

text: for

text: the

text: rest

text: of

text: this

text: and

text: the

text: next

text: section

text: we

text: are

text: not

text: talking

text: about

text: what

text: happens

text: to

text: infalling

text: information,

text: I−

text: we

text: are

text: only

text: considering

text: the

text: case

text: in

text: which

text: the

text: ﬁeld

text: is

text: in

text: the

text: vacuum

text: state

text: at

text: If

text: we

text: make

text: a

text: plane-wave

text: ansatz

text: for

text: Φ,

text: the

text: reﬂecting

text: condition

text: will

text: convert

text: I

text: ΦT

text: (exp(

text: iωu),

text: coming

text: from

text: to

text: away

text: Φ˜T

text: (ie

text: iωu/

text: ie

text: iω

text: ln(v/κ)/

text: moving

text: towards

text: S

text: As

text: anticipated,

text: the

text: phase

text: shift

text: from

text: y

text: is

text: cancelled

text: by

text: Y.

text: This

text: means

text: the

text: boundary

text: condition

text: converts

text: both

text: out-modes

text: and

text: dn-modes

text: into

text: combinations

text: of

text: out-

text: and

text: dn-modes

text: at

text: the

text: same

text: frequency.

text: Note

text: that

text: the

text: location

text: of

text: S

text: is

text: essential

text: for

text: this

text: to

text: work.

text: Needless

text: to

text: say,

text: this

text: particular

text: scattering

text: process

text: would

text: not

text: preserve

text: momentum

text: because

text: part

text: of

text: the

text: wave

text: is

text: reﬂected.

text: Alas,

text: in

text: the

text: vacuum

text: state

text: there

text: will

text: be

text: another

text: contribution

text: from

text: φ

text: that

text: provides

text: the

text: missing

text: momentum.

text: The

text: requirement

text: of

text: dn

text: momentum

text: conservation

text: in

text: the

text: chosen

text: base

text: means

text: that

text: the

text: mixture

text: of

text: modes

text: dependent

text: on

text: u

text: and

text: v

text: has

text: to

text: stay

text: the

text: same

text: at

text: the

text: boundary.

text: Since

text: C

text: mixes

text: them

text: in

text: equal

text: parts,

text: this

text: means

text: that

text: the

text: state

text: ingoing

text: to

text: the

text: boundary

text: must

text: also

text: have

text: had

text: contributions

text: in

text: equal

text: parts,

text: ie

text: Φ˜T

text: (φ

text: φ

text: the

text: relevant

text: point

text: being

text: that

text: the

text: prefactors

text: in

text: both

text: out

text: dn

text: components

text: are

text: identical.

text: For

text: the

text: annihilation

text: and

text: creation

text: operators,

text: the

text: boundary

text: condition

text: performs

text: the

text: fol-

text: lowing

text: mixing

text: a˜

text: a

text: C¯

text: T

text: a˜

text: a

text: CT

text: out

text: out

text: out

text: out

text: The

text: physical

text: picture

text: is

text: the

text: following.

text: We

text: take

text: the

text: in-modes

text: and

text: move

text: them

text: for-

text: I

text: ward

text: to

text: or

text: to

text: the

text: inside

text: of

text: the

text: horizon

text: respectively.

text: On

text: the

text: way,

text: we

text: have

text: to

text: pass

text: S

text: and

text: have

text: to

text: take

text: into

text: account

text: the

text: boundary

text: condition.

text: By

text: this,

text: the

text: solutions

text: of

text: the

text: wave-equation

text: change.

text: If

text: we

text: start

text: with

text: deﬁning

text: the

text: in-vacuum

text: as

text: the

text: physically

text: real

text: vacuum,

text: then

text: the

text: out-state

text: that

text: corresponds

text: to

text: the

text: in-vacuum

text: is

text: now

text: a

text: different

text: state.

text: It

text: differs

text: from

text: the

text: ‘normal’

text: out-vacuum

text: that

text: contains

text: thermal

text: radiation

text: (without

text: the

text: bound-

text: ary

text: condition)

text: by

text: the

text: operation

text: induced

text: by

text: C.

text: The

text: early

text: ingoing

text: modes

text: do

text: not

text: encounter

text: a

text: boundary

text: condition.

text: This

text: seems

text: natural,

text: because

text: in

text: the

text: early

text: phase

text: the

text: matter

text: distribu-

text: tion

text: is

text: nothing

text: like

text: a

text: black

text: hole.

text: The

text: dominant

text: contribution

text: for

text: the

text: Hawking-radiation

text: comes

text: from

text: the

text: near-horizon

text: region

text: which

text: is

text: why

text: the

text: approximation

text: around

text: these

text: rapidly

text: oscillating

text: modes

text: is

text: excellent.

text: Therefore

text: it

text: does

text: not

text: really

text: matter

text: what

text: happens

text: with

text: the

text: early

text: ingoing

text: modes

text: and

text: we

text: will

text: not

text: further

text: this

text: In

text: the

text: new

text: basis

text: T

text: BB

text: C¯

text: BB¯TCT

text: ie

text: the

text: spectrum

text: remains

text: unchanged

text: as

text: And

text: further

text: eiW

text: exp

text: a

text: V(a

text: )T

text: in

text: out

text: out

text: out

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 0.2982 0.292 0.298 0.29 m) 0 0 km 0.288 L=50 0.2978 o @ 6 0.286 m) / P( 0.2976 nt rati 0.284 0000 0.2974 e 6 v = e L otal 0.282 P( 0.2972 t 0.28 0.297 0.278 0.2968 0.276 0 2 4 6 8 10 0 2 4 6 8 10 12 θ14 / ° θ14 / ° FIG. 5: The total event ratio at the baseline of 60 km and the relative event rate P(L = 60000 m)/P(L = ◦ 500 m) as the functions of θ , where θ = 8.9 has been chosen as an input. 14 13 Figure 5 shows the total event ratio at the baseline of 60 km and the relative total event rate P (L = 60000 m)/P (L = 500 m) as the functions of θ 3. In order to get some information on the 14 active-sterile mixing angle θ or to put an upper limit on it, the total event ratio should be precisely 14 O − determined to the level of (10 3), or the relative event rate should be precisely measured to the O − level of (10 4) at the third baseline around 60 km in future precision reactor experiments, and the uncertainties from other mixing parameters (θ , θ and ∆m2 ) should also be well reduced 12 13 21 to have the same precision. B. Spectral Analysis One can see from Eq. (7) that three new oscillatory terms P , P and P (i.e., three new 41 42 43 oscillatory frequencies) are included in due to the existence of one light sterile neutrino ν . A 4 direct measurement of the oscillatory behaviors of these three terms will certainly provide the direct evidence of the existence of such light sterile neutrinos. However the amplitudes of all these three oscillations are rather small (proportional to s2 ). It has been found that comparing 14 to a normal L/E spectrum analysis, the Fourier analysis naturally separates the mass hierarchy information from uncertainties of the reactor antineutrino spectra and other mixing parameters, which is critical for very small oscillations. The frequency spectrum can be obtained by applying 3 The small peeks/dips in Figs. 4 and 5 are caused by the numerical errors, which arise from the integrations. This kind of numerical errors might be accidentally ampliﬁed when calculating the ratio of two integrated event rates. 12 ##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##

text: m)

text: km

text: o

text: m)

text: P(

text: nt

text: rati

text: e

text: v

text: e

text: L

text: otal

text: P(

text: t

text: FIG.

text: The

text: total

text: event

text: ratio

text: at

text: the

text: baseline

text: of

text: km

text: and

text: the

text: relative

text: event

text: rate

text: P(L

text: m)/P(L

text: m)

text: as

text: the

text: functions

text: of

text: θ

text: where

text: θ

text: has

text: been

text: chosen

text: as

text: an

text: input.

text: Figure

text: shows

text: the

text: total

text: event

text: ratio

text: at

text: the

text: baseline

text: of

text: km

text: and

text: the

text: relative

text: total

text: event

text: rate

text: P

text: (L

text: m)/P

text: (L

text: m)

text: as

text: the

text: functions

text: of

text: θ

text: In

text: order

text: to

text: get

text: some

text: information

text: on

text: the

text: active-sterile

text: mixing

text: angle

text: θ

text: or

text: to

text: put

text: an

text: upper

text: limit

text: on

text: it,

text: the

text: total

text: event

text: ratio

text: should

text: be

text: precisely

text: O

text: determined

text: to

text: the

text: level

text: of

text: or

text: the

text: relative

text: event

text: rate

text: should

text: be

text: precisely

text: measured

text: to

text: the

text: O

text: level

text: of

text: at

text: the

text: third

text: baseline

text: around

text: km

text: in

text: future

text: precision

text: reactor

text: experiments,

text: and

text: the

text: uncertainties

text: from

text: other

text: mixing

text: parameters

text: (θ

text: θ

text: and

text: should

text: also

text: be

text: well

text: reduced

text: to

text: have

text: the

text: same

text: precision.

text: B.

text: Spectral

text: Analysis

text: One

text: can

text: see

text: from

text: Eq.

text: that

text: three

text: new

text: oscillatory

text: terms

text: P

text: P

text: and

text: P

text: (i.e.,

text: three

text: new

text: oscillatory

text: frequencies)

text: are

text: included

text: in

text: due

text: to

text: the

text: existence

text: of

text: one

text: light

text: sterile

text: neutrino

text: ν

text: A

text: direct

text: measurement

text: of

text: the

text: oscillatory

text: behaviors

text: of

text: these

text: three

text: terms

text: will

text: certainly

text: provide

text: the

text: direct

text: evidence

text: of

text: the

text: existence

text: of

text: such

text: light

text: sterile

text: neutrinos.

text: However

text: the

text: amplitudes

text: of

text: all

text: these

text: three

text: oscillations

text: are

text: rather

text: small

text: (proportional

text: to

text: It

text: has

text: been

text: found

text: that

text: comparing

text: to

text: a

text: normal

text: L/E

text: spectrum

text: analysis,

text: the

text: Fourier

text: analysis

text: naturally

text: separates

text: the

text: mass

text: hierarchy

text: information

text: from

text: uncertainties

text: of

text: the

text: reactor

text: antineutrino

text: spectra

text: and

text: other

text: mixing

text: parameters,

text: which

text: is

text: critical

text: for

text: very

text: small

text: oscillations.

text: The

text: frequency

text: spectrum

text: can

text: be

text: obtained

text: by

text: applying

text: The

text: small

text: peeks/dips

text: in

text: Figs.

text: and

text: are

text: caused

text: by

text: the

text: numerical

text: errors,

text: which

text: arise

text: from

text: the

text: integrations.

text: This

text: kind

text: of

text: numerical

text: errors

text: might

text: be

text: accidentally

text: ampliﬁed

text: when

text: calculating

text: the

text: ratio

text: of

text: two

text: integrated

text: event

text: rates.

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text: requirements of energy resolution and statistic are required. Therefore, it should be considered only as a complementary to the measurement by the analysis of the frequency spectra of three standard oscillatory terms P , P and P [20, 24, 25]. 21 31 32 III. SUMMARY Even though there have been many positive hints of the possible existence of sterile neutrinos and small unitarity violation in the MNSP matrix from both the theoretical and the experimental sides, there is currently no deﬁnite constraint on the mass of these particles. It is one of the important jobs to determine or constrain the number of sterile neutrinos and their mass and mixing properties in future precision experiments. The existence of sterile neutrinos can produce various kinds of eﬀects on neutrino oscillations depending on the properties of the sterile neutrinos (e.g., the scale of the sterile neutrino mass, the magnitude and the structure of the active-sterile mixing) as well as the conﬁgurations of the experiments (e.g., the oscillation channel, the energy spectrum of the neutrino ﬂux, the baseline L, whether the matter eﬀect need to be taken into account). In this paper we studied the possibility of searching for sub-eV sterile neutrinos in the precision reactor antineutrino oscillation experiments with three diﬀerent baselines at around 500 m, 2 km and 60 km respectively. The strategy of placing functionally identical detectors at diﬀerent baselines and carrying out a combined analysis can oﬀer a “clean” measurement of the electron antineutrino survival probabilities which is CP-phases independent as well as antineutrino ﬂux independent. We found that the active-sterile mixing angle θ could be determined or constrained by the precision 14 measurement of the relative event rate P (L = 60000 m)/P (L = 500 m), provided that θ , θ 13 12 and ∆m2 were well determined. The mass-squared diﬀerence ∆m2 could be obtained from the 21 41 Fourier transformation to the L/E spectrum at the near detector. We underline that the antineutrino survival probabilities obtained in reactor experiments are sensitive only to the direct unitarity violation which is induced by the existence of light sterile neutrinos but independent of the indirect unitarity violation parameters. More speciﬁcally, the reactor experiments oﬀer very concentrated sensitivity only to two of the direct unitarity violation parameters θ and ∆m2 . This means if any signals of unitarity violation are observed, we can then 14 41 draw some deﬁnite informations on the mass and mixing properties of the light sterile neutrinos. On the contrary, if no observable eﬀect of the unitarity violation are found in the reactor experiments, strong constraints on θ and ∆m2 should be obtained without the possibilities of cancelations 14 41 16

text: requirements

text: of

text: energy

text: resolution

text: and

text: statistic

text: are

text: required.

text: Therefore,

text: it

text: should

text: be

text: considered

text: only

text: as

text: a

text: complementary

text: to

text: the

text: measurement

text: by

text: the

text: analysis

text: of

text: the

text: frequency

text: spectra

text: of

text: three

text: standard

text: oscillatory

text: terms

text: P

text: P

text: and

text: P

text: III.

label: section

text: SUMMARY

text: Even

text: though

text: there

text: have

text: been

text: many

text: positive

text: hints

text: of

text: the

text: possible

text: existence

text: of

text: sterile

text: neutrinos

text: and

text: small

text: unitarity

text: violation

text: in

text: the

text: MNSP

text: matrix

text: from

text: both

text: the

text: theoretical

text: and

text: the

text: experimental

text: sides,

text: there

text: is

text: currently

text: no

text: deﬁnite

text: constraint

text: on

text: the

text: mass

text: of

text: these

text: particles.

text: It

text: is

text: one

text: of

text: the

text: important

text: jobs

text: to

text: determine

text: or

text: constrain

text: the

text: number

text: of

text: sterile

text: neutrinos

text: and

text: their

text: mass

text: and

text: mixing

text: properties

text: in

text: future

text: precision

text: experiments.

text: The

text: existence

text: of

text: sterile

text: neutrinos

text: can

text: produce

text: various

text: kinds

text: of

text: eﬀects

text: on

text: neutrino

text: oscillations

text: depending

text: on

text: the

text: properties

text: of

text: the

text: sterile

text: neutrinos

text: (e.g.,

text: the

text: scale

text: of

text: the

text: sterile

text: neutrino

text: mass,

text: the

text: magnitude

text: and

text: the

text: structure

text: of

text: the

text: active-sterile

text: mixing)

text: as

text: well

text: as

text: the

text: conﬁgurations

text: of

text: the

text: experiments

text: (e.g.,

text: the

text: oscillation

text: channel,

text: the

text: energy

text: spectrum

text: of

text: the

text: neutrino

text: ﬂux,

text: the

text: baseline

text: L,

text: whether

text: the

text: matter

text: eﬀect

text: need

text: to

text: be

text: taken

text: into

text: account).

text: In

text: this

text: paper

text: we

text: studied

text: the

text: possibility

text: of

text: searching

text: for

text: sub-eV

text: sterile

text: neutrinos

text: in

text: the

text: precision

text: reactor

text: antineutrino

text: oscillation

text: experiments

text: with

text: three

text: diﬀerent

text: baselines

text: at

text: around

text: m,

text: km

text: and

text: km

text: respectively.

text: The

text: strategy

text: of

text: placing

text: functionally

text: identical

text: detectors

text: at

text: diﬀerent

text: baselines

text: and

text: carrying

text: out

text: a

text: combined

text: analysis

text: can

text: oﬀer

text: a

text: “clean”

text: measurement

text: of

text: the

text: electron

text: antineutrino

text: survival

text: probabilities

text: which

text: is

text: CP-phases

text: independent

text: as

text: well

text: as

text: antineutrino

text: ﬂux

text: independent.

text: We

text: found

text: that

text: the

text: active-sterile

text: mixing

text: angle

text: θ

text: could

text: be

text: determined

text: or

text: constrained

text: by

text: the

text: precision

text: measurement

text: of

text: the

text: relative

text: event

text: rate

text: P

text: (L

text: m)/P

text: (L

text: m),

text: provided

text: that

text: θ

text: θ

text: and

text: were

text: well

text: determined.

text: The

text: mass-squared

text: diﬀerence

text: could

text: be

text: obtained

text: from

text: the

text: Fourier

text: transformation

text: to

text: the

text: L/E

text: spectrum

text: at

text: the

text: near

text: detector.

text: We

text: underline

text: that

text: the

text: antineutrino

text: survival

text: probabilities

text: obtained

text: in

text: reactor

text: experiments

text: are

text: sensitive

text: only

text: to

text: the

text: direct

text: unitarity

text: violation

text: which

text: is

text: induced

text: by

text: the

text: existence

text: of

text: light

text: sterile

text: neutrinos

text: but

text: independent

text: of

text: the

text: indirect

text: unitarity

text: violation

text: parameters.

text: More

text: speciﬁcally,

text: the

text: reactor

text: experiments

text: oﬀer

text: very

text: concentrated

text: sensitivity

text: only

text: to

text: two

text: of

text: the

text: direct

text: unitarity

text: violation

text: parameters

text: θ

text: and

text: This

text: means

text: if

text: any

text: signals

text: of

text: unitarity

text: violation

text: are

text: observed,

text: we

text: can

text: then

text: draw

text: some

text: deﬁnite

text: informations

text: on

text: the

text: mass

text: and

text: mixing

text: properties

text: of

text: the

text: light

text: sterile

text: neutrinos.

text: On

text: the

text: contrary,

text: if

text: no

text: observable

text: eﬀect

text: of

text: the

text: unitarity

text: violation

text: are

text: found

text: in

text: the

text: reactor

text: experiments,

text: strong

text: constraints

text: on

text: θ

text: and

text: should

text: be

text: obtained

text: without

text: the

text: possibilities

text: of

text: cancelations


text: are parametrized as     1 0 O   = , 45 0 U  0 A R O O O O O O   = , 35 25 15 34 24 14 S B   V 0 O O O 0 = , (5) 23 13 12 0 1 O ≤ ≤ where ten (for 1 i < j 5) are two-dimensional rotation matrices in the ﬁve-dimensional ij complex space whose explicit expressions can be found in Ref. [15]. One can easily see from this parametrization that the matrices V and U are unitary while A, B, R, S are not. The production 0 0 × AV can be regarded as the eﬀective 3 3 MNSP matrix in this (3+1+1) scenario which is in 0 general non-unitary. | | An apparent advantage of this parametrization is that all the ﬁve moduli V that are involved ei in Eq. (3) have very concise expressions: | | V = c c c c , e1 12 13 14 15 | | V = s c c c , e2 12 13 14 15 | | V = s c c , e3 13 14 15 | | V = s c , e4 14 15 | | V = s , (6) e5 15 ≡ ≡ where c cos θ and s sin θ with ij = 12, 13, 14 and 15. Here θ stands for the mixing ij ij ij ij 14 between the light sterile neutrino and the active neutrinos while θ stands for the mixing between 15 the heavy sterile neutrino and the active neutrinos. We can clearly ﬁnd in Eq. (6) that if θ = 0 14 | | | | then we have V = 0, while V = 0 can be easily obtained by taking θ = 0. The survival e4 e5 15 → probability P (ν¯ ν¯ ) is then given by e e 5

text: are

text: parametrized

text: as

label: equation

text: O

text: U

text: 0

text: A

text: R

text: O

text: O

text: O

text: O

text: O

text: O

text: S

text: B

text: V

text: O

text: O

text: O

text: O

text: where

text: ten

text: (for

text: i

text: j

text: are

text: two-dimensional

text: rotation

text: matrices

text: in

text: the

text: ﬁve-dimensional

text: ij

text: complex

text: space

text: whose

text: explicit

text: expressions

text: can

text: be

text: found

text: in

text: Ref.

text: One

text: can

text: easily

text: see

text: from

text: this

text: parametrization

text: that

text: the

text: matrices

text: V

text: and

text: U

text: are

text: unitary

text: while

text: A,

text: B,

text: R,

text: S

text: are

text: not.

text: The

text: production

text: AV

text: can

text: be

text: regarded

text: as

text: the

text: eﬀective

text: MNSP

text: matrix

text: in

text: this

text: scenario

text: which

text: is

text: in

text: general

text: non-unitary.

text: An

text: apparent

text: advantage

text: of

text: this

text: parametrization

text: is

text: that

text: all

text: the

text: ﬁve

text: moduli

text: V

text: that

text: are

text: involved

text: ei

text: in

text: Eq.

text: have

text: very

text: concise

text: expressions:

text: V

text: c

text: c

text: c

text: c

text: V

text: s

text: c

text: c

text: c

text: V

text: s

text: c

text: c

text: V

text: s

text: c

text: V

text: s

text: where

text: c

text: cos

text: θ

text: and

text: s

text: sin

text: θ

text: with

text: ij

text: and

text: Here

text: θ

text: stands

text: for

text: the

text: mixing

text: ij

text: ij

text: ij

text: ij

text: between

text: the

text: light

text: sterile

text: neutrino

text: and

text: the

text: active

text: neutrinos

text: while

text: θ

text: stands

text: for

text: the

text: mixing

text: between

text: the

text: heavy

text: sterile

text: neutrino

text: and

text: the

text: active

text: neutrinos.

text: We

text: can

text: clearly

text: ﬁnd

text: in

text: Eq.

text: that

text: if

text: θ

text: then

text: we

text: have

text: V

text: while

text: V

text: can

text: be

text: easily

text: obtained

text: by

text: taking

text: θ

text: The

text: survival

text: probability

text: P

text: (ν¯

text: ν¯

text: is

text: then

text: given

text: by

text: e

text: e


text: Uniqueness of Conservative Solutions to the Camassa-Holm Equation via Characteristics ∗ ∗∗ ∗ Alberto Bressan( ), Geng Chen( ), and Qingtian Zhang( ) (*) Department of Mathematics, Penn State University, University Park, Pa. 16802, U.S.A. (**) School of Mathematics, Georgia Institute of Technology, Atlanta, Ga. 30332, U.S.A. e-mails: bressan@math.psu.edu, gchen73@math.gatech.edu, zhang q@math.psu.edu May 26, 2020 Abstract The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution u = u(t, x), an equation is introduced which singles out a unique characteristic curve through each initial point. By studying the evolution of the quantities u and v = 2 arctan u along each characteristic, x ∈ it is proved that the Cauchy problem with general initial data u H1(IR) has a unique 0 solution, globally in time. 1 Introduction The Cauchy problem for the Camassa-Holm equation [5] can be written in the form u + (u2/2) + P = 0 , (1.1) t x x u(0, x) = u (x). (1.2) 0 (cid:18) (cid:19) The nonlocal source term P is here deﬁned as a convolution: . 1 −| | ∗ u2 P = e x u2 + x . (1.3) 2 2 ∈ For any initial data u H1(IR), various papers have studied the global existence of solutions 0 [2, 3, 10, 11, 12]. In particular, in [2] a new set of variables was introduced, transforming the equation 1.1 into a semilinear system. This yields a group of conservative (i.e., energy- preserving) solutions u(t) = S u , depending continuously on the initial data w.r.t. suitable t 0 norms. In [3] a similar approach was used to construct a semigroup of dissipative solutions. Solutions obtained by this particular transformation of variables are clearly unique. However, 1 ##LTLine## ##LTLine## ##LTLine##

text: Uniqueness

label: title

text: of

text: Conservative

text: Solutions

text: to

text: the

text: Camassa-Holm

text: Equation

text: via

text: Characteristics

label: author

text: Alberto

text: Bressan(

text: Geng

text: Chen(

text: and

text: Qingtian

text: Zhang(

text: Department

text: of

text: Mathematics,

text: Penn

text: State

text: University,

text: University

text: Park,

text: Pa.

text: U.S.A.

text: School

text: of

text: Mathematics,

text: Georgia

text: Institute

text: of

text: Technology,

text: Atlanta,

text: Ga.

text: U.S.A.

text: e-mails:

text: bressan@math.psu.edu,

text: zhang

text: q@math.psu.edu

text: May

text: Abstract

text: The

label: abstract

text: paper

text: provides

text: a

text: direct

text: proof

text: the

text: uniqueness

text: of

text: solutions

text: to

text: the

text: Camassa-Holm

text: equation,

text: based

text: on

text: characteristics.

text: Given

text: a

text: conservative

text: solution

text: u

text: u(t,

text: x),

text: an

text: equation

text: is

text: introduced

text: which

text: singles

text: out

text: a

text: unique

text: characteristic

text: curve

text: through

text: each

text: initial

text: point.

text: By

text: studying

text: the

text: evolution

text: of

text: the

text: quantities

text: u

text: and

text: v

text: arctan

text: u

text: along

text: each

text: characteristic,

text: x

text: it

text: is

text: proved

text: that

text: the

text: Cauchy

text: problem

text: with

text: general

text: initial

text: data

text: u

text: has

text: a

text: unique

text: solution,

text: globally

text: in

text: time.

text: Introduction

text: The

text: Cauchy

text: problem

text: for

text: the

text: Camassa-Holm

text: equation

text: can

text: be

text: written

text: in

text: the

text: form

text: u

text: P

text: t

text: x

text: x

text: x)

text: u

text: (x).

text: The

text: nonlocal

text: source

text: term

text: P

text: is

text: here

text: deﬁned

text: as

text: a

text: convolution:

text: P

text: e

text: x

text: x

text: For

text: any

text: initial

text: data

text: u

text: various

text: papers

text: have

text: studied

text: the

text: global

text: existence

text: of

text: solutions

text: In

text: particular,

text: in

text: a

text: new

text: set

text: of

text: variables

text: was

text: introduced,

text: transforming

text: the

text: equation

text: into

text: a

text: semilinear

text: system.

text: This

text: yields

text: a

text: group

text: of

text: conservative

text: (i.e.,

text: energy-

text: preserving)

text: solutions

text: u(t)

text: S

text: u

text: depending

text: continuously

text: on

text: the

text: initial

text: data

text: w.r.t.

text: suitable

text: t

text: norms.

text: In

text: a

text: similar

text: approach

text: was

text: used

text: to

text: construct

text: a

text: semigroup

text: of

text: dissipative

text: solutions.

text: Solutions

text: obtained

text: by

text: this

text: particular

text: transformation

text: of

text: variables

text: are

text: clearly

text: unique.

text: However,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Since the family of measures µ depends continuously on t in the topology of weak conver- (cid:90) ((cid:90)t) → ∈ (cid:90)N (cid:90) gence, taking the limit of (3.16) as (cid:15) 0, for τ, t / we obtain x(τ) − x+(t) t x+(s) − 0 = −∞ u2x(τ,(cid:90)x) d(cid:90)x −∞ u2x(t, x) dx + −∞ [2u2ux 2uxP ] dxds τ t x+(s)+(cid:15) + lim u2(ϕ(cid:15) + uϕ(cid:15) ) dxds (3.18) (cid:90) (cid:15)→0 τ x+(s(cid:90))−(cid:15) x t x (cid:90) (cid:90) ≥ x(τ) − x+(t) t x+(s) − u2(τ, x) dx u2(t, x) dx + [2u2u 2u P ] dxds −∞ x −∞ x −∞ x x τ (cid:90) (cid:90) (cid:90) (cid:90) In turn, (3.18) implies x+(t) ≥ x(τ) t x+(s) − −∞(cid:90) u2x(t, x) dx −(cid:90)∞ (cid:90)u2x(τ, x) dx + −∞ [2u2ux 2uxP ] dxds τ x+(t) t x(s) − − = u2(t, x) dx + [2u2u 2u P ] dxds + o (t τ). −∞ x −∞ x x 1 τ − → → Notice that the last (cid:12)term is a higher order inﬁnite(cid:12)simal, satisfying o1(−t τ) 0 as t τ. (cid:12)(cid:90) (cid:90) (cid:12) (cid:90) (cid:90) t τ Indeed (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) | − | t x(s) − ≤ t x(s) | − | | | o (t τ) = [2u2y 2P u ] dxds 2u2 2P u dxds 1 (cid:90) (cid:90) x x x τ x+(s) τ x+(s) ≤ (cid:107) − (cid:107) · t x(s) | | 2 u2 P L∞ (cid:90) ux dxds τ x+(s) ≤ (cid:107) − (cid:107) t − (cid:107) · (cid:107) ≤ · − 2 u2 P ∞ (x(s) x+(s))1/2 u (s, ) ds C (t τ)3/2. L x L2 τ (cid:34) (cid:35) (cid:90) On the other hand, by (3.8) and (3.10) a linear approximation yields − 2 − x(τ) − β(t) = β(τ) + (t τ) u(τ, x(τ)) + u3(τ, x(τ)) 2u P dx + o (t τ) , (3.19) 3 −∞ x 2 − → → with o2(−t τ) 0 as t τ. t τ 10 ##LTLine## ##LTLine## ##LTLine##

text: Since

text: the

text: family

text: of

text: measures

text: µ

text: depends

text: continuously

text: on

text: t

text: in

text: the

text: topology

text: of

text: weak

text: conver-

text: gence,

text: taking

text: the

text: limit

text: of

text: as

text: for

text: τ,

text: t

text: we

text: obtain

text: x(τ)

text: x+(t)

text: t

text: x+(s)

text: x)

text: dx

text: dxds

text: τ

text: t

text: lim

text: dxds

text: τ

text: x

text: t

text: x

text: x(τ)

text: x+(t)

text: t

text: x+(s)

text: x)

text: dx

text: x)

text: dx

text: P

text: dxds

text: x

text: x

text: x

text: x

text: τ

text: In

text: turn,

text: implies

text: x+(t)

text: x(τ)

text: t

text: x+(s)

text: x)

text: dx

text: x)

text: dx

text: dxds

text: τ

text: x+(t)

text: t

text: x(s)

text: x)

text: dx

text: P

text: dxds

text: o

text: (t

text: τ).

text: x

text: x

text: x

text: τ

text: Notice

text: that

text: the

text: last

text: is

text: a

text: higher

text: order

text: satisfying

text: τ)

text: as

text: t

text: τ.

text: t

text: τ

text: Indeed

text: t

text: x(s)

text: t

text: x(s)

text: o

text: (t

text: τ)

text: u

text: dxds

text: u

text: dxds

text: x

text: x

text: x

text: τ

text: x+(s)

text: τ

text: x+(s)

text: t

text: x(s)

text: P

text: L∞

text: ux

text: dxds

text: τ

text: x+(s)

text: t

text: P

text: (x(s)

text: u

text: (s,

text: ds

text: C

text: (t

text: L

text: x

text: τ

text: On

text: the

text: other

text: hand,

text: by

text: and

text: a

text: linear

text: approximation

text: yields

text: x(τ)

text: β(t)

text: β(τ)

text: (t

text: τ)

text: u(τ,

text: x(τ))

text: x(τ))

text: P

text: dx

text: o

text: (t

text: τ)

text: x

text: with

text: τ)

text: as

text: t

text: τ.

text: t

text: τ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ Current-voltage characteristics of asymmetric double-barrier Josephson junctions E. V. Bezuglyi,1, † E. N. Bratus’,1 and V. S. Shumeiko1 We develop a theory for the current-voltage characteristics of diffusive superconductor-normal metal-super- conductor Josephson junctions with resistive interfaces and the distance between the electrodes smaller than the superconducting coherence length. The theory allows for a quantitative analytical and numerical analysis in the whole range of the interface transparencies and asymmetry. We focus on the regime of large interface resistance compared to the resistance of the normal region, when the electron-hole dephasing in the normal region is signiﬁcant and the ﬁnite length of the junction plays a role. In the limit of strong asymmetry we ﬁnd pronounced current structures at the combination subharmonics of ∆+∆g, where ∆g is the proximity minigap in the normal region, in addition to the subharmonics of the energy gap 2∆ in the electrodes. In the limit of rather transparent interfaces, our theory recovers a known formula for the current in a short mesoscopic connector – a convolution of the current through a single-channel point contact with the transparency distribution for an asymmetric double-barrier potential. PACS numbers: 74.50.+r, 74.45.+c I. INTRODUCTION where the coherent MAR problem has been analytically solved for a ﬁnite-length SNS junction with highly resis- During last few years a large number of experimental re- tive interfaces (I), SINIS. In this model, the length of the searches has been done on the proximity effect in semiconduc- junction is assumed to be small, but it cannot be put equal tor nanowires connected to superconducting electrodes.1–11 to zero because of signiﬁcant dwell time of quasiparticles in Hybrid devices of the nanowires have demonstrated Andreev the normal region conﬁned by the strong interface barriers. subgap conductance, Josephson ﬁeld effect, and Cooper-pair As it was shown, the parameter that deﬁnes the short junc- beam splitting. More recently, the nanowire hybrid devices at- (cid:28) tracted new attention following theoretical predictions of Ma- tion lim(cid:28)it in this case is γ = (R/RN)(∆/ETh) 1, rather than jorana bound states in nanowire proximity structures. ∆/ETh 1, where R is the interface resistance and RN is the resistance of the normal region. Therefore even if the latter From a theory viewpoint, the majority of investigated de- (cid:29) condition is fulﬁlled, the parameter γ can be large, γ 1. vices fall into the category of mesoscopic diffusive supercon- This is the most interesting regime, the physics of which is ductor-normal metal-superconductor (SNS) junctions with the characterized, qualitatively similar to the long junction case, length smaller or comparable to the superconducting coher- by large electron-hole dephasing in the normal region, leading ence length. These devices typically have about 100 conduct- ∼ − to signiﬁcant length dependence of the transport. ing channels, impurity mean free path 30 50 nm, and the length varying from tens to hundreds nanometers, i.e. the Solution of this problem is also important for understand- − Thouless energy ETh in the range of 10 0.1 meV. This is ing the properties of the coherent current transport in pla- larger or comparable to the energy gap ∆ in superconducting nar Nb/Al O /Al/Al O /Nb tunnel junctions which can be 2 3 2 3 Al, which is used as the electrode material. The most interest- used as basic elements of practical superconducting electron- ing regime of a strong proximity effect, manifested by consid- ics: rapid single ﬂux quantum devices,19 voltage standards,20 erable Josephson current, is achieved in junctions with rather high-frequency mixers,21 SQUIDs22 (see also a review in transparent nanowire-superconductor interfaces, whose trans- Ref. 23). As a rule, in such devices, the thickness of the Al parencies typically exceed 0.1. layer is about 10 nm, and the parameter γ may achieve the − The physics of the equilibrium proximity effect in such values of the order of 102 104. junctions is qualitatively well understood, and a quantitative In this paper we discuss the extension of theory developed theory for the dc Josephson transport has been developed by in Ref.18 to a practically important case of asymmetric junc- many authors on the basis of the quasiclassical Greens func- tions, namely junctions with different interface resistances. As tion technique, see, e.g., Ref. 12 and references therein. we show, the asymmetry leads to a qualitative change of the Quantitative description of the ac Josephson effect is more IVC characteristics. In these junctions a novel set of current challenging. The difﬁculty here arises from the presence of the time dependence of the dynamics in the normal region, in features appears at subharmonics of ∆ + ∆g, where ∆g is the proximity induced minigap in the normal region. addition to the spatial inhomogeneity and nonlinearity. The problem was found solvable in diffusive point contacts,13,14 The structure of the paper is as follows. A formal solution where approximation of the zero contact length is appropriate. for the Keldysh-Green’s function equation is presented in Sec- In that case, the problem can be reduced to the single channel tion II. Section III is devoted to demonstration of computation coherent multiple Andreev reﬂection (MAR) problem15–17. of equilibrium Josephson current using the found solution. In Such an approximation is not suitable for the interpretation Section IV, a general non-equilibrium case is considered, and of the nanowire experiments, which show pronounced length the dc current-voltage characteristics are computed in Section dependence of the transport characteristics. V; there we present the numerical results and analytical ex- A step towards solving this problem was made in Ref.18 pressions for the partial MAR currents and the excess current.

text: Current-voltage

text: characteristics

text: of

text: asymmetric

text: double-barrier

text: Josephson

text: junctions

text: E.

text: V.

text: E.

text: N.

text: and

text: V.

text: S.

text: We

text: develop

text: a

text: theory

text: for

text: the

text: current-voltage

text: characteristics

text: of

text: diffusive

text: superconductor-normal

text: metal-super-

text: conductor

text: Josephson

text: junctions

text: with

text: resistive

text: interfaces

text: and

text: the

text: distance

text: between

text: the

text: electrodes

text: smaller

text: than

text: the

text: superconducting

text: coherence

text: length.

text: The

text: theory

text: allows

text: for

text: a

text: quantitative

text: analytical

text: and

text: numerical

text: analysis

text: in

text: the

text: whole

text: range

text: of

text: the

text: interface

text: transparencies

text: and

text: asymmetry.

text: We

text: focus

text: on

text: the

text: regime

text: of

text: large

text: interface

text: resistance

text: compared

text: to

text: the

text: resistance

text: of

text: the

text: normal

text: region,

text: when

text: the

text: electron-hole

text: dephasing

text: in

text: the

text: normal

text: region

text: is

text: signiﬁcant

text: and

text: the

text: ﬁnite

text: length

text: of

text: the

text: junction

text: plays

text: a

text: role.

text: In

text: the

text: limit

text: of

text: strong

text: asymmetry

text: we

text: ﬁnd

text: pronounced

text: current

text: structures

text: at

text: the

text: combination

text: subharmonics

text: of

text: ∆+∆g,

text: where

text: ∆g

text: is

text: the

text: proximity

text: minigap

text: in

text: the

text: normal

text: region,

text: in

text: addition

text: to

text: the

text: subharmonics

text: of

text: the

text: energy

text: gap

text: 2∆

text: in

text: the

text: electrodes.

text: In

text: the

text: limit

text: of

text: rather

text: transparent

text: interfaces,

text: our

text: theory

text: recovers

text: a

text: known

text: formula

text: for

text: the

text: current

text: in

text: a

text: short

text: mesoscopic

text: connector

text: a

text: convolution

text: of

text: the

text: current

text: through

text: a

text: single-channel

text: point

text: contact

text: with

text: the

text: transparency

text: distribution

text: for

text: an

text: asymmetric

text: double-barrier

text: potential.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: where

text: the

text: coherent

text: MAR

text: problem

text: has

text: been

text: analytically

text: solved

text: for

text: a

text: ﬁnite-length

text: SNS

text: junction

text: with

text: highly

text: resis-

text: During

text: last

text: few

text: years

text: a

text: large

text: number

text: of

text: experimental

text: re-

text: tive

text: interfaces

text: (I),

text: SINIS.

text: In

text: this

text: model,

text: the

text: length

text: of

text: the

text: searches

text: has

text: been

text: done

text: on

text: the

text: proximity

text: effect

text: in

text: semiconduc-

text: junction

text: is

text: assumed

text: to

text: be

text: small,

text: but

text: it

text: cannot

text: be

text: put

text: equal

text: tor

text: nanowires

text: connected

text: to

text: superconducting

text: to

text: zero

text: because

text: of

text: signiﬁcant

text: dwell

text: time

text: of

text: quasiparticles

text: in

text: Hybrid

text: devices

text: of

text: the

text: nanowires

text: have

text: demonstrated

text: Andreev

text: the

text: normal

text: region

text: conﬁned

text: by

text: the

text: strong

text: interface

text: barriers.

text: subgap

text: conductance,

text: Josephson

text: ﬁeld

text: effect,

text: and

text: Cooper-pair

text: As

text: it

text: was

text: shown,

text: the

text: parameter

text: that

text: deﬁnes

text: the

text: short

text: junc-

text: beam

text: splitting.

text: More

text: recently,

text: the

text: nanowire

text: hybrid

text: devices

text: at-

text: tracted

text: new

text: attention

text: following

text: theoretical

text: predictions

text: of

text: Ma-

text: tion

text: in

text: this

text: case

text: is

text: γ

text: (R/RN)(∆/ETh)

text: rather

text: than

text: jorana

text: bound

text: states

text: in

text: nanowire

text: proximity

text: structures.

text: ∆/ETh

text: where

text: R

text: is

text: the

text: interface

text: resistance

text: and

text: RN

text: is

text: the

text: resistance

text: of

text: the

text: normal

text: region.

text: Therefore

text: even

text: if

text: the

text: latter

text: From

text: a

text: theory

text: viewpoint,

text: the

text: majority

text: of

text: investigated

text: de-

text: condition

text: is

text: fulﬁlled,

text: the

text: parameter

text: γ

text: can

text: be

text: large,

text: γ

text: vices

text: fall

text: into

text: the

text: category

text: of

text: mesoscopic

text: diffusive

text: supercon-

text: This

text: is

text: the

text: most

text: interesting

text: regime,

text: the

text: physics

text: of

text: which

text: is

text: ductor-normal

text: metal-superconductor

text: (SNS)

text: junctions

text: with

text: the

text: characterized,

text: qualitatively

text: similar

text: to

text: the

text: long

text: junction

text: case,

text: length

text: smaller

text: or

text: comparable

text: to

text: the

text: superconducting

text: coher-

text: by

text: large

text: electron-hole

text: dephasing

text: in

text: the

text: normal

text: region,

text: leading

text: ence

text: length.

text: These

text: devices

text: typically

text: have

text: about

text: conduct-

text: to

text: signiﬁcant

text: length

text: dependence

text: of

text: the

text: transport.

text: ing

text: channels,

text: impurity

text: mean

text: free

text: path

text: nm,

text: and

text: the

text: length

text: varying

text: from

text: tens

text: to

text: hundreds

text: nanometers,

text: i.e.

text: the

text: Solution

text: of

text: this

text: problem

text: is

text: also

text: important

text: for

text: understand-

text: Thouless

text: energy

text: ETh

text: in

text: the

text: range

text: of

text: meV.

text: This

text: is

text: ing

text: the

text: properties

text: of

text: the

text: coherent

text: current

text: transport

text: in

text: pla-

text: larger

text: or

text: comparable

text: to

text: the

text: energy

text: gap

text: in

text: superconducting

text: nar

text: Nb/Al

text: O

text: O

text: /Nb

text: tunnel

text: junctions

text: which

text: can

text: be

text: Al,

text: which

text: is

text: used

text: as

text: the

text: electrode

text: material.

text: The

text: most

text: interest-

text: used

text: as

text: basic

text: elements

text: of

text: practical

text: superconducting

text: electron-

text: ing

text: regime

text: of

text: a

text: strong

text: proximity

text: effect,

text: manifested

text: by

text: consid-

text: ics:

text: rapid

text: single

text: ﬂux

text: quantum

text: voltage

text: erable

text: Josephson

text: current,

text: is

text: achieved

text: in

text: junctions

text: with

text: rather

text: high-frequency

text: (see

text: also

text: a

text: review

text: in

text: transparent

text: nanowire-superconductor

text: interfaces,

text: whose

text: trans-

text: Ref.

text: As

text: a

text: rule,

text: in

text: such

text: devices,

text: the

text: thickness

text: of

text: the

text: Al

text: parencies

text: typically

text: exceed

text: layer

text: is

text: about

text: nm,

text: and

text: the

text: parameter

text: γ

text: may

text: achieve

text: the

text: The

text: physics

text: of

text: the

text: equilibrium

text: proximity

text: effect

text: in

text: such

text: values

text: of

text: the

text: order

text: of

text: junctions

text: is

text: qualitatively

text: well

text: understood,

text: and

text: a

text: quantitative

text: In

text: this

text: paper

text: we

text: discuss

text: the

text: extension

text: of

text: theory

text: developed

text: theory

text: for

text: the

text: dc

text: Josephson

text: transport

text: has

text: been

text: developed

text: by

text: in

text: to

text: a

text: practically

text: important

text: case

text: of

text: asymmetric

text: junc-

text: many

text: authors

text: on

text: the

text: basis

text: of

text: the

text: quasiclassical

text: Greens

text: func-

text: tions,

text: namely

text: junctions

text: with

text: different

text: interface

text: resistances.

text: As

text: tion

text: technique,

text: see,

text: e.g.,

text: Ref.

text: and

text: references

text: therein.

text: we

text: show,

text: the

text: asymmetry

text: leads

text: to

text: a

text: qualitative

text: change

text: of

text: the

text: Quantitative

text: description

text: of

text: the

text: ac

text: Josephson

text: effect

text: is

text: more

text: IVC

text: characteristics.

text: In

text: these

text: junctions

text: a

text: novel

text: set

text: of

text: current

text: challenging.

text: The

text: difﬁculty

text: here

text: arises

text: from

text: the

text: presence

text: of

text: the

text: time

text: dependence

text: of

text: the

text: dynamics

text: in

text: the

text: normal

text: region,

text: in

text: features

text: appears

text: at

text: subharmonics

text: of

text: ∆g,

text: where

text: ∆g

text: is

text: the

text: proximity

text: induced

text: minigap

text: in

text: the

text: normal

text: region.

text: addition

text: to

text: the

text: spatial

text: inhomogeneity

text: and

text: nonlinearity.

text: The

text: problem

text: was

text: found

text: solvable

text: in

text: diffusive

text: point

text: The

text: structure

text: of

text: the

text: paper

text: is

text: as

text: follows.

text: A

text: formal

text: solution

text: where

text: approximation

text: of

text: the

text: zero

text: contact

text: length

text: is

text: appropriate.

text: for

text: the

text: Keldysh-Green’s

text: function

text: equation

text: is

text: presented

text: in

text: Sec-

text: In

text: that

text: case,

text: the

text: problem

text: can

text: be

text: reduced

text: to

text: the

text: single

text: channel

text: tion

text: II.

text: Section

text: III

text: is

text: devoted

text: to

text: demonstration

text: of

text: computation

text: coherent

text: multiple

text: Andreev

text: reﬂection

text: (MAR)

text: of

text: equilibrium

text: Josephson

text: current

text: using

text: the

text: found

text: solution.

text: In

text: Such

text: an

text: approximation

text: is

text: not

text: suitable

text: for

text: the

text: interpretation

text: Section

text: IV,

text: a

text: general

text: non-equilibrium

text: case

text: is

text: considered,

text: and

text: of

text: the

text: nanowire

text: experiments,

text: which

text: show

text: pronounced

text: length

text: the

text: dc

text: current-voltage

text: characteristics

text: are

text: computed

text: in

text: Section

text: dependence

text: of

text: the

text: transport

text: characteristics.

text: V;

text: there

text: we

text: present

text: the

text: numerical

text: results

text: and

text: analytical

text: ex-

text: A

text: step

text: towards

text: solving

text: this

text: problem

text: was

text: made

text: in

text: pressions

text: for

text: the

text: partial

text: MAR

text: currents

text: and

text: the

text: excess

text: current.


text: 3 (cid:46)(cid:113) (cid:101) (cid:101) Equations (15) and (16) are the main technical result of the paper; they describe short asymmetric double-barrier SNS g(cid:101)ˆ = A(cid:113)ˆg Aˆ2g = σzu+iv(cid:101)exp((cid:113)iσzΦ(cid:101))σy, (21) junctions for all values of parameter γ. In what follows we will (cid:101) (cid:101) apply these equations for calculation of the dc current-voltage u = E , v = ∆(E,φ)(cid:16) , (cid:17) (22) characteristics. (cid:101) E2 −∆2(E,φ) E2 −∆2(E,φ) The chosen form for the solution is justiﬁed by the limit ∆η(φ) φ of vanishing dephasing parameter, γ = 0, when Eq. (16) re- ∆(E,φ) = − , Φ(φ) = arctan κ tan . (23) duces to a known universal formula for the current through a 1 iγ/v(E) 2 short connector.30,31 Indeed, in this case, reducing the integral in E−q. (16) to the positive axis, we have Kˇ(λ) = 2Gˇ+(Gˇ2+ + According to Eq. (22), the minigap ∆g(φ) in the spectrum λ2) 1, and then, after simple algebra, we obtain the commu- of the normal region is the solution of equation tator in Eq.(16) in terms of the functions Gˇ : (cid:101) 1,2 (cid:2) (cid:3) (cid:2) (cid:3) (cid:0) (cid:8) (cid:9)(cid:1) ∆g = ∆(∆g,φ). (24) Kˇ(λ),Gˇ− = 1 r1r2 Gˇ2,Gˇ1 . (17) (cid:29) ≈ 2 λ2 + 14 r12 +r22 +r1r2 Gˇ1,Gˇ2 As follows from Eq.(23), at γ 1 and φ = 0, ∆g ∆/(1+ γ). In strongly asymmetric junctions with essentially different (cid:29) Substituting Eq.(17) to Eq.(16), using the equality r1+r2 = resistances≈of the barriers, Rmax Rmin, the transparency pa- 2, and introducing the transparency variable D = r1r2/(λ2 + rameter γ 2γmin = 2(Rmin/RN)(∆/ETh)→is determined b→y the 1), we arrive at a convolution of a non-resonant single-channel smallest barrier strength. In this case, κ 1 and η(φ) 1, current with the transparency distribution ρ(D) for an asym- therefore the minigap weakly depends on the phase difference metric double-barrier junction,32,33 and approaches its value at φ = 0, while in the symmetric case | | (cid:90) (cid:2) (cid:3) the minigap oscillates with the phase as ∆g(0) cos(φ/2) . The (cid:0)(cid:8) (cid:9) (cid:1) physical explanation is as follows. In the main approxima- I(t) = 4eπRT 0Dmax TrτK 1D+ρD4(D)GˇGˇ12,G,ˇGˇ21 −dD2 (t,t), (18) twtihoaenll,i,mtthhaeegresetfmroorenetghtehoredb,sapisrercisetirmrucmilaanrofbtotehtechoNenosriendgeeirofeondr, caaasnlciemuflfpaeetcentdievtuersaibSnlIge- ρ(D) = πD3/2√1Dmax −D, Dmax = r1r2 = 4RR12TR2 . (19) NaγnIdSNijnuhntahcvteiionensgtiwdmoitauhtbelIeordfelf∆eenr)rg.itnShgi(nwtcoehtibhcoehtmhisoSmreealnetricfatenrsostpdeaedrsbeinyntdsbouaucrbhrileaernd, min g effective junction originate from the single S electrode, the ef- where RT = R1 +R2 is the net resistance of the tunnel bar- fective phase difference is zero within this approximation. riers. Expression for the current follows from Eq.(12) in energy representation, III. EQUILIBRIUM JOSEPHSON CURRENT AND THE (cid:90) MINIGAP FUNCTION ∞ dE it PisrioinrsttoructhtieveditsocudsseimononosftraategehnoewraltonounseequEilqibsr.iu(1m5)caasned, I = −∞ 16eR Tr Gˇ [Gˇ−,τK](E). (25) (16) for evaluation of the equilibrium Josephson current. − In this case, the distribution function is equilibrium, f = Using Eqs. (21)-(23) and GK = (gR gA)tanh(E/2T), we f2 = f1 = tanh(E/2T), and we need to calculate only the get (cid:90) Green’s func±tions. In the reservoirs, they are given by gˆ1,2 = (cid:101) σzu + iexp( iσzφ/2)σyv; the solution for the Green’s func- tdiioangogˆnianl t(hreetanrodremdaal nredgaiodnvahnacsetdh)ecfoomrmpoonfenEtq.Aˆ(g15o)fwthitehftuhlel I = 4eRisTinηφ(φ) −∞∞ dE vRvR tanh 2ET −(R → A), matrix Aˇ: or in the Matsubara representation, (cid:112) (cid:112) − Aˆg = σz(u iτE)+iv[σy cos(φ/2)+σxκ sin(φ/2)], (20) 2πT (cid:113)∑ 1 ∆2 sinφ Aˆ2g = (u−iτE)2 −v2η2, η2(φ) = cos2 φ2 +κ2 sin2 φ2 , I = eRT ωn>0 ωn2 +∆2 ωn2q2n +∆2η2(φ), (26) qn = 1+τ ωn2 +∆2, iωn = iπT(2n+1). − where κ = (R2 R1)/(R2 +R1). As the result, we obtain ##LTLine## ##LTLine## ##LTLine## ##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: Equations

text: and

text: are

text: the

text: main

text: technical

text: result

text: of

text: the

text: paper;

text: they

text: describe

text: short

text: asymmetric

text: double-barrier

text: SNS

text: junctions

text: for

text: all

text: values

text: of

text: parameter

text: γ.

text: In

text: what

text: follows

text: we

text: will

text: apply

text: these

text: equations

text: for

text: calculation

text: of

text: the

text: dc

text: current-voltage

text: u

text: E

text: v

text: characteristics.

text: The

text: chosen

text: form

text: for

text: the

text: solution

text: is

text: justiﬁed

text: by

text: the

text: limit

text: ∆η(φ)

text: φ

text: of

text: vanishing

text: dephasing

text: parameter,

text: γ

text: when

text: Eq.

text: re-

text: ∆(E,φ)

text: Φ(φ)

text: arctan

text: κ

text: tan

text: duces

text: to

text: a

text: known

text: universal

text: formula

text: for

text: the

text: current

text: through

text: a

text: iγ/v(E)

text: short

text: Indeed,

text: in

text: this

text: case,

text: reducing

text: the

text: integral

text: in

text: E−q.

text: to

text: the

text: positive

text: axis,

text: we

text: have

text: Kˇ(λ)

text: According

text: to

text: Eq.

text: the

text: minigap

text: ∆g(φ)

text: in

text: the

text: spectrum

text: and

text: then,

text: after

text: simple

text: algebra,

text: we

text: obtain

text: the

text: commu-

text: of

text: the

text: normal

text: region

text: is

text: the

text: solution

text: of

text: equation

text: tator

text: in

text: in

text: terms

text: of

text: the

text: functions

text: Gˇ

text: ∆g

text: ∆(∆g,φ).

text: Kˇ(λ),Gˇ−

text: λ2

text: As

text: follows

text: from

text: at

text: γ

text: and

text: φ

text: ∆g

text: γ).

text: In

text: strongly

text: asymmetric

text: junctions

text: with

text: essentially

text: different

text: Substituting

text: to

text: using

text: the

text: equality

text: resistances≈of

text: the

text: barriers,

text: Rmax

text: Rmin,

text: the

text: transparency

text: pa-

text: and

text: introducing

text: the

text: transparency

text: variable

text: D

text: rameter

text: γ

text: determined

text: b→y

text: the

text: we

text: arrive

text: at

text: a

text: convolution

text: of

text: a

text: non-resonant

text: single-channel

text: smallest

text: barrier

text: strength.

text: In

text: this

text: case,

text: κ

text: and

text: η(φ)

text: current

text: with

text: the

text: transparency

text: distribution

text: ρ(D)

text: for

text: an

text: asym-

text: therefore

text: the

text: minigap

text: weakly

text: depends

text: on

text: the

text: phase

text: difference

text: metric

text: double-barrier

text: and

text: approaches

text: its

text: value

text: at

text: φ

text: while

text: in

text: the

text: symmetric

text: case

text: the

text: minigap

text: oscillates

text: with

text: the

text: phase

text: as

text: The

text: physical

text: explanation

text: is

text: as

text: follows.

text: In

text: the

text: main

text: approxima-

text: I(t)

text: TrτK

text: (t,t),

text: ρ(D)

text: −D,

text: Dmax

text: min

text: g

text: effective

text: junction

text: originate

text: from

text: the

text: single

text: S

text: electrode,

text: the

text: ef-

text: where

text: RT

text: is

text: the

text: net

text: resistance

text: of

text: the

text: tunnel

text: bar-

text: fective

text: phase

text: difference

text: is

text: zero

text: within

text: this

text: approximation.

text: riers.

text: Expression

text: for

text: the

text: current

text: follows

text: from

text: in

text: energy

text: representation,

text: III.

text: EQUILIBRIUM

text: JOSEPHSON

text: CURRENT

text: AND

text: THE

text: MINIGAP

text: FUNCTION

text: dE

text: it

text: I

text: Tr

text: Gˇ

text: [Gˇ−,τK](E).

text: for

text: evaluation

text: of

text: the

text: equilibrium

text: Josephson

text: current.

text: In

text: this

text: case,

text: the

text: distribution

text: function

text: is

text: equilibrium,

text: f

text: Using

text: Eqs.

text: and

text: GK

text: (gR

text: we

text: and

text: we

text: need

text: to

text: calculate

text: only

text: the

text: get

text: Green’s

text: func±tions.

text: In

text: the

text: reservoirs,

text: they

text: are

text: given

text: by

text: σzu

text: iexp(

text: the

text: solution

text: for

text: the

text: Green’s

text: func-

text: tdiioangogˆnianl

text: t(hreetanrodremdaal

text: I

text: dE

text: vRvR

text: tanh

text: −(R

text: A),

text: matrix

text: Aˇ:

text: or

text: in

text: the

text: Matsubara

text: representation,

text: Aˆg

text: σz(u

text: iτE)+iv[σy

text: ∆2

text: sinφ

text: φ2

text: φ2

text: I

text: eRT

text: qn

text: iωn

text: where

text: κ

text: As

text: the

text: result,

text: we

text: obtain

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 12 1B ∞ = 1 di Σcj=1Bθ∗jdθj +−ΣNj=c+1 1+wdθjθj dθj . 2Bi,∞ =ΣNn=η1dn di(11++dγi)α i, ΣNn=1dn 1 + di 1 β1 (33) Σc B∗ d (1 +iγ )α + ΣN wθj d (1 + γ )α j=1 θj θj θj − j=c+1 1+dθj θj θj This concludes the proof. (cid:4) 1 β2 (38) E PROOF FOR THEOREM III.4 (cid:4) This concludes the proof. The proof of Theorem III.4 will be similar to that of F PROOF FOR THEOREM III.3 Theorem III.2. Based on Equation (5) and the deﬁnition of (cid:48) { (cid:48) (cid:48) (cid:48) } control vector V, in a network G of N nodes with control Consider an arbitrary con{trol set C = }θ1, θ2, . . . , θc strategy, the expected belief of an arbitrary non-control node with corresponding degree≥s dθ(cid:48)≥, d·θ(cid:48)·,·.≥. . , dθ(cid:48) . Without loss i, where i (cid:54)∈ C, at an arbitrary time step T, where T ∈ Z+, of gene{rality, we set dθ(cid:48) } dθ(cid:48) 1 2 dθ(cid:48) c. For control set will be: Co =≥ θo1, θo2, .≤. . , θoc1 ≤, the2c≤orrespondincg de≤gree≤s satisfy d d if i j, 1 i, j N, so, for 1 i c, we θoi θoj have − ∗ ∗ × 2B =ΣT 1Σc B (A M)t i,t +ΣtTt==−001ΣjNj==1c+1θwj ∗θj(A∗ × Mθj),itθj,i. (34) dθoi ≥ dθi(cid:48). (39) ∗× ∗ The matrix element (A Mt)i,j, with Ai,j = Ai,j/(1+dj) Since w = 0, for i = 1, . . . , N, B∗ = 1, for 1 ≤ j ≤ c, and the deﬁnition of control matrix M, may be written as: i j according to Equation (12), 1cp can be rewritten as: ∗ × (A M)ti,j d − d (cid:48) 2 =ΣNk1=c+1ΣNk2=c+1 . . . ΣNkt−1=c+1 1A+i,dk1k1 1A+k1d,kk22 . . . 1A+kt−d1k,jj . 1cp = ΣNi=1 1 +idi (Σcj=1dθj(cid:48) )/(ΣNk=1dk ΣNm=c+1 1 +θmdθ(m(cid:48)40).) (35) (A∗ × M)ti,j is then calculated from the expected value derSivinactieveΣoNi=f 1co1+nditdriolapnodwΣerNk=1c1pdkwiatrherepsopseitcitvetoctohnestdaengtrse,ethoef of Ai,j, which equates 2Pi,j as shown in Equation (15), and a controlled node d (cid:48) is: Equation (35): θ i (A∗ × M)ti,j ∂1cp = D ΣNk=c+1 1+dθdk(cid:48)θk(cid:48) + (Σ1cj+=d1θdi(cid:48)θ)j(cid:48)2 , (41) = d(cid:32)i(1 + γi)αdj(1 + γj)α(ΣNn=1dn) (cid:33) ∂dθi(cid:48) 1 D22 ΣN (d(θ1k(+1 +dj)γηθk)α)2 ΣN d t−1 . (36) AswDhere>D0,1o=neΣcNja=n1g1+edtjd:j , D2 = ΣNk=1 1+dkdk + ΣNi=c+1 1+dθdi(cid:48)θi(cid:48) . k=c+1 (1 + d )η n=1 n 1 θk ∂1cp > 0. (42) Let ∂d (cid:48) ΣN (dθk(1+γθk)α)2 θi β2 = k=c+1 η 1+dθk ΣNn=1dn. From Equations (39) and (42), one can get: Again inserting Equation (36) into Equation (34), and ≥ (cid:48) ∞ 1cp(C ) 1cp(C ). (43) setting T = yields o (cid:4) This concludes the proof. 2B ∞ i, ∗ G PROOF FOR THEOREM III.5 = Σcj=1 Bθjdi(1 + γi)α(d1θj+(1d+)ηγθj)α(ΣNn=1dn)Σ∞t=0β1t−1 Deﬁne the weight ξi of node i as(cid:48): ξi ={ (cid:48)di(1(cid:48) + γi)(cid:48)α}. i Consider an arbitrary control set C = θ , θ , . . . , θ + Σcj=1 1+wdθjθj di(1 + γi)(α1d+θj(d1i)+η γθj)α(ΣNn=1dn)Σ∞t=0β(317t−) 1. wo2fCithge=cnoe{rrraθel†istpy,o,θn†wdie,n.gs.e.wt, θed†iξg1(cid:48)h}t,≥t{hdedξξ1(cid:48)c2(cid:48),o·dr·rξe2·(cid:48)s,p≥.o.n.d,dξdicn(cid:48)ξ.gc(cid:48) }Fw.o1eWrigciht2ohtnsotursotaltliosscfseyst o≥ o1 o2≤ ≤oc ≤ ≤ ≤ ξ † ξ † if i j, 1 i, j N, so, for 1 i c, we have If β < 1, we can rewrite Σ∞ β t−1 as −1 , and Equation θoi θoj (37) b2ecomes t=0 2 1 β2 ξ † ≥ ξ (cid:48). (44) θoi θi ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: di

text: dθj

text: i,

text: di

text: β1

text: Σc

text: B∗

text: d

text: +iγ

text: )α

text: ΣN

text: wθj

text: d

text: γ

text: )α

text: θj

text: θj

text: θj

text: θj

text: θj

text: This

text: concludes

text: the

text: proof.

text: β2

text: E

text: PROOF

text: FOR

text: THEOREM

text: This

text: concludes

text: the

text: proof.

text: The

text: proof

text: of

text: Theorem

text: will

text: be

text: similar

text: to

text: that

text: of

text: F

text: PROOF

text: FOR

text: THEOREM

text: Theorem

text: Based

text: on

text: Equation

text: and

text: the

text: deﬁnition

text: of

text: control

text: vector

text: V,

text: in

text: a

text: network

text: G

text: of

text: N

text: nodes

text: with

text: control

text: Consider

text: an

text: arbitrary

text: con{trol

text: set

text: C

text: θc

text: strategy,

text: the

text: expected

text: belief

text: of

text: an

text: arbitrary

text: non-control

text: node

text: with

text: corresponding

text: degree≥s

text: Without

text: loss

text: i,

text: where

text: i

text: C,

text: at

text: an

text: arbitrary

text: time

text: step

text: T,

text: where

text: T

text: Z+,

text: of

text: gene{rality,

text: we

text: set

text: c.

text: For

text: control

text: set

text: will

text: be:

text: Co

text: de≤gree≤s

text: satisfy

text: d

text: d

text: if

text: i

text: j,

text: i,

text: j

text: N,

text: so,

text: for

text: i

text: c,

text: we

text: θoi

text: θoj

text: have

text: =ΣT

text: B

text: (A

text: M)t

text: i,t

text: ∗θj(A∗

text: Mθj),itθj,i.

text: dθoi

text: The

text: matrix

text: element

text: (A

text: Mt)i,j,

text: with

text: Ai,j

text: Since

text: w

text: for

text: i

text: N,

text: B∗

text: for

text: j

text: c,

text: and

text: the

text: deﬁnition

text: of

text: control

text: matrix

text: M,

text: may

text: be

text: written

text: as:

text: i

text: j

text: according

text: to

text: Equation

text: can

text: be

text: rewritten

text: as:

text: (A

text: M)ti,j

text: d

text: d

text: +idi

text: (A∗

text: M)ti,j

text: is

text: then

text: calculated

text: from

text: the

text: expected

text: value

text: derSivinactieveΣoNi=f

text: of

text: Ai,j,

text: which

text: equates

text: as

text: shown

text: in

text: Equation

text: and

text: a

text: controlled

text: node

text: d

text: is:

text: Equation

text: θ

text: i

text: (A∗

text: M)ti,j

text: D

text: ΣN

text: ΣN

text: d

text: d

text: )η

text: n

text: θk

text: Let

text: ∂d

text: ΣN

text: θi

text: β2

text: η

text: From

text: Equations

text: and

text: one

text: can

text: get:

text: Again

text: inserting

text: Equation

text: into

text: Equation

text: and

text: setting

text: T

text: yields

text: o

text: This

text: concludes

text: the

text: proof.

text: i,

text: G

text: PROOF

text: FOR

text: THEOREM

text: Deﬁne

text: the

text: weight

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text: 14 Yannis Viniotis received his Ph.D. from the Uni- versity of Maryland, College Park, in 1988 and is currently Professor, Department of Electrical and Computer Engineering at North Carolina State Uni- versity. Dr. Viniotis is the author of over one hundred technical publications, including two engineering textbooks. He has served as the cochair of two in- ternational conferences in computer networking. His research interests include Service-Oriented Architec- tures, service engineering, and design and analysis of stochastic algorithms. Dr. Viniotis was the cofounder of Orologic, a successful startup networking company in Research Triangle Park, NC, that specialized in ASIC implementation of integrated trafﬁc management solutions for high-speed networks. ##LTFigure## ##LTFigure## ##LTFigure##

text: Yannis

text: Viniotis

text: received

text: his

text: Ph.D.

text: from

text: the

text: Uni-

text: versity

text: of

text: Maryland,

text: College

text: Park,

text: in

text: and

text: is

text: currently

text: Professor,

text: Department

text: of

text: Electrical

text: and

text: Computer

text: Engineering

text: at

text: North

text: Carolina

text: State

text: Uni-

text: versity.

text: Dr.

text: Viniotis

text: is

text: the

text: author

text: of

text: over

text: one

text: hundred

text: technical

text: publications,

text: including

text: two

text: engineering

text: textbooks.

text: He

text: has

text: served

text: as

text: the

text: cochair

text: of

text: two

text: in-

text: ternational

text: conferences

text: in

text: computer

text: networking.

text: His

text: research

text: interests

text: include

text: Service-Oriented

text: Architec-

text: tures,

text: service

text: engineering,

text: and

text: design

text: and

text: analysis

text: of

text: stochastic

text: algorithms.

text: Dr.

text: Viniotis

text: was

text: the

text: cofounder

text: of

text: Orologic,

text: a

text: successful

text: startup

text: networking

text: company

text: in

text: Research

text: Triangle

text: Park,

text: NC,

text: that

text: specialized

text: in

text: ASIC

text: implementation

text: of

text: integrated

text: trafﬁc

text: management

text: solutions

text: for

text: high-speed

text: networks.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##