text: 2 C sym can be expressed as a linear combination of basis V which yields average projected cubic elastic constants tensors i as5 (cid:88) of N C V sym = a , (5) Ctric + Ctric + Ctric i i C¯1c1ub = 11 232 33 = 439.0 GPa, i=1 Ctric + Ctric + Ctric C¯cub = 12 13 23 = 160.3 GPa, where ai are constant coeﬃcients and N is the size of 12 3 the basis. For the elasticity of cubic materials, there are Ctric + Ctric + Ctric 3 independent elastic constants and therefore N = 3. C¯cub = 44 55 66 = 187.7 GPa, (4) C 44 3 Now, as I have discussed, a rotation of does not aﬀect its properties, only its representation, and thus I (cid:107)C −C (cid:107) can deﬁne the general rotation operator and Euclidean distance tric cub = 91.0 GPa. The problem with this approach is that it does not take into ≡ account the rotational degrees of freedom. The cubic R(θ , θ , θ ) R (θ )R (θ )R (θ ). (6) x y z z z y y x x elastic tensor takes the form indicated in Eq. (2) only if the crystallographic axes coincide with the Cartesian axes in which the representation of the tensor is carried In matrix form, each individual rotation takes the fol- out. Because the material properties are not aﬀected lowing form:   by rotations, one can further reduce the Euclidean dis- tance between Ctric and Ccub by rotating either one of   them so as to maximize the projection. In fact, Moakher 1 0 0 −   and Norris’s projector scheme implemented as is would Rx = 0 cos θx sin θx , lead to the inconsistency that a perfectly cubic elastic  0 sin θx cos θx  tensor, rotated with respect to the reference frame in cos θ 0 sin θ y y which the projector is obtained, would diﬀer from its   R = 0 1 0 , own cubic projection in that reference frame. The is- y − sin θ 0 cos θ  y y  sue of system orientation was already highlighted in their − original work.5−If the t−riclinic stiﬀness tensor of Eq. (2) cos θz sin θz 0 is rotated by 1.89, 1.83 and +6−.37 degrees (where Rz = sin θz cos θz 0 . (7) “+” means counterclockwise and “ ” clockwise) with 0 0 1 respect to the ﬁrst, second and third Cartesian axes re- C spectively, in that order, then the projection of tric onto The diﬀerent θ give counterclockwise rotation angles its closest cubic elastic tensor yields C¯1c1ub = 436.8 GPa, around the Cartesiian axes. Note that the order in which Ct¯a1cn2utbly=, th16e1E.4uGclPidaeaannddiCs¯t4ca4unbce=b1e8tw8.e7eGnPC˜at.ricMaonrde Cimcupboris- ttheremdiinﬀienrgenRt.roTtahteiroenfsoraerethceargreienderoaultfoisrmimpofortteannstorinCdeis- further reduced to 83.7 GPa, where the tilde denotes the given by rotation performed. Note that in this case the diﬀerence between the two diﬀerent approaches is small because Ctric was already “almost” cubic (the rotation angles to C˜(θ , θ , θ ) ≡ R(θ , θ , θ )C. (8) x y z x y z correct the structure are correspondingly small). In a more general case, the input tensor might be in a form that does not closely resemble the symmetry of interest, In the particular case of the stiﬀness tensor (rank-4), leading to even larger discrepancies. Based on similar the matrix elements(cid:88)of C˜ are given by considerations, Diner et al. have recently presented a rotation-based method to identify the symmetry class of a tensor which is not expressed in its natural coordinate C˜ijkl = RimRjnRkoRlpCmnop, (9) system.6 m,n,o,p Since, as previously mentioned, rotations do not af- fect the properties of materials, only their mathematical where all the indices run from 1 to 3. I could have representation, the procedure introduced here allows to equivalently made the rotation operator act on the dif- V obtain a better (“closer”) higher-symmetry projection of ferent basis components , however this would lead to i the original tensor. In the following I deal with the de- complicated angle-dependent projectors. It seems best tails of the present approach in the context of Moakher and simplest to keep Moakher and Norris’s original for- and Norris’s method, whose details are given in Ref. 5. mulation for the projectors and introduce the angular C C C Let and sym be material tensors of arbitrary sym- dependence on instead. The projector (see Ref. 5) for metry and speciﬁc symmetry “sym”, respectively. Then a given symmetry sym links C˜ and Csym as ##LTLine## ##LTLine## ##LTLine##

text: C

text: sym

text: can

text: be

text: expressed

text: as

text: a

text: linear

text: combination

text: of

text: basis

text: V

text: which

text: yields

text: average

text: projected

text: cubic

text: elastic

text: constants

text: tensors

text: i

text: of

text: N

text: C

text: V

text: sym

text: a

text: Ctric

text: Ctric

text: Ctric

text: i

text: i

text: GPa,

text: Ctric

text: Ctric

text: Ctric

text: C¯cub

text: GPa,

text: where

text: ai

text: are

text: constant

text: coeﬃcients

text: and

text: N

text: is

text: the

text: size

text: of

text: the

text: basis.

text: For

text: the

text: elasticity

text: of

text: cubic

text: materials,

text: there

text: are

text: Ctric

text: Ctric

text: Ctric

text: independent

text: elastic

text: constants

text: and

text: therefore

text: N

text: C¯cub

text: GPa,

text: C

text: Now,

text: as

text: I

text: have

text: discussed,

text: a

text: rotation

text: of

text: does

text: not

text: aﬀect

text: its

text: properties,

text: only

text: its

text: representation,

text: and

text: thus

text: I

text: −C

text: can

text: deﬁne

text: the

text: general

text: rotation

text: operator

text: and

text: Euclidean

text: distance

text: tric

text: cub

text: GPa.

text: The

text: problem

text: with

text: this

text: approach

text: is

text: that

text: it

text: does

text: not

text: take

text: into

text: account

text: the

text: rotational

text: degrees

text: of

text: freedom.

text: The

text: cubic

text: R(θ

text: θ

text: θ

text: R

text: (θ

text: )R

text: (θ

text: )R

text: (θ

text: x

text: y

text: z

text: z

text: z

text: y

text: y

text: x

text: x

text: elastic

text: tensor

text: takes

text: the

text: form

text: indicated

text: in

text: Eq.

text: only

text: if

text: the

text: crystallographic

text: axes

text: coincide

text: with

text: the

text: Cartesian

text: axes

text: in

text: which

text: the

text: representation

text: of

text: the

text: tensor

text: is

text: carried

text: In

text: matrix

text: form,

text: each

text: individual

text: rotation

text: takes

text: the

text: fol-

text: out.

text: Because

text: the

text: material

text: properties

text: are

text: not

text: aﬀected

text: lowing

text: form:

text: by

text: rotations,

text: one

text: can

text: further

text: reduce

text: the

text: Euclidean

text: dis-

text: tance

text: between

text: Ctric

text: and

text: Ccub

text: by

text: rotating

text: either

text: one

text: of

text: them

text: so

text: as

text: to

text: maximize

text: the

text: projection.

text: In

text: fact,

text: Moakher

text: and

text: Norris’s

text: projector

text: scheme

text: implemented

text: as

text: is

text: would

text: Rx

text: cos

text: θx

text: sin

text: θx

text: lead

text: to

text: the

text: inconsistency

text: that

text: a

text: perfectly

text: cubic

text: elastic

text: sin

text: θx

text: cos

text: θx

text: tensor,

text: rotated

text: with

text: respect

text: to

text: the

text: reference

text: frame

text: in

text: cos

text: θ

text: sin

text: θ

text: y

text: y

text: which

text: the

text: projector

text: is

text: obtained,

text: would

text: diﬀer

text: from

text: its

text: R

text: own

text: cubic

text: projection

text: in

text: that

text: reference

text: frame.

text: The

text: is-

text: y

text: sin

text: θ

text: cos

text: θ

text: y

text: y

text: sue

text: of

text: system

text: orientation

text: was

text: already

text: highlighted

text: in

text: their

text: original

text: the

text: t−riclinic

text: stiﬀness

text: tensor

text: of

text: Eq.

text: cos

text: θz

text: sin

text: θz

text: is

text: rotated

text: by

text: and

text: degrees

text: (where

text: Rz

text: sin

text: θz

text: cos

text: θz

text: means

text: counterclockwise

text: and

text: clockwise)

text: with

text: respect

text: to

text: the

text: ﬁrst,

text: second

text: and

text: third

text: Cartesian

text: axes

text: re-

text: C

text: spectively,

text: in

text: that

text: order,

text: then

text: the

text: projection

text: of

text: tric

text: onto

text: The

text: diﬀerent

text: θ

text: give

text: counterclockwise

text: rotation

text: angles

text: its

text: closest

text: cubic

text: elastic

text: tensor

text: yields

text: GPa,

text: around

text: the

text: Cartesiian

text: axes.

text: Note

text: that

text: the

text: order

text: in

text: which

text: Cimcupboris-

text: further

text: reduced

text: to

text: GPa,

text: where

text: the

text: tilde

text: denotes

text: the

text: given

text: by

text: rotation

text: performed.

text: Note

text: that

text: in

text: this

text: case

text: the

text: diﬀerence

text: between

text: the

text: two

text: diﬀerent

text: approaches

text: is

text: small

text: because

text: Ctric

text: was

text: already

text: “almost”

text: cubic

text: (the

text: rotation

text: angles

text: to

text: C˜(θ

text: θ

text: θ

text: R(θ

text: θ

text: θ

text: )C.

text: x

text: y

text: z

text: x

text: y

text: z

text: correct

text: the

text: structure

text: are

text: correspondingly

text: small).

text: In

text: a

text: more

text: general

text: case,

text: the

text: input

text: tensor

text: might

text: be

text: in

text: a

text: form

text: that

text: does

text: not

text: closely

text: resemble

text: the

text: symmetry

text: of

text: interest,

text: In

text: the

text: particular

text: case

text: of

text: the

text: stiﬀness

text: tensor

text: leading

text: to

text: even

text: larger

text: discrepancies.

text: Based

text: on

text: similar

text: the

text: matrix

text: C˜

text: are

text: given

text: by

text: considerations,

text: Diner

text: et

text: al.

text: have

text: recently

text: presented

text: a

text: rotation-based

text: method

text: to

text: identify

text: the

text: symmetry

text: class

text: of

text: a

text: tensor

text: which

text: is

text: not

text: expressed

text: in

text: its

text: natural

text: coordinate

text: C˜ijkl

text: RimRjnRkoRlpCmnop,

text: m,n,o,p

text: Since,

text: as

text: previously

text: mentioned,

text: rotations

text: do

text: not

text: af-

text: fect

text: the

text: properties

text: of

text: materials,

text: only

text: their

text: mathematical

text: where

text: all

text: the

text: indices

text: run

text: from

text: to

text: I

text: could

text: have

text: representation,

text: the

text: procedure

text: introduced

text: here

text: allows

text: to

text: equivalently

text: made

text: the

text: rotation

text: operator

text: act

text: on

text: the

text: dif-

text: V

text: obtain

text: a

text: better

text: (“closer”)

text: higher-symmetry

text: projection

text: of

text: ferent

text: basis

text: components

text: however

text: this

text: would

text: lead

text: to

text: i

text: the

text: original

text: tensor.

text: In

text: the

text: following

text: I

text: deal

text: with

text: the

text: de-

text: complicated

text: angle-dependent

text: projectors.

text: It

text: seems

text: best

text: tails

text: of

text: the

text: present

text: approach

text: in

text: the

text: context

text: of

text: Moakher

text: and

text: simplest

text: to

text: keep

text: Moakher

text: and

text: Norris’s

text: original

text: for-

text: and

text: Norris’s

text: method,

text: whose

text: details

text: are

text: given

text: in

text: Ref.

text: mulation

text: for

text: the

text: projectors

text: and

text: introduce

text: the

text: angular

text: C

text: C

text: C

text: Let

text: and

text: sym

text: be

text: material

text: tensors

text: of

text: arbitrary

text: sym-

text: dependence

text: on

text: instead.

text: The

text: projector

text: (see

text: Ref.

text: for

text: metry

text: and

text: speciﬁc

text: symmetry

text: “sym”,

text: respectively.

text: Then

text: a

text: given

text: symmetry

text: sym

text: links

text: C˜

text: and

text: Csym

text: as

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ≥ or at a certain time T 0, called maturity. There are two basic types of an option: a call option which gives a holder a right to buy and a put option which allows an owner to sell an underlying asset. Also one distinguishes between European options, where exercise is only permitted at maturity T and American options which can be exercised at any time before an expiration time T. We will denote the price of the ∈ R ≥ underlying asset by S = S , where τ 0 is the time to maturity T. τ + In standard option pricing models, e.g., the Black-Scholes model [1, 2, 25], a price of the underlying asset S follows a stochastic process, governed by the following τ stochastic diﬀerential equation dS = ιS dt + σ S dW , (2.1) τ τ τ τ τ with a Wiener process W , a drift ι and a volatility σ > 0. One of the main τ τ limitations of these models is the assumption that the volatility of the return on the underlying asset is constant σ = σ, while in ﬁnancial markets, the volatility is not a τ constant, but a stochastic variable. The Heston stochastic volatility model [21] takes into account the randomness of the volatility and is based on the following stock price and variance dynamics √ dS = ιS dt + v S d√W1, (2.2) τ τ − τ τ τ dv = κ(γ v )dτ + ξ v dW2, (2.3) τ τ τ τ where v = σ2 follows a square root process (known as a Cox-Ingersoll-Ross (CIR) τ process) with the mean variance γ > 0, rate of mean reversion κ > 0 and so called volatility of volatility ξ > 0. The Wiener processes W1 and W2 are correlated with ∈ − τ τ the correlation parameter ρ [ 1, 1]. With the use of Ito’s formula [25], each model can be formulated in terms of a partial diﬀerential equation (PDE). For purposes of brevity, we omit the derivation of the equations. The reader is referred to [1, 21, 22] for further details. We deﬁne the spatial diﬀerential operators corresponding to the Black-Scholes and Heston model as follows L 1 − − BSP : = σ2S2∂ P + rS∂ P (r q)P, (2.4) 2 SS S L 1 1 − − HP : = ξ2v∂ P + ρξvS∂ P + vS2∂ P + κ(γ v)∂ P + rS∂ P rP, 2 vv vS 2 SS v S (2.5) where r is the interest rate and q is the dividend payment. Then the value of an option P(τ, S) in the Black-Scholes model and P(τ, v, S) in the Heston model paying P (S) at maturity time T must satisfy the following partial diﬀerential equation 0 L ∂ P + P = 0, (2.6) τ 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: or

text: at

text: a

text: certain

text: time

text: T

text: called

text: maturity.

text: There

text: are

text: two

text: basic

text: types

text: of

text: an

text: option:

text: a

text: call

text: option

text: which

text: gives

text: a

text: holder

text: a

text: right

text: to

text: buy

text: and

text: a

text: put

text: option

text: which

text: allows

text: an

text: owner

text: to

text: sell

text: an

text: underlying

text: asset.

text: Also

text: one

text: distinguishes

text: between

text: European

text: options,

text: where

text: exercise

text: is

text: only

text: permitted

text: at

text: maturity

text: T

text: and

text: American

text: options

text: which

text: can

text: be

text: exercised

text: at

text: any

text: time

text: before

text: an

text: expiration

text: time

text: T.

text: We

text: will

text: denote

text: the

text: price

text: of

text: the

text: R

text: underlying

text: asset

text: by

text: S

text: S

text: where

text: τ

text: is

text: the

text: time

text: to

text: maturity

text: T.

text: τ

text: In

text: standard

text: option

text: pricing

text: models,

text: e.g.,

text: the

text: Black-Scholes

text: model

text: a

text: price

text: of

text: the

text: underlying

text: asset

text: S

text: follows

text: a

text: stochastic

text: process,

text: governed

text: by

text: the

text: following

text: τ

text: stochastic

text: diﬀerential

text: equation

text: dS

text: ιS

text: dt

text: σ

text: S

text: dW

text: τ

text: τ

text: τ

text: τ

text: τ

text: with

text: a

text: Wiener

text: process

text: W

text: a

text: drift

text: ι

text: and

text: a

text: volatility

text: σ

text: One

text: of

text: the

text: main

text: τ

text: τ

text: limitations

text: of

text: these

text: models

text: is

text: the

text: assumption

text: that

text: the

text: volatility

text: of

text: the

text: return

text: on

text: the

text: underlying

text: asset

text: is

text: constant

text: σ

text: σ,

text: while

text: in

text: ﬁnancial

text: markets,

text: the

text: volatility

text: is

text: not

text: a

text: τ

text: constant,

text: but

text: a

text: stochastic

text: variable.

text: The

text: Heston

text: stochastic

text: volatility

text: model

text: takes

text: into

text: account

text: the

text: randomness

text: of

text: the

text: volatility

text: and

text: is

text: based

text: on

text: the

text: following

text: stock

text: price

text: and

text: variance

text: dynamics

text: dS

text: ιS

text: dt

text: v

text: S

text: τ

text: τ

text: τ

text: τ

text: τ

text: dv

text: κ(γ

text: v

text: )dτ

text: ξ

text: v

text: τ

text: τ

text: τ

text: τ

text: where

text: v

text: σ2

text: follows

text: a

text: square

text: root

text: process

text: (known

text: as

text: a

text: Cox-Ingersoll-Ross

text: (CIR)

text: τ

text: process)

text: with

text: the

text: mean

text: variance

text: γ

text: rate

text: of

text: mean

text: reversion

text: κ

text: and

text: so

text: called

text: volatility

text: of

text: volatility

text: ξ

text: The

text: Wiener

text: processes

text: and

text: are

text: correlated

text: with

text: τ

text: τ

text: the

text: correlation

text: parameter

text: ρ

text: With

text: the

text: use

text: of

text: Ito’s

text: formula

text: each

text: model

text: can

text: be

text: formulated

text: in

text: terms

text: of

text: a

text: partial

text: diﬀerential

text: equation

text: (PDE).

text: For

text: purposes

text: of

text: brevity,

text: we

text: omit

text: the

text: derivation

text: of

text: the

text: equations.

text: The

text: reader

text: is

text: referred

text: to

text: for

text: further

text: details.

text: We

text: deﬁne

text: the

text: spatial

text: diﬀerential

text: operators

text: corresponding

text: to

text: the

text: Black-Scholes

text: and

text: Heston

text: model

text: as

text: follows

text: L

text: BSP

text: P

text: rS∂

text: P

text: (r

text: q)P,

text: SS

text: S

text: L

text: HP

text: ξ2v∂

text: P

text: ρξvS∂

text: P

text: P

text: κ(γ

text: v)∂

text: P

text: rS∂

text: P

text: rP,

text: vv

text: vS

text: SS

text: v

text: S

text: where

text: r

text: is

text: the

text: interest

text: rate

text: and

text: q

text: is

text: the

text: dividend

text: payment.

text: Then

text: the

text: value

text: of

text: an

text: option

text: P(τ,

text: S)

text: in

text: the

text: Black-Scholes

text: model

text: and

text: P(τ,

text: v,

text: S)

text: in

text: the

text: Heston

text: model

text: paying

text: P

text: (S)

text: at

text: maturity

text: time

text: T

text: must

text: satisfy

text: the

text: following

text: partial

text: diﬀerential

text: equation

text: L

text: P

text: P

text: τ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Again, to simplify the presentation, we omit the parameter vector µ in the no- tation of a, fn, gn. In the same way, we only consider the case θ = 1, that is, an implicit Euler time discretization. However the analysis presented hereafter holds for ∈ any θ (0, 1], up to technical supplementary computations. We start by introducing relevant residuals and preliminary results. In order to evaluate the approximation errors induced by our method, we deﬁne, · · · − for n = 0, (cid:28), L 1, the equ(cid:29)ality and inequality residuals by: − un+1 un − − ∀ ∈ rn(v) := N N , v + a(un+1, v) b(λn+1, v) fn(v), v V ∆t N N L2(Ω) − ∀ ∈ sn(η) := b(η, un+1) gn(η), η M. N We also introduce the primal and dual errors − − en = un un, en = λn λn. (4.1) u N λ N Using the linearity (cid:28)of rn, one ﬁnd(cid:29)s that: − en+1 en − rn(v) = u u , v + a(en+1, v) b(en+1, v). (4.2) ∆t u λ L2(Ω) As a consequence of inf-sup stability, one can bound the dual error by the primal error, as stated in the next lemma. · · · − Lemma 4.1 (Primal/Dual Error Relation). For n = 0, , L 1, the dual error at time step tn can be bou(cid:18)nded by the primal error as (cid:19) (cid:107) (cid:107) ≤ 1 C (cid:107) − (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) en+1 Ω en+1 en + γ en+1 + rn (cid:48) . (4.3) λ W β ∆t u u L2(Ω) a u V V · · ∈ (cid:54) Proof. The inf-sup stability of b( , ) guarantees the existence of a v(cid:63) V, v(cid:63) = 0 such that (cid:107) (cid:107) (cid:107) (cid:107) ≤ β v(cid:63) en+1 b(v(cid:63), en+1). V λ W λ Using (4.2), we ﬁnd(cid:28)that: (cid:29) − (cid:107) (cid:107) (cid:107) (cid:107) ≤ en+1 en − β v(cid:63) en+1 u u , v(cid:63) + a(en+1, v(cid:63)) rn(v(cid:63)) V λ W ∆t u L2(Ω) ≤ 1 (cid:107) − (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) en+1 en v(cid:63) + γ en+1 v(cid:63) + rn (cid:48) v(cid:63) ∆t u u L2(Ω) L2(Ω) a u V V V V and the result then follows from (2.8). 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Again,

text: to

text: simplify

text: the

text: presentation,

text: we

text: omit

text: the

text: parameter

text: vector

text: µ

text: in

text: the

text: no-

text: tation

text: of

text: a,

text: fn,

text: gn.

text: In

text: the

text: same

text: way,

text: we

text: only

text: consider

text: the

text: case

text: θ

text: that

text: is,

text: an

text: implicit

text: Euler

text: time

text: discretization.

text: However

text: the

text: analysis

text: presented

text: hereafter

text: holds

text: for

text: any

text: θ

text: up

text: to

text: technical

text: supplementary

text: computations.

text: We

text: start

text: by

text: introducing

text: relevant

text: residuals

text: and

text: preliminary

text: results.

text: In

text: order

text: to

text: evaluate

text: the

text: approximation

text: errors

text: induced

text: by

text: our

text: method,

text: we

text: deﬁne,

text: for

text: n

label: equation

text: L

text: the

text: and

text: inequality

text: residuals

text: by:

text: un

text: rn(v)

text: N

text: N

text: v

text: v)

text: v)

text: fn(v),

text: v

text: V

text: ∆t

text: N

text: N

text: sn(η)

text: b(η,

text: gn(η),

text: η

text: M.

text: N

text: We

text: also

text: introduce

text: the

text: primal

text: and

text: dual

text: errors

text: en

text: un

text: un,

text: en

text: λn

text: λn.

text: u

text: N

text: λ

text: N

text: Using

text: the

text: linearity

text: rn,

text: one

text: that:

text: en

text: rn(v)

text: u

text: u

text: v

text: v)

text: v).

text: ∆t

text: u

text: λ

text: As

text: a

text: consequence

text: of

text: inf-sup

text: stability,

text: one

text: can

text: bound

text: the

text: dual

text: error

text: by

text: the

text: primal

text: error,

text: as

text: stated

text: in

text: the

text: next

text: lemma.

text: Lemma

text: (Primal/Dual

text: Error

text: Relation).

text: For

text: n

text: L

text: the

text: dual

text: error

text: at

text: time

text: step

text: tn

text: can

text: be

text: by

text: the

text: primal

text: error

text: as

text: C

text: Ω

text: en

text: γ

text: rn

text: λ

text: W

text: β

text: ∆t

text: u

text: u

text: a

text: u

text: V

text: V

text: Proof.

text: The

text: inf-sup

text: stability

text: of

text: b(

text: guarantees

text: the

text: existence

text: of

text: a

text: V,

text: such

text: that

text: β

text: V

text: λ

text: W

text: λ

text: Using

text: we

text: en

text: β

text: u

text: u

text: V

text: λ

text: W

text: ∆t

text: u

text: en

text: γ

text: rn

text: ∆t

text: u

text: u

text: a

text: u

text: V

text: V

text: V

text: V

text: and

text: the

text: result

text: then

text: follows

text: from

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 assume that the nodes use IEEE 802.15.4 CSMA/CA for medium access. Moreover, all the nodes are assumed to be in the carrier sense range of one another. We shall refer to this assumption as the No Hidden Nodes (NH) assumption. Justiﬁcation for the NH assumption While the NH assumption may seem restrictive, observe that by using a cluster based network topology where within each cluster, the nodes satisfy the NH assumption, and diﬀerent channels are used across adjacent clusters (we recall that there are 16 channels available for IEEE 802.15.4 CSMA/CA [4]), moderate to large areas can be covered. One way of constructing such a cluster based topology is explained below. We assume that each cluster will cover a regular hexagonal region in the plane, and within each cluster, there will be a single BS placed at the center of the hexagon. Suppose the carrier sense range of each node is r , and, for a ﬁxed packet size and a given path cs loss model, the maximum allowed link length to ensure the target link PER l is r. Further suppose that r is chosen such that r = 2mr, for some integer m > 0. cs Having thus chosen r, we let the length of each side of a hexagonal cluster be mr. Then, it follows from triangle inequality that all nodes within a hexagonal cluster are within CS range of each other. R BS1 BS2 ≤ mr ≤ mr s N1 N2 Fig. 3. Computation of the minimum distance between nodes in two diﬀerent clusters Now consider two distinct clusters. Let R be the distance between the BSs in these two clusters. Consider a pair of nodes, one picked from each of these clusters. Let s be the distance between these two nodes. See Figure 3. It follows from triangle inequality that ≤ R s + 2mr ⇒ ≥ − s R 2mr − Thus, the distance between any pair of points across the two clusters is at least R 2mr. We can use the same channel in these two clusters if the nodes in one cluster are completely hidden fr−om th≥ose in the other, so that they do not inte≥rfere with one another. This is ensured if R 2mr r = 2mr. Thus, it suﬃces to have R 4mr. In other words, cs we can use the same channel in two diﬀerent clusters if the distance between the BSs in these two clusters is at least 4mr. Now consider the channel assignment in the hexagonal cluster layout shown in Figure 4, where the length of each side of each hexagon is mr. It can be veriﬁed using elementary geometry that the BSs of any two co-channel clusters are separated by more than 4mr, and hence with this channel assignment, there is no interference ##LTFigure## ##LTLine## ##LTLine##

font: MGBBCV+URWPalladioL-Roma

text: assume

text: that

text: the

text: nodes

text: use

text: IEEE

text: CSMA/CA

text: for

text: medium

text: access.

text: Moreover,

text: all

text: the

text: nodes

text: are

text: assumed

text: to

text: be

text: in

text: the

text: carrier

text: sense

text: range

text: of

text: one

text: another.

text: We

text: shall

text: refer

text: to

text: this

text: assumption

text: as

text: the

text: No

font: YTJVXT+URWPalladioL-Ital

text: Hidden

text: Nodes

text: (NH)

text: assumption.

text: Justiﬁcation

font: ETUBGR+URWPalladioL-Bold

text: for

text: the

text: NH

text: assumption

text: While

text: the

text: NH

text: assumption

text: may

text: seem

text: restrictive,

text: observe

text: that

text: by

text: using

text: a

text: cluster

text: based

text: network

text: topology

text: where

text: within

text: each

text: cluster,

text: the

text: nodes

text: satisfy

text: the

text: NH

text: assumption,

text: and

text: diﬀerent

text: channels

text: are

text: used

text: across

text: adjacent

text: clusters

text: (we

text: recall

text: that

text: there

text: are

text: channels

text: available

text: for

text: IEEE

text: CSMA/CA

text: moderate

text: to

text: large

text: areas

text: can

text: be

text: covered.

text: One

text: way

text: of

text: constructing

text: such

text: a

text: cluster

text: based

text: topology

text: is

text: explained

text: below.

text: We

text: assume

text: that

text: each

text: cluster

text: will

text: cover

text: a

text: regular

text: hexagonal

text: region

text: in

text: the

text: plane,

text: and

text: within

text: each

text: cluster,

text: there

text: will

text: be

text: a

text: single

text: BS

text: placed

text: at

text: the

text: center

text: of

text: the

text: hexagon.

text: Suppose

text: the

text: carrier

text: sense

text: range

text: of

text: each

text: node

text: is

text: r

text: and,

text: for

text: a

text: ﬁxed

text: packet

text: size

text: and

text: a

text: given

text: path

text: cs

text: loss

text: model,

text: the

text: maximum

text: allowed

text: link

text: length

text: to

text: ensure

text: the

text: target

text: link

text: PER

text: l

text: is

text: r.

text: Further

text: suppose

text: that

text: r

text: is

text: chosen

text: such

text: that

text: r

text: for

text: some

text: integer

text: m

text: cs

text: Having

text: thus

text: chosen

text: r,

text: we

text: let

text: the

text: length

text: of

text: each

text: side

text: of

text: a

text: hexagonal

text: cluster

text: be

text: mr.

text: Then,

text: it

text: follows

text: from

text: triangle

text: inequality

text: that

text: all

text: nodes

text: within

text: a

text: hexagonal

text: cluster

text: are

text: within

text: CS

text: range

text: of

text: each

text: other.

text: R

text: mr

text: mr

text: s

text: Fig.

text: Computation

text: of

text: the

text: minimum

text: distance

text: between

text: nodes

text: in

text: two

text: diﬀerent

text: clusters

text: Now

text: consider

text: two

text: distinct

text: clusters.

text: Let

text: R

text: be

text: the

text: distance

text: between

text: the

text: BSs

text: in

text: these

text: two

text: clusters.

text: Consider

text: a

text: pair

text: of

text: nodes,

text: one

text: picked

text: from

text: each

text: of

text: these

text: clusters.

text: Let

text: s

text: be

text: the

text: distance

text: between

text: these

text: two

text: nodes.

text: See

text: Figure

text: It

text: follows

text: from

text: triangle

text: inequality

text: that

text: R

text: s

text: s

text: R

text: Thus,

text: the

text: distance

text: between

text: any

text: pair

text: of

text: points

text: across

text: the

text: two

text: clusters

text: is

text: at

text: least

text: R

text: We

text: can

text: use

text: the

text: same

text: channel

text: in

text: these

text: two

text: clusters

text: if

text: the

text: nodes

text: in

text: one

text: cluster

text: are

text: completely

text: hidden

text: fr−om

text: th≥ose

text: in

text: the

text: other,

text: so

text: that

text: they

text: do

text: not

text: inte≥rfere

text: with

text: one

text: another.

text: This

text: is

text: ensured

text: if

text: R

text: r

text: Thus,

text: it

text: suﬃces

text: to

text: have

text: R

text: In

text: other

text: words,

text: cs

text: we

text: can

text: use

text: the

text: same

text: channel

text: in

text: two

text: diﬀerent

text: clusters

text: if

text: the

text: distance

text: between

text: the

text: BSs

text: in

text: these

text: two

text: clusters

text: is

text: at

text: least

text: Now

text: consider

text: the

text: channel

text: assignment

text: in

text: the

text: hexagonal

text: cluster

text: layout

text: shown

text: in

text: Figure

text: where

text: the

text: length

text: of

text: each

text: side

text: of

text: each

text: hexagon

text: is

text: mr.

text: It

text: can

text: be

text: veriﬁed

text: using

text: elementary

text: geometry

text: that

text: the

text: BSs

text: of

text: any

text: two

text: co-channel

text: clusters

text: are

text: separated

text: by

text: more

text: than

text: and

text: hence

text: with

text: this

text: channel

text: assignment,

text: there

text: is

text: no

text: interference

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: 10 II. Modeling CSMA/CA for a Network with No Hidden Nodes A. The beaconless IEEE 802.15.4 CSMA/CA protocol[3] When a node has data to send (i.e., has a non-empty queue), it initiates a rand−om back-oﬀ with the ﬁrst back-oﬀ period being sampled uniformly from 0 to 2macminBE 1, where macminBE is a parameter ﬁxed by the standard. For each node, the back-oﬀ period is speciﬁed in terms of slots where a slot equals 20 symbol times (T ), and a s symbol time equals 16 µs. The node then performs a CCA (Clear Channel Assessment) to determine whether the channel is idle. We note that, unlike the IEEE 802.11 CSMA/CA, the back-oﬀ timer of a node is not “frozen” during the transmissions of other nodes. If the CCA succeeds, the node does a Rx-to-Tx turnaround, which is 12 symbol times, and starts transmitting on the channel. The failure of the CCA starts a new back-oﬀ process with the back-oﬀ exponent raised by one, i.e., to macminBE+1, provided that this is less than its maximum value, macmaxBE. The maximum number of successive CCA failures for the same packet is governed by macMaxCSMABackoﬀs, exceeding which the packet is discarded at the MAC layer. The standard allows the inclusion of acknowledgements (ACKs) which are sent by the intended receivers on a successful packet reception. Once the packet is received, the receiver performs a Rx-to-Tx turnaround, which is again 12 symbol times, and sends a 22 symbol ﬁxed size ACK packet. A successful transmission is followed by an InterFrame Spacing(IFS) before sending another packet. When a transmitted packet collides or is corrupted by the PHY layer noise, the ACK packet is not generated, which is interpreted by the transmitter as failure in delivery. The node retransmits the same packet for a maximum of aMaxFrameRetries times before discarding it at the MAC layer. After transmitting a packet, the node turns to the Rx-mode and waits for the ACK. The macAckWaitDuration determines the maximum amount of time a node must wait for in order to receive the ACK before concluding that the packet (or the ACK) has collided. The default values of macminBE, macmaxBE, macMaxCSMABackoﬀs, and aMaxFrameRetries are 3, 5, 4, and 3 respectively. B. Detailed ﬁxed point equations for NH case Although there has been considerable research on analytical modeling of CSMA/CA, the work reported in [13], [3], in the context of beacon-less IEEE 802.15.4 CSMA/CA, appears to be the most comprehensive (capturing aspects such as multi-hopping, hop- by-hop queueing, presence of acknowledgements, packet collisions due to presence of hidden terminals, and dilation of perceived channel activity period due to hidden terminals, etc.), and also very accurate. We base our work in this paper on a simpliﬁed version of the detailed ﬁxed point analysis reported in [13], [3]. For ease of reference, we brieﬂy describe the modeling philosophy of their analysis here, and extract from that analysis the main equations for the NH case. In [13], [3], akin to the approach in [15] and [16], a “decoupling” approximation is made, whereby each node is modeled separately, incorporating the inﬂuence of the other nodes in the network by their average statistics, and as if these nodes were independent of the tagged node. In the literature, such an approach has also been called a “mean ﬁeld approximation,” and formal justiﬁcation has been provided in, for example, [14]. Modeling the activity of a tagged node, say i: All packets entering node i are assumed to have the same ﬁxed length, and hence they take the same amount of time when being transmitted over the medium (denoted by T ). In a network with no hidden nodes, the tx channel activity perceived by node i is due to all the other nodes in the network. Each node alternates between periods during which its transmitter queue is empty and those during which the queue is non-empty. During periods in which its transmitter queue

text: II.

label: section

text: Modeling

font: PAAMKC+Rpxsc

text: CSMA/CA

text: for

text: a

text: Network

text: with

text: No

text: Hidden

text: Nodes

text: A.

text: The

text: beaconless

text: IEEE

text: CSMA/CA

text: When

text: a

text: node

text: has

text: data

text: to

text: send

text: (i.e.,

text: has

text: a

text: non-empty

text: queue),

text: it

text: initiates

text: a

text: rand−om

text: back-oﬀ

text: with

text: the

text: ﬁrst

text: back-oﬀ

text: period

text: being

text: sampled

text: uniformly

text: from

text: to

text: where

text: macminBE

text: is

text: a

text: parameter

text: ﬁxed

text: by

text: the

text: standard.

text: For

text: each

text: node,

text: the

text: back-oﬀ

text: period

text: is

text: speciﬁed

text: in

text: terms

text: of

text: slots

text: where

text: a

text: slot

text: equals

text: symbol

text: times

text: (T

text: and

text: a

text: s

text: symbol

text: time

text: equals

text: µs.

text: The

text: node

text: then

text: performs

text: a

text: CCA

text: (Clear

text: Channel

text: Assessment)

text: to

text: determine

text: whether

text: the

text: channel

text: is

text: idle.

text: We

text: note

text: that,

text: unlike

text: the

text: IEEE

text: CSMA/CA,

text: the

text: back-oﬀ

text: timer

text: of

text: a

text: node

text: is

text: not

text: “frozen”

text: during

text: the

text: transmissions

text: of

text: other

text: nodes.

text: If

text: the

text: CCA

text: succeeds,

text: the

text: node

text: does

text: a

text: Rx-to-Tx

text: turnaround,

text: which

text: is

text: symbol

text: times,

text: and

text: starts

text: transmitting

text: on

text: the

text: channel.

text: The

text: failure

text: of

text: the

text: CCA

text: starts

text: a

text: new

text: back-oﬀ

text: process

text: with

text: the

text: back-oﬀ

text: exponent

text: raised

text: by

text: one,

text: i.e.,

text: to

text: provided

text: that

text: this

text: is

text: less

text: than

text: its

text: maximum

text: value,

text: macmaxBE.

text: The

text: maximum

text: number

text: of

text: successive

text: CCA

text: failures

text: for

text: the

text: same

text: packet

text: is

text: governed

text: by

text: macMaxCSMABackoﬀs,

text: exceeding

text: which

text: the

text: packet

text: is

text: discarded

text: at

text: the

text: MAC

text: layer.

text: The

text: standard

text: allows

text: the

text: inclusion

text: of

text: acknowledgements

text: (ACKs)

text: which

text: are

text: sent

text: by

text: the

text: intended

text: receivers

text: on

text: a

text: successful

text: packet

text: reception.

text: Once

text: the

text: packet

text: is

text: received,

text: the

text: receiver

text: performs

text: a

text: Rx-to-Tx

text: turnaround,

text: which

text: is

text: again

text: symbol

text: times,

text: and

text: sends

text: a

text: symbol

text: ﬁxed

text: size

text: ACK

text: packet.

text: A

text: successful

text: transmission

text: is

text: followed

text: by

text: an

text: InterFrame

text: Spacing(IFS)

text: before

text: sending

text: another

text: packet.

text: When

text: a

text: transmitted

text: packet

text: collides

text: or

text: is

text: corrupted

text: by

text: the

text: PHY

text: layer

text: noise,

text: the

text: ACK

text: packet

text: is

text: not

text: generated,

text: which

text: is

text: interpreted

text: by

text: the

text: transmitter

text: as

text: failure

text: in

text: delivery.

text: The

text: node

text: retransmits

text: the

text: same

text: packet

text: for

text: a

text: maximum

text: of

text: aMaxFrameRetries

text: times

text: before

text: discarding

text: it

text: at

text: the

text: MAC

text: layer.

text: After

text: transmitting

text: a

text: packet,

text: the

text: node

text: turns

text: to

text: the

text: Rx-mode

text: and

text: waits

text: for

text: the

text: ACK.

text: The

text: macAckWaitDuration

text: determines

text: the

text: maximum

text: amount

text: of

text: time

text: a

text: node

text: must

text: wait

text: for

text: in

text: order

text: to

text: receive

text: the

text: ACK

text: before

text: concluding

text: that

text: the

text: packet

text: (or

text: the

text: ACK)

text: has

text: collided.

text: The

text: default

text: values

text: of

text: macminBE,

text: macmaxBE,

text: macMaxCSMABackoﬀs,

text: and

text: aMaxFrameRetries

text: are

text: and

text: respectively.

text: B.

text: Detailed

text: ﬁxed

text: point

text: equations

text: for

text: NH

text: case

text: Although

text: there

text: has

text: been

text: considerable

text: research

text: on

text: analytical

text: modeling

text: of

text: CSMA/CA,

text: the

text: work

text: reported

text: in

text: in

text: the

text: context

text: of

text: beacon-less

text: IEEE

text: CSMA/CA,

text: appears

text: to

text: be

text: the

text: most

text: comprehensive

text: (capturing

text: aspects

text: such

text: as

text: multi-hopping,

text: hop-

text: by-hop

text: queueing,

text: presence

text: of

text: acknowledgements,

text: packet

text: collisions

text: due

text: to

text: presence

text: of

text: hidden

text: terminals,

text: and

text: dilation

text: of

text: perceived

text: channel

text: activity

text: period

text: due

text: to

text: hidden

text: terminals,

text: etc.),

text: and

text: also

text: very

text: accurate.

text: We

text: base

text: our

text: work

text: in

text: this

text: paper

text: on

text: a

text: simpliﬁed

text: version

text: of

text: the

text: detailed

text: ﬁxed

text: point

text: analysis

text: reported

text: in

text: For

text: ease

text: of

text: reference,

text: we

text: brieﬂy

text: describe

text: the

text: modeling

text: philosophy

text: of

text: their

text: analysis

text: here,

text: and

text: extract

text: from

text: that

text: analysis

text: the

text: main

text: equations

text: for

text: the

text: NH

text: case.

text: In

text: akin

text: to

text: the

text: approach

text: in

text: and

text: a

text: “decoupling”

text: approximation

text: is

text: made,

text: whereby

text: each

text: node

text: is

text: modeled

text: separately,

text: incorporating

text: the

text: inﬂuence

text: of

text: the

text: other

text: nodes

text: in

text: the

text: network

text: by

text: their

text: average

text: statistics,

text: and

text: as

text: if

text: these

text: nodes

text: were

text: independent

text: of

text: the

text: tagged

text: node.

text: In

text: the

text: literature,

text: such

text: an

text: approach

text: has

text: also

text: been

text: called

text: a

text: “mean

text: ﬁeld

text: approximation,”

text: and

text: formal

text: justiﬁcation

text: has

text: been

text: provided

text: in,

text: for

text: example,

text: Modeling

text: the

text: activity

text: of

text: a

text: tagged

text: node,

text: say

text: i:

text: All

text: packets

text: entering

text: node

text: i

text: are

text: assumed

text: to

text: have

text: the

text: same

text: ﬁxed

text: length,

text: and

text: hence

text: they

text: take

text: the

text: same

text: amount

text: of

text: time

text: when

text: being

text: transmitted

text: over

text: the

text: medium

text: (denoted

text: by

text: T

text: In

text: a

text: network

text: with

text: no

text: hidden

text: nodes,

text: the

text: tx

text: channel

text: activity

text: perceived

text: by

text: node

text: i

text: is

text: due

text: to

text: all

text: the

text: other

text: nodes

text: in

text: the

text: network.

text: Each

text: node

text: alternates

text: between

text: periods

text: during

text: which

text: its

text: transmitter

text: queue

text: is

text: empty

text: and

text: those

text: during

text: which

text: the

text: queue

text: is

text: non-empty.

text: During

text: periods

text: in

text: which

text: its

text: transmitter

text: queue


text: 14 P P relation (4) or violate it. Moreover, by we denote those elements of for which ∆ the value M(p) is precisely 4 + ∆, i.e., P { ∈ P| } = p M(p) = 4 + ∆ . (53) ∆ P P Clearly, and are polytopes whose vertices can easily be found, and, in ∆ P P P particular, the vertices of belong to either or . → R 0 2 Let ﬁnally φ : B 6 be a vector-valued function associating a vector of six correlators c = (x1 , y1 , x0 , y0 , x1 , y1 ) to any element of B. With the aid of this A A B B B B mapping we can associate to P the following polytope ∆ Q { ∈ R | ∈ P } = (φ(p), ∆) 7 p . (54) ∆ (cid:101) ∆ Q On the other hand, let us introduce(cid:101)the polytope of vectors of the form (c, ∆) ∆ with c s(cid:101)atisfying the inequalities (26)-(29) for some ﬁxed ∆ along with the trivial Q ⊆ Q conditions (51). By deﬁnition, for any ∆ and our aim now is to prove that Q Q ∆ ∆ ∈ Q ≥ = . In particular, we want to show that any c with some ﬁxed ∆ 0 ∆ ∆ ∆ ∈ P can always be completed to a full probability distribution p violating (4) by ∆ ∆. With the above goal we deﬁne two additional polytopes Q { − ∈ R | ∈ P} = (φ(p), M(p) 4) 7 p , (55) v (cid:91) and (cid:101) (cid:101) Q Q = . (56) v ∆ ∈ ∆ [0,2] P Q Direct numeric(cid:101)al computation shows that, analogous(cid:101)ly to , the vertices of Q Q v belong to either or . In the same way one shows that the vertices of both Q Q0 2 Q Q polytopes (cid:101) and overlap, which implies that = . Using then the deﬁnition v v → v v of these sets and the fact that the mapping p (φ(p), M(p)) is linear, one obtains Q Q that = for any ∆. ∆ ∆ Appendix B: Analytical computation of C 2 Here we determine analytically the capacity C in the case when the monogamy ∆ relation (4) is violated maximally, i.e., for ∆ = 2. From Ineqs. (26)-(29) it immedi- − ately follows that x0 = x1 = 1, y0 = x1 , and y1 = y1 , and the problem of B B B A B A determining C considerably simpliﬁes(cid:101)to (cid:101) (cid:101) 2 { − } C = min max C(1, α), C(1, β), C(α, β) , (57) 2 − ≤ ≤ (cid:101) 1 α,β 1 where we have substituted y0 = α and y1 = β and have denoted C(α, β) = B B C((1 + x)/2, (1 + y)/2) with C deﬁned in Eq. (19). To compute the above, it is ##LTLine##

text: P

text: P

text: relation

text: or

text: violate

text: it.

text: Moreover,

text: by

text: we

text: denote

text: those

text: elements

text: of

text: for

text: which

text: the

text: value

text: M(p)

text: is

text: precisely

text: i.e.,

text: P

text: P|

text: p

text: M(p)

text: P

text: P

text: Clearly,

text: and

text: are

text: polytopes

text: whose

text: vertices

text: can

text: easily

text: be

text: found,

text: and,

text: in

text: P

text: P

text: P

text: particular,

text: the

text: vertices

text: of

text: belong

text: to

text: either

text: or

text: R

text: Let

text: ﬁnally

text: φ

text: B

text: be

text: a

text: vector-valued

text: function

text: associating

text: a

text: vector

text: of

text: six

text: correlators

text: c

text: to

text: any

text: element

text: of

text: B.

text: With

text: the

text: aid

text: of

text: this

text: A

text: A

text: B

text: B

text: B

text: B

text: mapping

text: we

text: can

text: associate

text: to

text: P

text: the

text: following

text: polytope

text: Q

text: R

text: P

text: (φ(p),

text: p

text: Q

text: On

text: the

text: other

text: hand,

text: let

text: us

text: polytope

text: of

text: vectors

text: of

text: the

text: form

text: (c,

text: with

text: c

text: the

text: inequalities

text: for

text: some

text: ﬁxed

text: along

text: with

text: the

text: trivial

text: Q

text: Q

text: conditions

text: By

text: deﬁnition,

text: for

text: any

text: and

text: our

text: aim

text: now

text: is

text: to

text: prove

text: that

text: Q

text: Q

text: Q

text: In

text: particular,

text: we

text: want

text: to

text: show

text: that

text: any

text: c

text: with

text: some

text: ﬁxed

text: P

text: can

text: always

text: be

text: completed

text: to

text: a

text: full

text: probability

text: distribution

text: p

text: violating

text: by

text: With

text: the

text: above

text: goal

text: we

text: deﬁne

text: two

text: additional

text: polytopes

text: Q

text: R

text: P}

text: (φ(p),

text: M(p)

text: p

text: v

text: and

text: Q

text: Q

text: v

text: P

text: Q

text: Direct

text: computation

text: shows

text: that,

text: to

text: the

text: vertices

text: of

text: Q

text: Q

text: v

text: belong

text: to

text: either

text: or

text: In

text: the

text: same

text: way

text: one

text: shows

text: that

text: the

text: vertices

text: of

text: both

text: Q

text: Q

text: Q

text: polytopes

text: and

text: overlap,

text: which

text: implies

text: that

text: Using

text: then

text: the

text: deﬁnition

text: v

text: v

text: v

text: v

text: of

text: these

text: sets

text: and

text: the

text: fact

text: that

text: the

text: mapping

text: p

text: (φ(p),

text: M(p))

text: is

text: linear,

text: one

text: obtains

text: Q

text: Q

text: that

text: for

text: any

text: Appendix

text: B:

text: Analytical

text: computation

text: of

text: C

text: Here

text: we

text: determine

text: analytically

text: the

text: capacity

text: C

text: in

text: the

text: case

text: when

text: the

text: monogamy

text: relation

text: is

text: violated

text: maximally,

text: i.e.,

text: for

text: From

text: Ineqs.

text: it

text: immedi-

text: ately

text: follows

text: that

text: and

text: and

text: the

text: problem

text: of

text: B

text: B

text: B

text: A

text: B

text: A

text: determining

text: C

text: considerably

text: C

text: min

text: max

text: α),

text: β),

text: C(α,

text: β)

text: α,β

text: where

text: we

text: have

text: substituted

text: α

text: and

text: β

text: and

text: have

text: denoted

text: C(α,

text: β)

text: B

text: B

text: with

text: C

text: deﬁned

text: in

text: Eq.

text: To

text: compute

text: the

text: above,

text: it

text: is

text: ##LTLine##


text: 4 last correlator, B is replaced by any A or B (for clarity, however, we proceed with 0 i i a ﬁxed measurement B ). Also, without any loss of generality we can assume that (cid:104) (cid:105) 0 both I and B E are positive; if this is not the case, we redeﬁne observables A , AB 0 → − → − → − 0 A and/or E in the following way: A A , A A , and/or E E. 1 0 0 1 1 Consequently, in what follows we omit the absolute values in (4). In order to prove (4), let us ﬁrst make the following observation. Suppose that for ± some random variables X, Y and Z taking values 1 there exists the joint probability distribution p(XY Z). Then, the latter fulﬁls the following inequalities − (cid:104) (cid:105) − (cid:104) (cid:105) − (cid:104) (cid:105) ≤ ( 1)i XY + ( 1)j Y Z + ( 1)k XZ 1, (6) Z X Y ⊕ ⊕ with i, j, k = 0, 1 such that i j k = 1, where addition is modulo 2. To prove (6), it sufﬁces to check it for the extremal values of correlators. Now, one notices that each triple of observables A , B and E is jointly mea- i j surable and therefore, for any pair i, j, there exists the joint probability distribution p(A B E) which must satisfy (6). This gives rise to the following four inequalities i j (cid:104) (cid:105) (cid:104) (cid:105) − (cid:104) (cid:105) ≤ A B + B E A E 1, (7) (cid:104) 0 0(cid:105)E (cid:104) 0 (cid:105)A0 − (cid:104) 0 (cid:105)B0 ≤ A B + B E A E 1, (8) (cid:104) 1 0(cid:105)E − (cid:104) 0 (cid:105)A1 (cid:104) 1 (cid:105)B0 ≤ A B B E + A E 1, (9) −(cid:104) 1 1(cid:105)E (cid:104) 1 (cid:105)A1 (cid:104) 1 (cid:105)B1 ≤ A B + B E + A E 1. (10) 0 1 E 1 A0 0 B1 (cid:104) (cid:105) By summing these up and using the fact that in a nonsignaling theory XY = (cid:104) (cid:105) (cid:54) (cid:48) Z XY (cid:48) for any Z = Z , one obtains (4). Z 3 Signaling boxes as classical channels (cid:104) (cid:105) (cid:104) (cid:105) Let us assume that correlators B E and B E are equal (later we will 0 A0 0 A1 show how this assumption can be relaxed). Then the monogamy relation (4) is well deﬁned. It bounds the possible correlations achievable in any no-signaling theory between outcomes of measurements performed by the three parties A, B and E. If it is violated by some probability distribution p, then the latter must violate the no- signalling principle, in which case we call such a box signaling. In other words, if p violates (4), then values of some bipartite correlators become dependent on the measurement choice made by the third party. This dependence allows one to use such signaling boxes to send information from a single party to the remaining two parties. To illustrate this idea, suppose that a box p violates the relation (4) by ∆ > 0, that is, ≡ (cid:104) (cid:105) R(p) I + 2 B E = 4 + ∆. (11) AB 0 Then, by adding the inequalities (7)-(10), one concludes that (cid:104) (cid:105) − (cid:104) (cid:105) (cid:104) (cid:105) − (cid:104) (cid:105) (cid:104) (cid:105) − (cid:104) (cid:105) ≥ A E A E + A E A E + B E B E ∆. (12) 0 B0 0 B1 1 B0 1 B1 1 A1 1 A0 ##LTLine##

text: last

text: correlator,

text: B

text: is

text: replaced

text: by

text: any

text: A

text: or

text: B

text: (for

text: clarity,

text: however,

text: we

text: proceed

text: with

text: i

text: i

text: a

text: ﬁxed

text: measurement

text: B

text: Also,

text: without

text: any

text: loss

text: of

text: generality

text: we

text: can

text: assume

text: that

text: both

text: I

text: and

text: B

text: E

text: are

text: positive;

text: if

text: this

text: is

text: not

text: the

text: case,

text: we

text: redeﬁne

text: observables

text: A

text: AB

text: A

text: and/or

text: E

text: in

text: the

text: following

text: way:

text: A

text: A

text: A

text: A

text: and/or

text: E

text: E.

text: Consequently,

text: in

text: what

text: follows

text: we

text: omit

text: the

text: absolute

text: values

text: in

text: In

text: order

text: to

text: prove

text: let

text: us

text: ﬁrst

text: make

text: the

text: following

text: observation.

text: Suppose

text: that

text: for

text: some

text: random

text: variables

text: X,

text: Y

text: and

text: Z

text: taking

text: values

text: there

text: exists

text: the

text: joint

text: probability

text: distribution

text: p(XY

text: Z).

text: Then,

text: the

text: latter

text: fulﬁls

text: the

text: following

text: inequalities

text: XY

text: Y

text: Z

text: XZ

text: Z

text: X

text: Y

text: with

text: i,

text: j,

text: k

text: such

text: that

text: i

text: j

text: k

text: where

text: addition

text: is

text: modulo

text: To

text: prove

text: it

text: sufﬁces

text: to

text: check

text: it

text: for

text: the

text: extremal

text: values

text: of

text: correlators.

text: Now,

text: one

text: notices

text: that

text: each

text: triple

text: of

text: observables

text: A

text: B

text: and

text: E

text: is

text: jointly

text: mea-

text: i

text: j

text: surable

text: and

text: therefore,

text: for

text: any

text: pair

text: i,

text: j,

text: there

text: exists

text: the

text: joint

text: probability

text: distribution

text: p(A

text: B

text: E)

text: which

text: must

text: satisfy

text: This

text: gives

text: rise

text: to

text: the

text: following

text: four

text: inequalities

text: i

text: j

text: A

text: B

text: B

text: E

text: A

text: E

text: A

text: B

text: B

text: E

text: A

text: E

text: A

text: B

text: B

text: E

text: A

text: E

text: A

text: B

text: B

text: E

text: A

text: E

text: E

text: By

text: summing

text: these

text: up

text: and

text: using

text: the

text: fact

text: that

text: in

text: a

text: nonsignaling

text: theory

text: XY

text: Z

text: XY

text: for

text: any

text: Z

text: Z

text: one

text: obtains

text: Z

text: Signaling

text: boxes

text: as

text: classical

text: channels

text: Let

text: us

text: assume

text: that

text: correlators

text: B

text: E

text: and

text: B

text: E

text: are

text: equal

text: (later

text: we

text: will

text: show

text: how

text: this

text: assumption

text: can

text: be

text: relaxed).

text: Then

text: the

text: monogamy

text: relation

text: is

text: well

text: deﬁned.

text: It

text: bounds

text: the

text: possible

text: correlations

text: achievable

text: in

text: any

text: no-signaling

text: theory

text: between

text: outcomes

text: of

text: measurements

text: performed

text: by

text: the

text: three

text: parties

text: A,

text: B

text: and

text: E.

text: If

text: it

text: is

text: violated

text: by

text: some

text: probability

text: distribution

text: p,

text: then

text: the

text: latter

text: must

text: violate

text: the

text: no-

text: signalling

text: principle,

text: in

text: which

text: case

text: we

text: call

text: such

text: a

text: box

text: signaling.

text: In

text: other

text: words,

text: if

text: p

text: violates

text: then

text: values

text: of

text: some

text: bipartite

text: correlators

text: become

text: dependent

text: on

text: the

text: measurement

text: choice

text: made

text: by

text: the

text: third

text: party.

text: This

text: dependence

text: allows

text: one

text: to

text: use

text: such

text: signaling

text: boxes

text: to

text: send

text: information

text: from

text: a

text: single

text: party

text: to

text: the

text: remaining

text: two

text: parties.

text: To

text: illustrate

text: this

text: idea,

text: suppose

text: that

text: a

text: box

text: p

text: violates

text: the

text: relation

text: by

text: that

text: is,

text: R(p)

text: I

text: B

text: E

text: AB

text: Then,

text: by

text: adding

text: the

text: inequalities

text: one

text: concludes

text: that

text: A

text: E

text: A

text: E

text: A

text: E

text: A

text: E

text: B

text: E

text: B

text: E

text: ##LTLine##


text: 9 Let us conclude by noting that one can also drop the assumption that the corre- (cid:104) (cid:105) (cid:104) (cid:105) lators B E and B E are equal, in which case the monogamy relation (4) 0 A0 0 A1 reads | | |(cid:104) (cid:105) (cid:104) (cid:105) | ≤ I + B E + B E 4. (33) AB 0 A0 0 A1 Then, following the above methodology one can associate another classical chan- {(cid:104) (cid:105) (cid:104) (cid:105) } nel to the pair S0→ = B E , B E . Our numerics shows, however, A BE 0 A0 0 A1 that an addition of this channel in the deﬁnition of C does not change its value; in ∆ (cid:104) (cid:105) (cid:104) (cid:105) particular, the box (32) realizes C and has the property that B E = B E . ∆ 0 A0 0 A1 5 Generalizing to the chained Bell inequality The above considerations can be applied to a monogamy relation for the gener- alization of the CHSH Bell inequality to any number of dichotomic measurements at both sites—the chained Bell inequality [11]. To recall the latter and the correspond- ing monogamy, let us assume that now A and B have M dichotomic measurements − at their disposal denoted A and B (k = 0, . . . , M 1). The chained Bell inequality k k reads [11]: (cid:88) − M 1 (cid:104) (cid:105) (cid:104) (cid:105) ≤ − IM := ( A B + A B ) 2M 2, (34) AB k k k+1 k k=0 − where we use the convention that A = A . As shown in Ref. [8], it obeys the M 0 following simple monogamy relation for any nonsignaling correlations | | |(cid:104) (cid:105)| ≤ IM + 2 B E 2M, (35) AB 0 where E stands for Eve’s measurement. As before, we can assume that both IM (cid:104) (cid:105) AB and B E are nonnegative, and hence, in what follows we omit the absolute values 0 in (35). To proceed with our considerations we ﬁrst note that analogously to (4), the monogamy (35) can be derived from (6). Precisely, as any three observables A , B i j and E are jointly measurable, due to (6) the following set of 2M inequalities (cid:104) (cid:105) (cid:104) (cid:105) − (cid:104) (cid:105) ≤ A B + B E A E 1, (36) (cid:104) 0 0(cid:105)E (cid:104) 0 (cid:105)A0 − (cid:104) 0 (cid:105)B0 ≤ A B + B E A E 1 (37) 1 0 E 0 A1 1 B0 and (cid:104) (cid:105) − − (cid:104) (cid:105) − (cid:104) (cid:105) ≤ A B ( 1)j B E + ( 1)j A E 1 (38) i+j i E i Ai+j i+j Bi ##LTLine##

text: Let

text: us

text: conclude

text: by

text: noting

text: that

text: one

text: can

text: also

text: drop

text: the

text: assumption

text: that

text: the

text: corre-

text: lators

text: B

text: E

text: and

text: B

text: E

text: are

text: equal,

text: in

text: which

text: case

text: the

text: monogamy

text: relation

text: reads

text: I

text: B

text: E

text: B

text: E

text: AB

text: Then,

text: following

text: the

text: above

text: methodology

text: one

text: can

text: associate

text: another

text: classical

text: chan-

text: nel

text: to

text: the

text: pair

text: B

text: E

text: B

text: E

text: Our

text: numerics

text: shows,

text: however,

text: A

text: BE

text: that

text: an

text: addition

text: of

text: this

text: channel

text: in

text: the

text: deﬁnition

text: of

text: C

text: does

text: not

text: change

text: its

text: value;

text: in

text: particular,

text: the

text: box

text: realizes

text: C

text: and

text: has

text: the

text: property

text: that

text: B

text: E

text: B

text: E

text: Generalizing

text: to

text: the

text: chained

text: Bell

text: inequality

text: The

text: above

text: considerations

text: can

text: be

text: applied

text: to

text: a

text: monogamy

text: relation

text: for

text: the

text: gener-

text: alization

text: of

text: the

text: CHSH

text: Bell

text: inequality

text: to

text: any

text: number

text: of

text: dichotomic

text: measurements

text: at

text: both

text: sites—the

text: chained

text: Bell

text: inequality

text: To

text: recall

text: the

text: latter

text: and

text: the

text: correspond-

text: ing

text: monogamy,

text: let

text: us

text: assume

text: that

text: now

text: A

text: and

text: B

text: have

text: M

text: dichotomic

text: measurements

text: at

text: their

text: disposal

text: denoted

text: A

text: and

text: B

text: (k

text: M

text: The

text: chained

text: Bell

text: inequality

text: k

text: k

text: reads

text: M

text: IM

text: A

text: B

text: A

text: B

text: AB

text: k

text: k

text: k

text: where

text: we

text: use

text: the

text: convention

text: that

text: A

text: A

text: As

text: shown

text: in

text: Ref.

text: it

text: obeys

text: the

text: M

text: following

text: simple

text: monogamy

text: relation

text: for

text: any

text: nonsignaling

text: correlations

text: IM

text: B

text: E

text: AB

text: where

text: E

text: stands

text: for

text: Eve’s

text: measurement.

text: As

text: before,

text: we

text: can

text: assume

text: that

text: both

text: IM

text: AB

text: and

text: B

text: E

text: are

text: nonnegative,

text: and

text: hence,

text: in

text: what

text: follows

text: we

text: omit

text: the

text: absolute

text: values

text: in

text: To

text: proceed

text: with

text: our

text: considerations

text: we

text: ﬁrst

text: note

text: that

text: analogously

text: to

text: the

text: monogamy

text: can

text: be

text: derived

text: from

text: Precisely,

text: as

text: any

text: three

text: observables

text: A

text: B

text: i

text: j

text: and

text: E

text: are

text: jointly

text: measurable,

text: due

text: to

text: the

text: following

text: set

text: of

text: inequalities

text: A

text: B

text: B

text: E

text: A

text: E

text: A

text: B

text: B

text: E

text: A

text: E

text: E

text: and

text: A

text: B

text: B

text: E

text: A

text: E

text: i+j

text: i

text: E

text: i

text: Ai+j

text: i+j

text: Bi

text: ##LTLine##


text: 11 γ(1−ε) W 1 W N + + and 0 1 2 ... N 0 γε W−1 W−N p ≡ 1/(1 + exp Fout). (68) FIG. 8. Uni-cyclic model. The transition rates between state 0 and 1 are associated with an information reservoir, whereas Using a diagrammatic method to obtain the stationary the other transition rates are related to a standard reservoir. Note that we have a cyclic system with N + 1 being the 0 probability distribution [55], we obtain state again. − p = τp + (1 τ)(cid:15) (69) τ A well known result in linear response theory is that the eﬃciency at maximum power for uni-cyclic machines ≡ (cid:48) (cid:48) (cid:48) (cid:48) ≡ (cid:48) ≡ is 1/2 [49]. We now calculate the IP-eﬃciencies at max- where τ k /(k + γ ), with k W−/C and γ −F imum power for uni-cyclic machines. The standard eﬃ- γ/[1 + exp( )]. Note that this formula is similar to out ciency contains the work entering the system in its de- the formula (2) for the two state model of Sec. , which (cid:48) nominator, which corresponds to the work to reset the corresponds to N = 1. The parameter k has dimen- tape appearing in the standard entropy production, as sion of a transition rate and is related to the thermal ≤ ≤ explained in Sec. . Since this work is larger than the transition rates. Therefore, the parameter 0 τ 1 is rate of Shannon entropy change, the IP-eﬃciency at max- dimensionless being 1 (0) if the transitions of the infor- imum power should not be smaller than 1/2. mation reservoir, which are proportional to γ, are much We consider the generic uni-cyclic machine with N + slower (faster) than thermal transitions. 1 states on a ring depicted in Fig. 8. The transition Up to ﬁrst order in the aﬃnities, the current (65) be- rates between states 0 and 1,−which are related to the comes information reservoir, are γ(1 (cid:15)) and γ(cid:15), with the ﬁrst being from 0 to 1. The other transition rates are related − ≈ F − F to standard reservoirs with inverse temperature β = 1, J = γ(P + P )τ(p (cid:15)) Γ( ), (70) 1 0 (cid:15) out and the transition rate from n (n + 1) to n + 1 (n) is Wn + (W−n). We assume that the aﬃnity ≡ where Γ γ(P + P )τ/4. Hence, within linear re- 0 1 F sponse, the rate of extracted work is = ln(W−/W ), (64) (cid:81) out + (cid:81) F F − F F w˙ = J = Γ( ) , (71) where W = N Wn and W− = N W−n, is re- out out (cid:15) out out + n=1 + n=1 lated to work extracted from the system. For the system to operate as a machine the probability and the rate of Shannon entropy change (62) is current from left to right in Fig. 8 must be positive. This probability current is F − F F − τ F − F h˙ = Γ( )[ ( )]. (72) (cid:15) out (cid:15) 2 (cid:15) out − − − J = γ(P (1 (cid:15)) P (cid:15)) = γ(P + P )(p (cid:15)), (65) 0 1 1 0 τ We now maximize the power w˙ with respect to the F F out ≡ oFu∗tput Fout for ﬁxed input (cid:15). The power is maximum at where pτ P0/(P0 + P1). The aﬃnity related to the out = (cid:15)/2, which gives the IP-eﬃciency at maximum information reservoir is power F − ∗ = ln[(1 (cid:15))/(cid:15)]. (66) ∗ ≡ w˙ 1 (cid:15) η h˙o∗ut = 2 − τ/2, (73) It is convenient to d(cid:32)eﬁne (cid:33) where w˙∗ and h˙ ∗ are obtained from (71) and (72) (cid:88) (cid:89) (cid:89) F∗ ouFt with = /2, respectively. The IP-eﬃciency at max- out (cid:15) → − imum power reaches its maximum value 2/3 for τ 1, ≡ N 1 a N C W−n Wm (67) where the transitions related to the information reservoir + are much slower than the thermal transitions. If we had a=0 n=1 m=a+2 ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: γ(1−ε)

font: default

text: W

text: W

text: N

text: and

text: N

text: γε

text: W−N

text: p

text: exp

text: Fout).

text: FIG.

text: Uni-cyclic

label: caption

text: model.

text: The

text: transition

text: rates

text: between

text: state

text: and

text: are

text: associated

text: with

text: an

text: information

text: reservoir,

text: whereas

text: Using

text: a

text: diagrammatic

text: method

text: to

text: obtain

text: the

text: stationary

text: the

text: other

text: transition

text: rates

text: are

text: related

text: to

text: a

text: standard

text: reservoir.

text: Note

text: that

text: we

text: have

text: a

text: cyclic

text: system

text: with

text: N

text: being

text: the

text: probability

text: distribution

text: we

text: obtain

text: state

text: again.

text: p

text: τp

text: τ

text: A

text: well

text: known

text: result

text: in

text: linear

text: response

text: theory

text: is

text: that

text: the

text: eﬃciency

text: at

text: maximum

text: power

text: for

text: uni-cyclic

text: machines

text: is

text: We

text: now

text: calculate

text: the

text: IP-eﬃciencies

text: at

text: max-

text: where

text: τ

text: k

text: /(k

text: γ

text: with

text: k

text: W−/C

text: and

text: γ

text: −F

text: imum

text: power

text: for

text: uni-cyclic

text: machines.

text: The

text: standard

text: eﬃ-

text: exp(

text: Note

text: that

text: this

text: formula

text: is

text: similar

text: to

text: out

text: ciency

text: contains

text: the

text: work

text: entering

text: the

text: system

text: in

text: its

text: de-

text: the

text: formula

text: for

text: the

text: two

text: state

text: model

text: of

text: Sec.

text: which

text: nominator,

text: which

text: corresponds

text: to

text: the

text: work

text: to

text: reset

text: the

text: corresponds

text: to

text: N

text: The

text: parameter

text: k

text: has

text: dimen-

text: tape

text: appearing

text: in

text: the

text: standard

text: entropy

text: production,

text: as

text: sion

text: of

text: a

text: transition

text: rate

text: and

text: is

text: related

text: to

text: the

text: thermal

text: explained

text: in

text: Sec.

text: Since

text: this

text: work

text: is

text: larger

text: than

text: the

text: transition

text: rates.

text: Therefore,

text: the

text: parameter

text: τ

text: is

text: rate

text: of

text: Shannon

text: entropy

text: change,

text: the

text: IP-eﬃciency

text: at

text: max-

text: dimensionless

text: being

text: if

text: the

text: transitions

text: of

text: the

text: infor-

text: imum

text: power

text: should

text: not

text: be

text: smaller

text: than

text: mation

text: reservoir,

text: which

text: are

text: proportional

text: to

text: γ,

text: are

text: much

text: We

text: consider

text: the

text: generic

text: uni-cyclic

text: machine

text: with

text: N

text: slower

text: (faster)

text: than

text: thermal

text: transitions.

text: states

text: on

text: a

text: ring

text: depicted

text: in

text: Fig.

text: The

text: transition

text: Up

text: to

text: ﬁrst

text: order

text: in

text: the

text: aﬃnities,

text: the

text: current

text: be-

text: rates

text: between

text: states

text: and

text: are

text: related

text: to

text: the

text: comes

text: information

text: reservoir,

text: are

text: and

text: with

text: the

text: ﬁrst

text: being

text: from

text: to

text: The

text: other

text: transition

text: rates

text: are

text: related

text: F

text: F

text: to

text: standard

text: reservoirs

text: with

text: inverse

text: temperature

text: β

text: J

text: γ(P

text: P

text: )τ(p

text: Γ(

text: out

text: and

text: the

text: transition

text: rate

text: from

text: n

text: (n

text: to

text: n

text: (n)

text: is

text: Wn

text: (W−n).

text: We

text: assume

text: that

text: the

text: aﬃnity

text: where

text: Γ

text: γ(P

text: P

text: Hence,

text: within

text: linear

text: re-

text: F

text: sponse,

text: the

text: rate

text: of

text: extracted

text: work

text: is

text: ln(W−/W

text: out

text: F

text: F

text: F

text: F

text: w˙

text: J

text: Γ(

text: where

text: W

text: N

text: Wn

text: and

text: W−

text: N

text: W−n,

text: is

text: re-

text: out

text: out

text: out

text: out

text: lated

text: to

text: work

text: extracted

text: from

text: the

text: system.

text: For

text: the

text: system

text: to

text: operate

text: as

text: a

text: machine

text: the

text: probability

text: and

text: the

text: rate

text: of

text: Shannon

text: entropy

text: change

text: is

text: current

text: from

text: left

text: to

text: right

text: in

text: Fig.

text: must

text: be

text: positive.

text: This

text: probability

text: current

text: is

text: F

text: F

text: F

text: τ

text: F

text: F

text: h˙

text: Γ(

text: out

text: out

text: J

text: γ(P

text: P

text: γ(P

text: P

text: )(p

text: τ

text: We

text: now

text: maximize

text: the

text: power

text: w˙

text: with

text: respect

text: to

text: the

text: F

text: F

text: out

text: oFu∗tput

text: Fout

text: for

text: ﬁxed

text: input

text: The

text: power

text: is

text: maximum

text: at

text: where

text: pτ

text: The

text: aﬃnity

text: related

text: to

text: the

text: out

text: which

text: gives

text: the

text: IP-eﬃciency

text: at

text: maximum

text: information

text: reservoir

text: is

text: power

text: F

text: w˙

text: η

text: h˙o∗ut

text: It

text: is

text: convenient

text: to

text: where

text: w˙∗

text: and

text: h˙

text: are

text: obtained

text: from

text: and

text: F∗

text: ouFt

text: with

text: respectively.

text: The

text: IP-eﬃciency

text: at

text: max-

text: out

text: imum

text: power

text: reaches

text: its

text: maximum

text: value

text: for

text: τ

text: N

text: a

text: N

text: C

text: W−n

text: Wm

text: where

text: the

text: transitions

text: related

text: to

text: the

text: information

text: reservoir

text: are

text: much

text: slower

text: than

text: the

text: thermal

text: transitions.

text: If

text: we

text: had

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 13   (cid:88) (cid:88)   Note that λ(i ) = λ(i ). A B ≡ K W K R A stochastic∈trajectory in the duplicated system for a Ω[XT ] ij[XT ] ln ij + ii[XT ] ln i . time interval t [0, T] is deno∈ted XT = (x0, τ0; x1, τ1; . . . ; xN, τN), i j(cid:54)=i Wji Ri where xn is the state for t [tn, tn + τn], with t0 = 0, (A8) t = t + τ , and t = T. The probability of a n+1 n n N+1 trajectory is If the constraint (A6) is satisﬁed, by choosing uniform (cid:32) (cid:33) distributions for both P(x ) and P˜(x ) in Eq. (A5), we (cid:89) (cid:89) 0 N obtain − P N 1 N − [X ] = P(x ) W exp( λ(x )τ ) P T 0 n=0 xnxn+1 n=0 n (nA3) P[[XX˜TT ]] = exp (Ω[XT ]) . (A9) where P(x0) denotes the initial probability. The prob- This relatio(cid:88)n then implies (cid:88) ability of the reversed trajectory X˜ = (x , τ ; . . . ; x , τ ; x , τ ) T N N 1 1 0 0 with modiﬁed transition rates Wij (or Ri is the jump is (cid:104) − (cid:105) ≡ − P P between iA and iB) reads exp( Ω) exp ( Ω[XT ]) [XT ] = [X˜T ] = 1, (cid:32) (cid:33) (cid:80) XT XT (cid:89) (cid:89) (A10) − P N 1 N − where represents an integral over all stochastic [X˜T ] = P˜(xN) Wxn+1xn exp( λ(xn)τn), trajectories.XTThis integral ﬂuctuation theorem leads to n=0 n=0 the inequality (A4) where P˜(x ) is the initial probability of the reversed (cid:104) (cid:105) ≥ N Ω /T 0. (A11) trajectory and λ(x ) is the escape rate for the modiﬁed n rates. The ratio of trajecto(cid:18)ry probabilities t(cid:19)hen becomes (cid:81) (cid:104) (cid:105) The above inequality is equivalent to (26), as Ω /T = P P[XT ] = (cid:81)P(x0) N−1 Wxnxn+1 ωs˙y.mWmeetnroicte, it.eh.a,titthcaenfnuontctbieonwarlitΩteins,aisnagseunmeraolf,pnrootbaanbtili-- [X˜T ] P˜(xN) n=0 Wxn+1xn ity currents. It does become antisymmetric if the auxil- × − (cid:104) (cid:105) N exp[(λ(x ) λ(x ))τ )]. (A5) iary rates are chosen so that Ω /T becomes the standard n=0 n n n entropy production but for auxiliary rates leading to the IP-entropy production it does not. − From Eq. (A2), we obtain that the term λ(x ) λ(x ) = n n Whereas the standard entropy production s˙ can be ob- 0 if tained from a ﬂuctuation theorem for the original system (cid:88) (cid:88) [49], in order to obtain the IP-entropy production s˙ we 1 need this ﬂuctuation theorem for the duplicated system. This duplication has a physical interpretation if we com- R + W = R + W , (A6) i ij i ij pare Fig. 3 with Fig. 4 for the paradigmatic model of (cid:54) (cid:54) j=i j=i Sec. . The duplication in Fig. 3 is necessary to include the possibility of transitions from u to u and d to A B A d , which corresponds to transitions where the new in- B which is the constraint (24). coming bit is in a state that couples to the state of the The activity for jumps from i to j and from i to system. Note that in the duplicated system of Fig 3 the A B B j is a functional of the the trajectory X deﬁned as states in the same replica are connected by the thermal A T transition link, which is diﬀerent from the duplication in (cid:88) (cid:0) (cid:1) this appendix. If a set of links is assumed to be related to standard reservoirs, then a duplicated system keeping K ≡ N these links connecting states in the same replica and not [X ] δ δ + δ δ . (A7) ij T xn,iA xn+1,jB xn,iB xn+1,jA in diﬀerent replicas suﬃces to obtain a ﬂuctuation theo- n=0 rem leading to the corresponding s˙ [46]. The derivation 1 of the ﬂuctuation theorem in this case is very similar. Here, we have chosen the duplication scheme above in With this activity we deﬁne the functional order to obtain the most general inequality. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Note

text: that

text: λ(i

text: λ(i

text: A

text: B

text: K

text: W

text: K

text: R

text: A

text: stochastic∈trajectory

text: in

text: the

text: duplicated

text: system

text: for

text: a

text: Ω[XT

text: ij[XT

text: ln

text: ij

text: ii[XT

text: ln

text: i

text: time

text: interval

text: t

text: T]

text: is

text: deno∈ted

text: XT

text: xN,

text: τN),

text: i

text: Wji

text: Ri

text: where

text: xn

text: is

text: the

text: state

text: for

text: t

text: [tn,

text: tn

text: τn],

text: with

text: t

text: t

text: τ

text: and

text: t

text: T.

text: The

text: probability

text: of

text: a

text: n

text: n

text: trajectory

text: is

text: If

text: the

text: constraint

text: is

text: satisﬁed,

text: by

text: choosing

text: uniform

text: distributions

text: for

text: both

text: P(x

text: and

text: P˜(x

text: in

text: Eq.

text: we

text: N

text: obtain

text: P

text: N

text: N

text: [X

text: P(x

text: W

text: exp(

text: λ(x

text: )τ

text: P

text: T

text: n

text: P[[XX˜TT

text: exp

text: (Ω[XT

text: where

text: denotes

text: the

text: initial

text: probability.

text: The

text: prob-

text: This

text: then

text: implies

text: ability

text: of

text: the

text: reversed

text: trajectory

text: X˜

text: (x

text: τ

text: x

text: τ

text: x

text: τ

text: T

text: N

text: N

text: with

text: modiﬁed

text: transition

text: rates

text: Wij

text: (or

text: Ri

text: is

text: the

text: jump

text: is

text: P

text: P

text: between

text: iA

text: and

text: iB)

text: reads

text: exp(

text: Ω)

text: exp

text: Ω[XT

text: [XT

text: [X˜T

text: XT

text: XT

text: P

text: N

text: N

text: where

text: represents

text: an

text: integral

text: over

text: all

text: stochastic

text: [X˜T

text: P˜(xN)

text: exp(

text: λ(xn)τn),

text: trajectories.XTThis

text: integral

text: ﬂuctuation

text: theorem

text: leads

text: to

text: the

text: inequality

text: where

text: P˜(x

text: is

text: the

text: initial

text: probability

text: of

text: the

text: reversed

text: N

text: Ω

text: /T

text: trajectory

text: and

text: λ(x

text: is

text: the

text: escape

text: rate

text: for

text: the

text: modiﬁed

text: n

text: rates.

text: The

text: ratio

text: of

text: probabilities

text: becomes

text: The

text: above

text: inequality

text: is

text: equivalent

text: to

text: as

text: Ω

text: /T

text: P

text: P[XT

text: ωs˙y.mWmeetnroicte,

text: [X˜T

text: P˜(xN)

text: ity

text: currents.

text: It

text: does

text: become

text: antisymmetric

text: if

text: the

text: auxil-

text: N

text: exp[(λ(x

text: λ(x

text: ))τ

text: iary

text: rates

text: are

text: chosen

text: so

text: that

text: Ω

text: /T

text: becomes

text: the

text: standard

text: n

text: n

text: n

text: entropy

text: production

text: but

text: for

text: auxiliary

text: rates

text: leading

text: to

text: the

text: IP-entropy

text: production

text: it

text: does

text: not.

text: From

text: Eq.

text: we

text: obtain

text: that

text: the

text: term

text: λ(x

text: λ(x

text: n

text: n

text: Whereas

text: the

text: standard

text: entropy

text: production

text: s˙

text: can

text: be

text: ob-

text: if

text: tained

text: from

text: a

text: ﬂuctuation

text: theorem

text: for

text: the

text: original

text: system

text: in

text: order

text: to

text: obtain

text: the

text: IP-entropy

text: production

text: s˙

text: we

text: need

text: this

text: ﬂuctuation

text: theorem

text: for

text: the

text: duplicated

text: system.

text: This

text: duplication

text: has

text: a

text: physical

text: interpretation

text: if

text: we

text: com-

text: R

text: W

text: R

text: W

text: i

text: ij

text: i

text: ij

text: pare

text: Fig.

text: with

text: Fig.

text: for

text: the

text: paradigmatic

text: model

text: of

text: j=i

text: j=i

text: Sec.

text: The

text: duplication

text: in

text: Fig.

text: is

text: necessary

text: to

text: include

text: the

text: possibility

text: of

text: transitions

text: from

text: u

text: to

text: u

text: and

text: d

text: to

text: A

text: B

text: A

text: d

text: which

text: corresponds

text: to

text: transitions

text: where

text: the

text: new

text: in-

text: B

text: which

text: is

text: the

text: constraint

text: coming

text: bit

text: is

text: in

text: a

text: state

text: that

text: couples

text: to

text: the

text: state

text: of

text: the

text: The

text: activity

text: for

text: jumps

text: from

text: i

text: to

text: j

text: and

text: from

text: i

text: to

text: system.

text: Note

text: that

text: in

text: the

text: duplicated

text: system

text: of

text: Fig

text: the

text: A

text: B

text: B

text: j

text: is

text: a

text: functional

text: of

text: the

text: the

text: trajectory

text: X

text: deﬁned

text: as

text: states

text: in

text: the

text: same

text: replica

text: are

text: connected

text: by

text: the

text: thermal

text: A

text: T

text: transition

text: link,

text: which

text: is

text: diﬀerent

text: from

text: the

text: duplication

text: in

text: this

text: appendix.

text: If

text: a

text: set

text: of

text: links

text: is

text: assumed

text: to

text: be

text: related

text: to

text: standard

text: reservoirs,

text: then

text: a

text: duplicated

text: system

text: keeping

text: K

text: N

text: these

text: links

text: connecting

text: states

text: in

text: the

text: same

text: replica

text: and

text: not

text: [X

text: δ

text: δ

text: δ

text: δ

text: ij

text: T

text: xn,iA

text: xn,iB

text: in

text: diﬀerent

text: replicas

text: suﬃces

text: to

text: obtain

text: a

text: ﬂuctuation

text: theo-

text: rem

text: leading

text: to

text: the

text: corresponding

text: s˙

text: The

text: derivation

text: of

text: the

text: ﬂuctuation

text: theorem

text: in

text: this

text: case

text: is

text: very

text: similar.

text: Here,

text: we

text: have

text: chosen

text: the

text: duplication

text: scheme

text: above

text: in

text: With

text: this

text: activity

text: we

text: deﬁne

text: the

text: functional

text: order

text: to

text: obtain

text: the

text: most

text: general

text: inequality.

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: 8 can move particles against the chemical potential. This ≡ − − − ∝ − corresponds to extracting work from a single heat bath, J γ[(1 (cid:15))Pc (cid:15)Pa] = γ(Pa+Pc)[pτ (cid:15)] (p (cid:15)), (36) which was also the case of the model from Sec. . Second, − for the case where the temperature gradient β β drives 2 1 the particles against ∆µ, the pseudo-eﬃciency where ≡ − − η w˙/( q˙ ) (41) ≡ 1 ≤ ps 1 p τli→m1 pτ = 1 + e(β1−β2)E 1/2. (37) ≤ becomes Restricting to (cid:15) 1/2, for p > (cid:15) the probability current in Eq. (36) is positive leading to heat ﬂowing from the cold to the hot reservoir. More precisely, the IP-entropy ≤ h˙ η η + − , (42) production (18) becomes ps c β ( q˙ ) 2 1 ≡ − − − − where η 1 β /β is the Carnot eﬃciency. Hence s˙ = γ(P + P )[H(p ) H((cid:15))] q˙(β β ), (38) c 1 2 1 a c τ 1 2 this pseudo-eﬃciency can exceeded the Carnot eﬃciency η . Actually, it can even exceed 1 as demonstrated below. c A relation similar to (42) has also been obtained in [29, where q˙ = JE is the rate at which heat ﬂows from 45] using diﬀerent frameworks. the cold to the hot reservoir. The refrigerator mode of operation (p > (cid:15)) is powered by the tape, which has As a speciﬁc model describing such situation we take its Shannon entropy increased from H((cid:15)) to H(p ). For the three-state model from Fig. 6. We now assume that τ p < (cid:15) the probability current J becomes negative and in state b the number of particles in the system is N = 1 heat ﬂows from the hot to the cold reservoir. In this and in states a and c it is N = 0. The local detailed case information is erased from the tape and the rate of balance relation must be modiﬁed to Shannon entropy decrease of the tape is compensated by the rate of entropy increase of the external environment − ≤ W − − W − d−ue to−the heat ﬂow, i.e., γ(Pa + Pc)[H((cid:15)) H(pτ)] ln ab = β (E µ ) and ln bc = β (E µ ), q˙(β1 β2). Wba 1 1 Wcb 2 2 (43) B. Thermoelectric machine interacting with a tape where µ and µ are chemical potentials. We set chaWnegensopwarctoicnlseisdewritthhethceassetawndhaerrde rtehseersvyositresm. Tahlseoseyxs-- tkhee(Ese−µt1r)aβn1s/1i2t,ioWnbcra=t2ekset(oE−Wµ2a)bβ2=/2,kaen−d(EW−cµb1=)β1k/2e,−(WE−baµ2=)β2/2. tem is in contact with a reservoir at inverse temperature The transition rates between a and c are mediated by an β and chemical potential µ , another reservoir charac- 1 1 information reservoir and are as in the model from Sec. terized by β and µ , and an information reservoir. The 2 2 ≡ − ≥ . chemical potentials fulﬁll ∆µ µ µ 0, where 1 is 2 1 ≥ Calculating the stationary distribution we obtain assumed to be the hot reservoir, i.e., β β . The ﬁrst 2 1 law (17) is reduced to − ≡ P C τp + C (1 τ)(cid:15) p c = 1 2 − , (44) − − − τ P + P C τ + C (1 τ) q˙ q˙ = w˙. (39) a c 1 2 2 1 ≡ ≡ − w≡here τ k/(k + γ), C1 eβ2E+β1µ1 + eβ1E+β2µ2, where w˙ = N˙ ∆µ is the rate of work extracted from C2 e(β1+β2)E/2(eβ2E/2+β1µ1/2 + eβ1E/2+β2µ2/2), and the system to move particles against the chemical poten- tial gradient ∆µ (from 1 to 2) at a rate N˙ , and q˙ (q˙ ) is 1 2 ≡ − − the rate of dissipated heat related to reservoir 1 (2). The p 1/(1 + eβ2[(µ2 µ1)+ηc(µ1 E)]). (45) IP-entropy production (18) for this case reads ≥ The probability current is again s˙ = h˙ + β q˙ + β q˙ 0, (40) 1 1 1 2 2 ≡ − − − ∝ − J γ[(1 (cid:15))P (cid:15)P ] = γ(P +P )[p (cid:15)] (p (cid:15)). (46) where h˙ is the rate of Shannon entropy change given c a a c τ in Eq. (19). First, we note that if β = β a positive h˙ 1 2 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: can

text: move

text: particles

text: against

text: the

text: chemical

text: potential.

text: This

text: corresponds

text: to

text: extracting

text: work

text: from

text: a

text: single

text: heat

text: bath,

text: J

text: γ(Pa+Pc)[pτ

text: (p

text: which

text: was

text: also

text: the

text: case

text: of

text: the

text: model

text: from

text: Sec.

text: Second,

text: for

text: the

text: case

text: where

text: the

text: temperature

text: gradient

text: β

text: β

text: drives

text: the

text: particles

text: against

text: ∆µ,

text: the

text: pseudo-eﬃciency

text: where

text: η

text: w˙/(

text: q˙

text: ps

text: p

text: pτ

text: becomes

text: Restricting

text: to

text: for

text: p

text: the

text: probability

text: current

text: in

text: Eq.

text: is

text: positive

text: leading

text: to

text: heat

text: ﬂowing

text: from

text: the

text: cold

text: to

text: the

text: hot

text: reservoir.

text: More

text: precisely,

text: the

text: IP-entropy

text: h˙

text: η

text: η

text: production

text: becomes

text: ps

text: c

text: β

text: q˙

text: where

text: η

text: β

text: /β

text: is

text: the

text: Carnot

text: eﬃciency.

text: Hence

text: s˙

text: γ(P

text: P

text: )[H(p

text: q˙(β

text: β

text: c

text: a

text: c

text: τ

text: this

text: pseudo-eﬃciency

text: can

text: exceeded

text: the

text: Carnot

text: eﬃciency

text: η

text: Actually,

text: it

text: can

text: even

text: exceed

text: as

text: demonstrated

text: below.

text: c

text: A

text: relation

text: similar

text: to

text: has

text: also

text: been

text: obtained

text: in

text: where

text: q˙

text: JE

text: is

text: the

text: rate

text: at

text: which

text: heat

text: ﬂows

text: from

text: using

text: diﬀerent

text: frameworks.

text: the

text: cold

text: to

text: the

text: hot

text: reservoir.

text: The

text: refrigerator

text: mode

text: of

text: operation

text: (p

text: is

text: powered

text: by

text: the

text: tape,

text: which

text: has

text: As

text: a

text: speciﬁc

text: model

text: describing

text: such

text: situation

text: we

text: take

text: its

text: Shannon

text: entropy

text: increased

text: from

text: to

text: H(p

text: For

text: the

text: three-state

text: model

text: from

text: Fig.

text: We

text: now

text: assume

text: that

text: τ

text: p

text: the

text: probability

text: current

text: J

text: becomes

text: negative

text: and

text: in

text: state

text: b

text: the

text: number

text: of

text: particles

text: in

text: the

text: system

text: is

text: N

text: heat

text: ﬂows

text: from

text: the

text: hot

text: to

text: the

text: cold

text: reservoir.

text: In

text: this

text: and

text: in

text: states

text: a

text: and

text: c

text: it

text: is

text: N

text: The

text: local

text: detailed

text: case

text: information

text: is

text: erased

text: from

text: the

text: tape

text: and

text: the

text: rate

text: of

text: balance

text: relation

text: must

text: be

text: modiﬁed

text: to

text: Shannon

text: entropy

text: decrease

text: of

text: the

text: tape

text: is

text: compensated

text: by

text: the

text: rate

text: of

text: entropy

text: increase

text: of

text: the

text: external

text: environment

text: W

text: W

text: d−ue

text: to−the

text: heat

text: ﬂow,

text: i.e.,

text: γ(Pa

text: H(pτ)]

text: ln

text: ab

text: β

text: (E

text: µ

text: and

text: ln

text: bc

text: β

text: (E

text: µ

text: Wba

text: Wcb

text: B.

text: Thermoelectric

text: machine

text: interacting

text: with

text: a

text: tape

text: where

text: µ

text: and

text: µ

text: are

text: chemical

text: potentials.

text: We

text: set

text: rtehseersvyositresm.

text: Tahlseoseyxs--

text: tem

text: is

text: in

text: contact

text: with

text: a

text: reservoir

text: at

text: inverse

text: temperature

text: The

text: transition

text: rates

text: between

text: a

text: and

text: c

text: are

text: mediated

text: by

text: an

text: β

text: and

text: chemical

text: potential

text: µ

text: another

text: reservoir

text: charac-

text: information

text: reservoir

text: and

text: are

text: as

text: in

text: the

text: model

text: from

text: Sec.

text: terized

text: by

text: β

text: and

text: µ

text: and

text: an

text: information

text: reservoir.

text: The

text: chemical

text: potentials

text: fulﬁll

text: ∆µ

text: µ

text: µ

text: where

text: is

text: Calculating

text: the

text: stationary

text: distribution

text: we

text: obtain

text: assumed

text: to

text: be

text: the

text: hot

text: reservoir,

text: i.e.,

text: β

text: β

text: The

text: ﬁrst

text: law

text: is

text: reduced

text: to

text: P

text: C

text: τp

text: C

text: p

text: c

text: τ

text: P

text: P

text: C

text: τ

text: C

text: τ)

text: q˙

text: q˙

text: w˙.

text: a

text: c

text: w≡here

text: τ

text: k/(k

text: γ),

text: where

text: w˙

text: N˙

text: ∆µ

text: is

text: the

text: rate

text: of

text: work

text: extracted

text: from

text: and

text: the

text: system

text: to

text: move

text: particles

text: against

text: the

text: chemical

text: poten-

text: tial

text: gradient

text: ∆µ

text: (from

text: to

text: at

text: a

text: rate

text: N˙

text: and

text: q˙

text: (q˙

text: is

text: the

text: rate

text: of

text: dissipated

text: heat

text: related

text: to

text: reservoir

text: The

text: p

text: E)]).

text: IP-entropy

text: production

text: for

text: this

text: case

text: reads

text: The

text: probability

text: current

text: is

text: again

text: s˙

text: h˙

text: β

text: q˙

text: β

text: q˙

text: J

text: γ(P

text: +P

text: )[p

text: (p

text: where

text: h˙

text: is

text: the

text: rate

text: of

text: Shannon

text: entropy

text: change

text: given

text: c

text: a

text: a

text: c

text: τ

text: in

text: Eq.

text: First,

text: we

text: note

text: that

text: if

text: β

text: β

text: a

text: positive

text: h˙

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##