text: EXTREMAL FUNCTIONS FOR REAL CONVEX BODIES 13 Since r (0, 0) = r (x(0), y(0)) = 0 and r (0, 0) = 1, the right-hand side of x x y (??) is the average of the radii of the osculating circles of ∂K at a = (0, 0) and b = (x(0), y(0)). We claim that the left-hand side of (??) is the average of the radii of the osculating (cid:0) (cid:1) circles of CT at a and b. To see this, note from (??) and (??) (and ∂∂2θF22 (θ0) = y0), that − − ∂∂Fθ1 (θ0) = ρ−( α sin θ0 cos ψ β cos θ0 sin ψ)2 . y0 (cid:0) (α cos θ(cid:1)0 sin ψ + β sin θ0 cos ψ) Let us rotate coordinates so that sin ψ = 1; i.e., ψ = π, and assume θ = 0. Then 2 0 ∂∂Fθ1 (θ0) 2 = ρβ2 . y α 0 On the other hand, C now has the parametrization T − F (θ) = ρβ sin θ + x , F (θ) = ρα cos θ + y , 1 0 2 0 and the curvature of C as a function of θ is T 1 · αβ κ(θ) = . ρ (β2 cos2(θ) + α2 sin2 θ)3/2 At θ = 0, π we get (cid:16) (cid:17) α κ(0) = κ(π) = β2ρ · as claimed (precisely, 1 1 + 1 = 1 2β2ρ = ρβ2 .) 2 κ(0) κ(π) 2 α α Remark 3.5. (cid:12) (i) Note that the(cid:12) radii of the osculating circles for ∂K at the points a, b are at least as large as those for C since C is inscribed in ∂K. Thus the T T condition ∂A = 0 fails if the curvature of C is strictly less than that ∂s s=0 T of ∂K at either a or b. (ii) A degenerate ellipse (i.e. line segment) occurs when β = 0. A careful examination of the preceding calculations shows that (??) always holds in this case. When ∂K is Cr, the implicit function theorem shows that s and ρ can be solved ∈ in terms of γ, ψ (equivalently, c H∞) as Cr functions. This implies that locally, the center (x (s), y (s)) is a Cr function of c, and must therefore coincide with a(c) 0 0 given in equation (??). Similarly, the scale factor ρ(c) is also a Cr function of c. Altogether, this shows that the foliation of extremal ellipses near C is Cr. ∈ C \ T The smoothness of the foliation at a point z n K in turn implies the smoothness of V at z, as the partial derivatives ∂/∂z may be computed explicitly K j in terms of foliation parameters using the chain rule. We summarize this in the following theorem. ∈ C \ Theorem 3.6. Suppose z 2 K lies on an extremal ellipse C for K with the following properties: ∩ (1) The intersection ∂K C is exactly two points. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: EXTREMAL

text: FUNCTIONS

text: FOR

text: REAL

text: CONVEX

text: BODIES

text: Since

text: r

text: r

text: and

text: r

text: the

text: right-hand

text: side

text: of

text: x

text: x

text: y

text: is

text: the

text: average

text: of

text: the

text: radii

text: of

text: the

text: osculating

text: circles

text: of

text: ∂K

text: at

text: a

text: and

text: b

text: We

text: claim

text: that

text: the

text: left-hand

text: side

text: of

text: is

text: the

text: average

text: of

text: the

text: radii

text: of

text: the

text: osculating

text: circles

text: of

text: CT

text: at

text: a

text: and

text: b.

text: To

text: see

text: this,

text: note

text: from

text: and

text: (and

text: that

text: ∂∂Fθ1

text: ρ−(

text: α

text: sin

text: θ0

text: cos

text: ψ

text: β

text: cos

text: θ0

text: sin

text: (α

text: cos

text: sin

text: ψ

text: β

text: sin

text: θ0

text: cos

text: ψ)

text: Let

text: us

text: rotate

text: coordinates

text: so

text: that

text: sin

text: ψ

text: i.e.,

text: ψ

text: π,

text: and

text: assume

text: θ

text: Then

text: ∂∂Fθ1

text: ρβ2

text: y

text: α

text: On

text: the

text: other

text: hand,

text: C

text: now

text: has

text: the

text: parametrization

text: T

text: F

text: (θ)

text: ρβ

text: sin

text: θ

text: x

text: F

text: (θ)

text: ρα

text: cos

text: θ

text: y

text: and

text: the

text: curvature

text: of

text: C

text: as

text: a

text: function

text: of

text: θ

text: is

text: T

text: αβ

text: κ(θ)

text: ρ

text: α2

text: At

text: θ

text: π

text: we

text: get

text: α

text: κ(π)

text: β2ρ

text: as

text: claimed

text: (precisely,

text: 2β2ρ

text: ρβ2

text: κ(π)

text: α

text: α

text: Remark

text: (i)

text: Note

text: that

text: radii

text: of

text: the

text: osculating

text: circles

text: for

text: ∂K

text: at

text: the

text: points

text: a,

text: b

text: are

text: at

text: least

text: as

text: large

text: as

text: those

text: for

text: C

text: since

text: C

text: is

text: inscribed

text: in

text: ∂K.

text: Thus

text: the

text: T

text: T

text: condition

text: ∂A

text: fails

text: if

text: the

text: curvature

text: of

text: C

text: is

text: strictly

text: less

text: than

text: that

text: ∂s

text: T

text: of

text: ∂K

text: at

text: either

text: a

text: or

text: b.

text: (ii)

text: A

text: degenerate

text: ellipse

text: (i.e.

text: line

text: segment)

text: occurs

text: when

text: β

text: A

text: careful

text: examination

text: of

text: the

text: preceding

text: calculations

text: shows

text: that

text: always

text: holds

text: in

text: this

text: case.

text: When

text: ∂K

text: is

text: Cr,

text: the

text: implicit

text: function

text: theorem

text: shows

text: that

text: s

text: and

text: ρ

text: can

text: be

text: solved

text: in

text: terms

text: of

text: γ,

text: ψ

text: (equivalently,

text: c

text: H∞)

text: as

text: Cr

text: functions.

text: This

text: implies

text: that

text: locally,

text: the

text: center

text: (x

text: (s),

text: y

text: (s))

text: is

text: a

text: Cr

text: function

text: of

text: c,

text: and

text: must

text: therefore

text: coincide

text: with

text: a(c)

text: given

text: in

text: equation

text: Similarly,

text: the

text: scale

text: factor

text: ρ(c)

text: is

text: also

text: a

text: Cr

text: function

text: of

text: c.

text: Altogether,

text: this

text: shows

text: that

text: the

text: foliation

text: of

text: extremal

text: ellipses

text: near

text: C

text: is

text: Cr.

text: C

text: T

text: The

text: smoothness

text: of

text: the

text: foliation

text: at

text: a

text: point

text: z

text: n

text: K

text: in

text: turn

text: implies

text: the

text: smoothness

text: of

text: V

text: at

text: z,

text: as

text: the

text: partial

text: derivatives

text: ∂/∂z

text: may

text: be

text: computed

text: explicitly

text: K

text: j

text: in

text: terms

text: of

text: foliation

text: parameters

text: using

text: the

text: chain

text: rule.

text: We

text: summarize

text: this

text: in

text: the

text: following

text: theorem.

text: C

text: Theorem

text: Suppose

text: z

text: K

text: lies

text: on

text: an

text: extremal

text: ellipse

text: C

text: for

text: K

text: with

text: the

text: following

text: properties:

text: The

text: intersection

text: ∂K

text: C

text: is

text: exactly

text: two

text: points.

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: 20 D. BURNS, N. LEVENBERG AND S. MA‘U Figure 4. The construction of K and K by locally modifying ∩ 1 2 { } ∂K. The real ellipse C = C K intersects ∂K in b, c, d and { } T 1 ∂K in a, b, c . Equation (??) holds near C because an extremal 2 ellipse for K whose parameters are suﬃciently close to those of C is also an extremal ellipse for one of the K ’s. j locally a maximum of smooth functions by considering enlargements of K to convex sets whose boundaries each meet C in three points, and then taking their extremal T functions. The same argument can also be used to get local smooth approximations: given ∈ C \ (cid:15) > 0 and z 2 K, one can construct a convex set K with the property that for 0 (cid:15) ≤ ≤ all z in some neighborhood of z , V is smooth and V (z) V (z) V (z) + (cid:15). 0 K(cid:15) K(cid:15) K K(cid:15) For any compact convex set K, it seems plausible that one can make a ﬁnite number of local boundary modiﬁcations as in Figure ?? to remove the ‘bad’ condi- tions listed in the proof of Theorem ??, at least q.e.1 This motivates the following conjecture. ⊂ R Conjecture. Let K 2 be a convex body with smooth boundary. { } (1) There is a ﬁnite collection K of convex bodies with the property that V C \ j Kj is smooth q.e. on 2 K for each j, and V = max V . j K j Kj (2) Given (cid:15) > 0 there is a convex body K such that V is smooth q.e. on C \ | − | (cid:15) ∈ C K(cid:15) 2 K and V (z) V (z) < (cid:15) for all z 2. (cid:15) K K(cid:15) 1Recall that a property holds q.e. = quasi-everywhere if it holds everywhere outside a (possibly nonempty) pluripolar set. ##LTFigure## ##LTLine##

label: paragraph

text: D.

text: BURNS,

text: N.

text: LEVENBERG

text: AND

text: S.

text: MA‘U

text: Figure

text: The

label: caption

text: construction

text: of

text: K

text: and

text: K

text: by

text: locally

text: modifying

text: ∂K.

text: The

text: real

text: ellipse

text: C

text: C

text: K

text: intersects

text: ∂K

text: in

text: b,

text: c,

text: d

text: and

text: T

text: ∂K

text: in

text: a,

text: b,

text: c

text: Equation

text: holds

text: near

text: C

text: because

text: an

text: extremal

text: ellipse

text: for

text: K

text: whose

text: parameters

text: are

text: suﬃciently

text: close

text: to

text: those

text: of

text: C

text: is

text: also

text: an

text: extremal

text: ellipse

text: for

text: one

text: of

text: the

text: K

text: ’s.

text: j

text: locally

text: a

text: maximum

text: of

text: smooth

text: functions

text: by

text: considering

text: enlargements

text: of

text: K

text: to

text: convex

text: sets

text: whose

text: boundaries

text: each

text: meet

text: C

text: in

text: three

text: points,

text: and

text: then

text: taking

text: their

text: extremal

text: T

text: functions.

text: The

text: same

text: argument

text: can

text: also

text: be

text: used

text: to

text: get

text: local

text: smooth

text: approximations:

text: given

text: C

text: and

text: z

text: K,

text: one

text: can

text: construct

text: a

text: convex

text: set

text: K

text: with

text: the

text: property

text: that

text: for

text: all

text: z

text: in

text: some

text: neighborhood

text: of

text: z

text: V

text: is

text: smooth

text: and

text: V

text: (z)

text: V

text: (z)

text: V

text: (z)

text: K

text: For

text: any

text: compact

text: convex

text: set

text: K,

text: it

text: seems

text: plausible

text: that

text: one

text: can

text: make

text: a

text: ﬁnite

text: number

text: of

text: local

text: boundary

text: modiﬁcations

text: as

text: in

text: Figure

text: to

text: remove

text: the

text: ‘bad’

text: condi-

text: tions

text: listed

text: in

text: the

text: proof

text: of

text: Theorem

text: at

text: least

text: This

text: motivates

text: the

text: following

text: conjecture.

text: R

text: Conjecture.

text: Let

text: K

text: be

text: a

text: convex

text: body

text: with

text: smooth

text: boundary.

text: There

text: is

text: a

text: ﬁnite

text: collection

text: K

text: of

text: convex

text: bodies

text: with

text: the

text: property

text: that

text: V

text: C

text: j

text: Kj

text: is

text: smooth

text: q.e.

text: on

text: K

text: for

text: each

text: j,

text: and

text: V

text: max

text: V

text: j

text: K

text: j

text: Kj

text: Given

text: there

text: is

text: a

text: convex

text: body

text: K

text: such

text: that

text: V

text: is

text: smooth

text: q.e.

text: on

text: C

text: C

text: K

text: and

text: V

text: (z)

text: V

text: (z)

text: for

text: all

text: z

text: K

text: that

text: a

text: property

text: holds

text: q.e.

text: quasi-everywhere

text: if

text: it

text: holds

text: everywhere

text: outside

text: a

text: (possibly

text: nonempty)

text: pluripolar

text: set.

text: ##LTFigure##

text: ##LTLine##


text: (cid:82) (cid:82) 28 D. BURNS, N. LEVENBERG AND S. MA‘U → (cid:3) Then ϕ dµ ϕ dµ. j j C We will apply this to Monge-Amp`ere measures of functions in L+( n), which have total mass (2π)n. We can now prove Theorem ??. (cid:26) Proof of Theorem ??. Let φ be a real-valued continuous function on K. Form the continuous function ∈ φ(z) if z K φ˜(z) = ∈ C \ φ(f(|ζ|)) if z = f(ζ) n K ζ where f parametrizes an extremal disk that goes through z. Note that φ˜ is con- tinuous since the foliation of extremals for V is continuous and each leaf extends K holomorphically across K (as a complex ellipse). (cid:90) (cid:90) (cid:90) Fix r > 1. By Theorem ??, we have ◦ ◦ (4.15) φ˜(ddcV )n = (φ˜ F )(ddcρ )n = (φ˜ F )(ddcρ )n, j,r j,r j,r j j,r where we use the observation in Remark ?? to get the second equality. The Monge-Amp`ere formula (??) applied to ρ , ρ shows that (ddcρ )n, (ddcρ )n { } { j,r} r j,r C \ r are supported on the sets ρ = log r , ρ = log r , which are contained in n K . → j ◦ → ◦ ρ Here we have F F uniformly; hence φ˜ F φ˜ F uniformly on a compact set S ⊂ Cn \(cid:90)Kj that contains a neigh(cid:90)borhoodj of {ρ = log r} and hence contains { } ρ ρ = log r for all suﬃciently large j. Applying Lemma ??, we have j ◦ −→ ◦ → ∞ (φ˜ F )(ddcρ )n (φ˜ F)(ddcρ )n as j . (cid:90) j j,r (cid:90) r The standard Monge-Amp`ere convergence also gives (cid:82) −→ (cid:82) → ∞ φ˜(ddcV )n φ˜(ddcV )n as j . j,r r ◦ Since (??) is true for all j, φ˜(ddcV )n = (φ˜ F)(ddcρ )n follows by taking the → ∞ r r limit as j . This latter formula is true for all r > 1. (cid:37) (cid:90)(cid:37) → (cid:90)− ∗ Since V V and ρ ρ as r 1 , we also have the weak- convergences →r r → → − (ddcV )n (ddcV )n and (ddcρ )n (ddcρ)n. Taking the limit as r 1 yields r r ◦ φ˜(ddcV )n = (φ˜ F)(ddcρ)n. Finally, note that on K, where (ddcV )n is supported, we have φ˜ = φ. Similarly, ◦ ◦ it is easy to check that φ˜ F = φ F on the support of (ddcρ)n. The theorem is (cid:3) proved. Remark 4.10. Note that the formula in Theorem ?? exhibits (ddcV )n as the K push-forward of the measure (ddcρ+ )n under the Robin exponential map: K F∗((ddcρ+ )n) = (ddcV )n. K K ##LTLine##

text: D.

text: BURNS,

text: N.

text: LEVENBERG

text: AND

text: S.

text: MA‘U

text: Then

text: ϕ

text: dµ

text: ϕ

text: dµ.

text: j

text: j

text: C

text: We

text: will

text: apply

text: this

text: to

text: Monge-Amp`ere

text: measures

text: of

text: functions

text: in

text: L+(

text: n),

text: which

text: have

text: total

text: mass

text: We

text: can

text: now

text: prove

text: Theorem

text: Proof

text: of

text: Theorem

text: Let

text: φ

text: be

text: a

text: real-valued

text: continuous

text: function

text: on

text: K.

text: Form

text: the

text: continuous

text: function

text: φ(z)

text: if

text: z

text: K

text: φ˜(z)

text: C

text: φ(f(|ζ|))

text: if

text: z

text: f(ζ)

text: n

text: K

text: ζ

text: where

text: f

text: parametrizes

text: an

text: extremal

text: disk

text: that

text: goes

text: through

text: z.

text: Note

text: that

text: φ˜

text: is

text: con-

text: tinuous

text: since

text: the

text: foliation

text: of

text: extremals

text: for

text: V

text: is

text: continuous

text: and

text: each

text: leaf

text: extends

text: K

text: holomorphically

text: across

text: K

text: (as

text: a

text: complex

text: ellipse).

text: Fix

text: r

text: By

text: Theorem

text: we

text: have

text: φ˜(ddcV

text: )n

text: (φ˜

text: F

text: )(ddcρ

text: )n

text: (φ˜

text: F

text: )(ddcρ

text: )n,

text: j,r

text: j,r

text: j,r

text: j

text: j,r

text: where

text: we

text: use

text: the

text: observation

text: in

text: Remark

text: to

text: get

text: the

text: second

text: equality.

text: The

text: Monge-Amp`ere

text: formula

text: applied

text: to

text: ρ

text: ρ

text: shows

text: that

text: (ddcρ

text: )n,

text: (ddcρ

text: )n

text: j,r}

text: r

text: j,r

text: C

text: r

text: are

text: supported

text: on

text: the

text: sets

text: ρ

text: log

text: r

text: ρ

text: log

text: r

text: which

text: are

text: contained

text: in

text: n

text: K

text: j

text: ρ

text: Here

text: we

text: have

text: F

text: F

text: uniformly;

text: hence

text: φ˜

text: F

text: φ˜

text: F

text: uniformly

text: on

text: a

text: compact

text: set

text: S

text: Cn

text: that

text: contains

text: a

text: of

text: {ρ

text: log

text: r}

text: and

text: hence

text: contains

text: ρ

text: ρ

text: log

text: r

text: for

text: all

text: suﬃciently

text: large

text: j.

text: Applying

text: Lemma

text: we

text: have

text: j

text: (φ˜

text: F

text: )(ddcρ

text: )n

text: (φ˜

text: F)(ddcρ

text: )n

text: as

text: j

text: j

text: j,r

text: r

text: The

text: standard

text: Monge-Amp`ere

text: convergence

text: also

text: gives

text: φ˜(ddcV

text: )n

text: φ˜(ddcV

text: )n

text: as

text: j

text: j,r

text: r

text: Since

text: is

text: true

text: for

text: all

text: j,

text: φ˜(ddcV

text: )n

text: (φ˜

text: F)(ddcρ

text: )n

text: follows

text: by

text: taking

text: the

text: r

text: r

text: limit

text: as

text: j

text: This

text: latter

text: formula

text: is

text: true

text: for

text: all

text: r

text: Since

text: V

text: V

text: and

text: ρ

text: ρ

text: as

text: r

text: we

text: also

text: have

text: the

text: weak-

text: convergences

text: →r

text: r

text: (ddcV

text: )n

text: (ddcV

text: )n

text: and

text: (ddcρ

text: )n

text: (ddcρ)n.

text: Taking

text: the

text: limit

text: as

text: r

text: yields

text: r

text: r

text: φ˜(ddcV

text: )n

text: (φ˜

text: F)(ddcρ)n.

text: Finally,

text: note

text: that

text: on

text: K,

text: where

text: (ddcV

text: )n

text: is

text: supported,

text: we

text: have

text: φ˜

text: φ.

text: Similarly,

text: it

text: is

text: easy

text: to

text: check

text: that

text: φ˜

text: F

text: φ

text: F

text: on

text: the

text: support

text: of

text: (ddcρ)n.

text: The

text: theorem

text: is

text: proved.

text: Remark

text: Note

text: that

text: the

text: formula

text: in

text: Theorem

text: exhibits

text: (ddcV

text: )n

text: as

text: the

text: K

text: push-forward

text: of

text: the

text: measure

text: (ddcρ+

text: )n

text: under

text: the

text: Robin

text: exponential

text: map:

text: K

text: F∗((ddcρ+

text: )n)

text: (ddcV

text: )n.

text: K

text: K

text: ##LTLine##


text: 6 D. BURNS, N. LEVENBERG AND S. MA‘U ± ∈ We verify that ρ = 1. Since C intersects the lines z = 1 there exist φ , φ − 2 1 1 2 [ π, π] such that − 1 = ρ(eiφ1 + e iφ1) = 2ρ cos φ , 1 − − 1 = ρ(eiφ2 + e iφ2) = 2ρ cos φ 2 ≥ Either equation immediately implies that ρ 1, and the reverse inequality follows 2 since the real points of C lie in S. Hence z is given by the Joukowski function: − ∈ C 1 z = 1(t + 1) = h 1(t) for t and 1 2 t | | | ◦ − | | | V (z) = log t = log h h 1(t) = log h(z ) . S 1 Conversely, suppose C does not intersect, say, z = 1. We want to show that | | 1 V (z) > log h(z ) . Now the real points of C, which are contained in S, tangen- S 1 − ∈ tially intersect the line z = 1 (cid:15) for some (cid:15) > 0. At a point (z , z ) C, the 1 1 2 parametrization of C yields z = a + ρ(t + 1). The z -coordinates of real points of ∈ R 1 1 t 1 C, given by t = eiθ (θ ) must satisfy ∈ − − ∀ a + 2ρ cos θ [ 1, 1 (cid:15)], θ. 1 ≥ ∈ We show that ρ 1 cannot hold. Suppose it does; then we must have cos θ − − − − 2 − (cid:54)∈ [ 1 a , 1 (cid:15) a ] =: I. If a < 0, then cos π = 1 I, a contradiction. But on 1 1 ≥ 1 (cid:54)∈ the other hand, if a 0 then cos 0 = 1 I, which is also a contradiction. Hence 1 ρ < 1. Finally, a calculation yields 2 | | | 1 | | | | | log h(z ) = log h(ρ(t + )) = log 2ρt < log t = V (z ). 1 t S 1 (cid:3) ∈ C \ The above lemma shows that if z 2 S lies on an extremal ellipse for S that does not intersect all four sides, then V is pluriharmonic in a neighborhood of z. S ⊂ R Theorem 3.2. Suppose K 2 contains a pair of parallel line segments. Suppose C is an extremal curve of K that intersects ∂K in the interior of these two line ∈ \ segments and in no other points. Then for any z C K, V is pluriharmonic in K a neighbourhood of z. Proof. First, we simplify the situation using Theorem ?? and the fact that pluri- harmonicity is unaﬀected by linear transformations. Hence we may assume that the − parallel line segments lie on the lines x = 1 and x = 1 and that C is centered 2 2 at the origin. By rescaling and translating the x -axis, we may further assume that ⊂ − × − 1 K S, S = [ 1, 1] [ 1, 1] as above, and that C intersects ∂K in the two points − − (α, 1) and ( α, 1) where 0 < α < 1. { ∈ R} Write C = F(eiθ) : θ for the real ellipse contained in K, where F(t) = T R ρ(bt + b/t) is the parametrization of C. By elementary topology in 2 there exists ∈ − (cid:15) > 0 such that for any s [ (cid:15), (cid:15)], the translated sets (s, 0) + C are contained in T K. ∈ \ R By construction, C is extremal for S as well as for K, so for any z C 2, we (cid:48) have V (z) = V (z). We show that for any suﬃciently close point z we also have (cid:48) K (cid:48) S V (z ) = V (z ). K S ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: D.

text: BURNS,

text: N.

text: LEVENBERG

text: AND

text: S.

text: MA‘U

text: We

text: verify

text: that

text: ρ

text: Since

text: C

text: intersects

text: the

text: lines

text: z

text: there

text: exist

text: φ

text: φ

text: π,

text: π]

text: such

text: that

text: e

text: 2ρ

text: cos

text: φ

text: e

text: 2ρ

text: cos

text: φ

text: Either

text: equation

text: immediately

text: implies

text: that

text: ρ

text: and

text: the

text: reverse

text: inequality

text: follows

text: since

text: the

text: real

text: points

text: of

text: C

text: lie

text: in

text: S.

text: Hence

text: z

text: is

text: given

text: by

text: the

text: Joukowski

text: function:

text: C

text: z

text: h

text: for

text: t

text: and

text: t

text: V

text: (z)

text: log

text: t

text: log

text: h

text: h

text: log

text: h(z

text: S

text: Conversely,

text: suppose

text: C

text: does

text: not

text: intersect,

text: say,

text: z

text: We

text: want

text: to

text: show

text: that

text: V

text: (z)

text: log

text: h(z

text: Now

text: the

text: real

text: points

text: of

text: C,

text: which

text: are

text: contained

text: in

text: S,

text: tangen-

text: S

text: tially

text: intersect

text: the

text: line

text: z

text: for

text: some

text: At

text: a

text: point

text: (z

text: z

text: C,

text: the

text: parametrization

text: of

text: C

text: yields

text: z

text: a

text: ρ(t

text: The

text: z

text: -coordinates

text: of

text: real

text: points

text: of

text: R

text: t

text: C,

text: given

text: by

text: t

text: eiθ

text: (θ

text: must

text: satisfy

text: a

text: 2ρ

text: cos

text: θ

text: θ.

text: We

text: show

text: that

text: ρ

text: cannot

text: hold.

text: Suppose

text: it

text: does;

text: then

text: we

text: must

text: have

text: cos

text: θ

text: a

text: a

text: I.

text: If

text: a

text: then

text: cos

text: π

text: I,

text: a

text: contradiction.

text: But

text: on

text: the

text: other

text: hand,

text: if

text: a

text: then

text: cos

text: I,

text: which

text: is

text: also

text: a

text: contradiction.

text: Hence

text: ρ

text: Finally,

text: a

text: calculation

text: yields

text: log

text: h(z

text: log

text: h(ρ(t

text: log

text: log

text: t

text: V

text: (z

text: t

text: S

text: C

text: The

text: above

text: lemma

text: shows

text: that

text: if

text: z

text: S

text: lies

text: on

text: an

text: extremal

text: ellipse

text: for

text: S

text: that

text: does

text: not

text: intersect

text: all

text: four

text: sides,

text: then

text: V

text: is

text: pluriharmonic

text: in

text: a

text: neighborhood

text: of

text: z.

text: S

text: R

text: Theorem

text: Suppose

text: K

text: contains

text: a

text: pair

text: of

text: parallel

text: line

text: segments.

text: Suppose

text: C

text: is

text: an

text: extremal

text: curve

text: of

text: K

text: that

text: intersects

text: ∂K

text: in

text: the

text: interior

text: of

text: these

text: two

text: line

text: segments

text: and

text: in

text: no

text: other

text: points.

text: Then

text: for

text: any

text: z

text: C

text: K,

text: V

text: is

text: pluriharmonic

text: in

text: K

text: a

text: neighbourhood

text: of

text: z.

text: Proof.

text: First,

text: we

text: simplify

text: the

text: situation

text: using

text: Theorem

text: and

text: the

text: fact

text: that

text: pluri-

text: harmonicity

text: is

text: unaﬀected

text: by

text: linear

text: transformations.

text: Hence

text: we

text: may

text: assume

text: that

text: the

text: parallel

text: line

text: segments

text: lie

text: on

text: the

text: lines

text: x

text: and

text: x

text: and

text: that

text: C

text: is

text: centered

text: at

text: the

text: origin.

text: By

text: rescaling

text: and

text: translating

text: the

text: x

text: -axis,

text: we

text: may

text: further

text: assume

text: that

text: K

text: S,

text: S

text: as

text: above,

text: and

text: that

text: C

text: intersects

text: ∂K

text: in

text: the

text: two

text: points

text: (α,

text: and

text: α,

text: where

text: α

text: R}

text: Write

text: C

text: F(eiθ)

text: θ

text: for

text: the

text: real

text: ellipse

text: contained

text: in

text: K,

text: where

text: F(t)

text: T

text: R

text: ρ(bt

text: b/t)

text: is

text: the

text: parametrization

text: of

text: C.

text: By

text: elementary

text: topology

text: in

text: there

text: exists

text: such

text: that

text: for

text: any

text: s

text: the

text: translated

text: sets

text: (s,

text: C

text: are

text: contained

text: in

text: T

text: K.

text: R

text: By

text: construction,

text: C

text: is

text: extremal

text: for

text: S

text: as

text: well

text: as

text: for

text: K,

text: so

text: for

text: any

text: z

text: C

text: we

text: have

text: V

text: (z)

text: V

text: (z).

text: We

text: show

text: that

text: for

text: any

text: suﬃciently

text: close

text: point

text: z

text: we

text: also

text: have

text: K

text: S

text: V

text: (z

text: V

text: (z

text: K

text: S

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: EXTREMAL FUNCTIONS FOR REAL CONVEX BODIES 7 Figure 1. Proof of Theorem ?? (cid:48) (cid:48) Consider an extremal ellipse C for S containing a point z that is given by the (cid:55)→ (cid:48) (cid:48) parametrization t b t + b /t. By the continuity of the foliation for V given by S extremal ellipses centered at the origin, then given (cid:15) > 0 there is δ > 0 such that | − (cid:48)| | − (cid:48)| | − (cid:48) | z z < δ implies b b < (cid:15)/2 and t t < (cid:15)/2 where 0 0 (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) z = ct + c/t and z = c t + c /t . 0 0 0 0 ⊂ (cid:48) Since K S, to show that C is extremal for K we only need to verify that (cid:48) (cid:48) ∩ ⊂ ∈ (cid:48) C = (C S) K. Let x = (x , x ) C and deﬁne y by the condition that T ∈ 1 2 T 1 y = (y , x ) C is the closest point to x with the same second coordinate. We 1 2 T have (cid:48) (cid:48) − − (cid:48) − (cid:48) − − x = b eiθ + b e iθ = b eiθ + b e iθ + (b b )eiθ + (b b )e iθ 1 1 1 1 1 1 1 1 1 (cid:48) − (cid:48) − − = y + (b b )eiθ + (b b )e iθ, 1 1 1 1 1 ∈ | − | ≤ | (cid:48) − | (cid:48) where y = (y , y ) for some y C . Hence x y 2 b b < (cid:15). So C is 1 2 − T 1 1 1 1 T contained in the convex hull of C ((cid:15), 0) and C +((cid:15), 0), which in turn is contained T (cid:48) T (cid:48) in K. (See Figure 1.) Therefore C is an extremal curve through z for both S and (cid:48) (cid:48) T K, and V (z ) = V (z ) follows. K S (cid:48) (cid:48) | (cid:48) | (cid:48) (cid:48) (cid:48) Applying the previous lemma, V (z ) = V (z ) = log h(z ) for all z = (z , z ) | − (cid:48)| K S 2 1 2(cid:3) with z z < δ. So V is pluriharmonic in a neighborhood of z. K ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: EXTREMAL

text: FUNCTIONS

text: FOR

text: REAL

text: CONVEX

text: BODIES

text: Figure

text: Proof

text: of

text: Theorem

text: Consider

text: an

text: extremal

text: ellipse

text: C

text: for

text: S

text: containing

text: a

text: point

text: z

text: that

text: is

text: given

text: by

text: the

text: parametrization

text: t

text: b

text: t

text: b

text: /t.

text: By

text: the

text: continuity

text: of

text: the

text: foliation

text: for

text: V

text: given

text: by

text: S

text: extremal

text: ellipses

text: centered

text: at

text: the

text: origin,

text: then

text: given

text: there

text: is

text: δ

text: such

text: that

text: z

text: z

text: δ

text: implies

text: b

text: b

text: and

text: t

text: t

text: where

text: z

text: ct

text: c/t

text: and

text: z

text: c

text: t

text: c

text: /t

text: Since

text: K

text: S,

text: to

text: show

text: that

text: C

text: is

text: extremal

text: for

text: K

text: we

text: only

text: need

text: to

text: verify

text: that

text: C

text: (C

text: S)

text: K.

text: Let

text: x

text: (x

text: x

text: C

text: and

text: deﬁne

text: y

text: by

text: the

text: condition

text: that

text: T

text: T

text: y

text: (y

text: x

text: C

text: is

text: the

text: closest

text: point

text: to

text: x

text: with

text: the

text: same

text: second

text: coordinate.

text: We

text: T

text: have

text: x

text: b

text: eiθ

text: b

text: e

text: iθ

text: b

text: eiθ

text: b

text: e

text: iθ

text: (b

text: b

text: )eiθ

text: (b

text: b

text: )e

text: iθ

text: y

text: (b

text: b

text: )eiθ

text: (b

text: b

text: )e

text: iθ,

text: where

text: y

text: (y

text: y

text: for

text: some

text: y

text: C

text: Hence

text: x

text: y

text: b

text: b

text: So

text: C

text: is

text: T

text: T

text: contained

text: in

text: the

text: convex

text: hull

text: of

text: C

text: and

text: C

text: which

text: in

text: turn

text: is

text: contained

text: T

text: T

text: in

text: K.

text: (See

text: Figure

text: Therefore

text: C

text: is

text: an

text: extremal

text: curve

text: through

text: z

text: for

text: both

text: S

text: and

text: T

text: K,

text: and

text: V

text: (z

text: V

text: (z

text: follows.

text: K

text: S

text: Applying

text: the

text: previous

text: lemma,

text: V

text: (z

text: V

text: (z

text: log

text: h(z

text: for

text: all

text: z

text: (z

text: z

text: K

text: S

text: with

text: z

text: z

text: δ.

text: So

text: V

text: is

text: pluriharmonic

text: in

text: a

text: neighborhood

text: of

text: z.

text: K

text: ##LTFigure##

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: 3 (cid:90) (cid:90) b (cid:48) 1 { | (cid:48) − | − | (cid:48) − | J2 = 2 − dzdz z2z(cid:48)2 λ[α2 ln L1(α2) + λ λ + α2 β2 ln L1(β2) + λ λ + β2 ] a (cid:48) | − (cid:48) | − | − (cid:48) | } + λ [α2 ln L (α2) + λ λ + α2 β2 ln L (β2) + λ λ + β2 ] , (14) 1 1 or, by the use of the fact that both of L (α2) and L (β2) remain unchanged [in virtue of the second identity in Eq. (cid:48) 1 1 (10)] upon exchanging(cid:90)z an(cid:90)d z , b (cid:48) 1 | (cid:48) − | − | (cid:48) − | J2 = 4 − dzdz z2z(cid:48)2 λ[α2 ln L1(α2) + λ λ + α2 β2 ln L1(β2) + λ λ + β2 ]; (15) a and (cid:90) (cid:90) Understandably, this term with null contribution might however cost a huge amount of labor in any further numerical or analytical calculations if not appreciated. J3 = 4 − b dzdz(cid:48) z21z(cid:48)2 λ[β2 ln |β2| − α2 ln |α2|]. (16) sucTcheessfeuxlyprreesdsiuocnedfoinr tIe(rkm)shoafstwaos-daimcoennssieoqnuaelnicnetebgreaeln. a Similar type of reduction had been accomplished or ap- plied in Refs. [18, 21–24, 26]. (Brosens et al. [18] in J can be evaluated to be 0 fact considered both of the static and dynamic cases.) A diﬀerent approach was adopted by Holas et al. [19] who − | | | | J0 = 8 ln b/a [ab(1 + 2 ln 2) ln b/a + k(3 + ln 2)], (17) essentially ﬁrst calculated the imaginary part of Π1(k, ω), and then the real part of it [with the static Π (k, 0) as 1 its special case] via the dispersion relation, which virtu- and accordingly ally also amounts to a two-dimensional integral over all. The remaining two-dimensional integral was then calcu- lated numerically in Refs. [18, 19, 21, 22, 26]; analytically − 1 { − | | (cid:88) | | with the aid of some ingenious method and mathemat- I(k) = 8 ln b/a [ab(1 + 2 ln 2) ln b/a 16k2 ical assumption in Ref. [23]; and analytically but with 3 } computer programs in Ref. [24]. We need point out that + k(3 + 2 ln 2)] + Ji . (18) there exists discrepance between our Eq. (18) and the i=1 corresponding expression in Ref. [24] [the one shown in the middle of the paragraph above Eq. (10) in Ref. [24]]. The J , J , and J are the same as the ones deﬁned III. EVALUATION FOR J 1 2 3 1 in Ref. [24]. The expression for J here appears to be 1 diﬀerent to that in Ref. [24] but is actually equal. There It turns out that the integral in Eq. (13) for J can be 1 is an(cid:90) extra(cid:90)term of the following form, fully carried through. We are about to achieve this step (cid:48) by step. Indeed, either the integral over z or z can be carried out ﬁrst. But before doing that it is worth point- b (cid:48) 1 (cid:48) − − −a dz dz z2z(cid:48)2 (λ λ)[L1(α2) L1(β2)], (19) iancgtuoaulltyabseo,mbyewdhiraetcthisdudbesntitbuuttiounseinfuEl tqr.u(t1h0.)L, s1i(mβp2)liﬁcaend to the following form: in the expression for J in Eq. (10) of Ref. [24], but 1 this extra term can be shown to be equal to zero. (We | | L (β2) = 2 β . (21) note that in our foregoing derivation this term did not 1 − (cid:48) arise at all.) In fact, the factor of λ λ changes sign − but that of L (α2) L (β2) [actually each of L (α2) and L (β2)] rema1ins uncha1nged upon exchanging z1and z(cid:48) as Accordingly J1, with some further algebra, can be writ- 1 ten as (cid:90) (cid:90) m(cid:90)entione(cid:90)d above. In consequence, (cid:112) b 1 b (cid:48) 1 (cid:48) | | − dz b dz(cid:48) z21z(cid:48)2 (λ(cid:48) − λ)[L1(α2) − L1(β2)] = 0. (20) J1 = 4 −a dz z2 −a dz z(cid:48) [α R(z, z ) + β β ], (22) a where ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: b

text: dzdz

text: λ[α2

text: ln

text: λ

text: λ

text: α2

text: β2

text: ln

text: λ

text: λ

text: β2

text: a

text: λ

text: ln

text: L

text: λ

text: λ

text: α2

text: β2

text: ln

text: L

text: λ

text: λ

text: β2

text: or,

text: by

text: the

text: use

text: of

text: the

text: fact

text: that

text: both

text: of

text: L

text: and

text: L

text: remain

text: unchanged

text: [in

text: virtue

text: of

text: the

text: second

text: identity

text: in

text: Eq.

text: upon

text: z

text: b

text: dzdz

text: λ[α2

text: ln

text: λ

text: λ

text: α2

text: β2

text: ln

text: λ

text: λ

text: β2

text: a

text: and

text: Understandably,

text: this

text: term

text: with

text: contribution

text: might

text: however

text: cost

text: a

text: huge

text: amount

text: of

text: labor

text: in

text: any

text: further

text: numerical

text: or

text: analytical

text: calculations

text: if

text: not

text: appreciated.

text: b

text: λ[β2

text: ln

text: α2

text: ln

text: sucTcheessfeuxlyprreesdsiuocnedfoinr

text: a

text: Similar

text: type

text: of

text: reduction

text: had

text: been

text: accomplished

text: or

text: ap-

text: plied

text: in

text: Refs.

text: (Brosens

text: et

text: al.

text: in

text: J

text: can

text: be

text: evaluated

text: to

text: be

text: fact

text: considered

text: both

text: of

text: the

text: static

text: and

text: dynamic

text: cases.)

text: A

text: diﬀerent

text: approach

text: was

text: adopted

text: by

text: Holas

text: et

text: al.

text: who

text: ln

text: b/a

text: ln

text: ln

text: b/a

text: ln

text: essentially

text: ﬁrst

text: calculated

text: the

text: imaginary

text: part

text: of

text: ω),

text: and

text: then

text: the

text: real

text: part

text: of

text: it

text: [with

text: the

text: static

text: Π

text: (k,

text: as

text: its

text: special

text: case]

text: via

text: the

text: dispersion

text: relation,

text: which

text: virtu-

text: and

text: accordingly

text: ally

text: also

text: amounts

text: to

text: a

text: two-dimensional

text: integral

text: over

text: all.

text: The

text: remaining

text: two-dimensional

text: integral

text: was

text: then

text: calcu-

text: lated

text: numerically

text: in

text: Refs.

text: analytically

text: with

text: the

text: aid

text: of

text: some

text: ingenious

text: method

text: and

text: mathemat-

text: I(k)

text: ln

text: b/a

text: ln

text: ln

text: b/a

text: ical

text: assumption

text: in

text: Ref.

text: and

text: analytically

text: but

text: with

text: computer

text: programs

text: in

text: Ref.

text: We

text: need

text: point

text: out

text: that

text: ln

text: Ji

text: there

text: exists

text: discrepance

text: between

text: our

text: Eq.

text: and

text: the

text: corresponding

text: expression

text: in

text: Ref.

text: [the

text: one

text: shown

text: in

text: the

text: middle

text: of

text: the

text: paragraph

text: above

text: Eq.

text: in

text: Ref.

text: The

text: J

text: J

text: and

text: J

text: are

text: the

text: same

text: as

text: the

text: ones

text: deﬁned

text: III.

text: EVALUATION

text: FOR

text: J

text: in

text: Ref.

text: The

text: expression

text: for

text: J

text: here

text: appears

text: to

text: be

text: diﬀerent

text: to

text: that

text: in

text: Ref.

text: but

text: is

text: actually

text: equal.

text: There

text: It

text: turns

text: out

text: that

text: the

text: integral

text: in

text: Eq.

text: for

text: J

text: can

text: be

text: is

text: of

text: the

text: following

text: form,

text: fully

text: carried

text: through.

text: We

text: are

text: about

text: to

text: achieve

text: this

text: step

text: by

text: step.

text: Indeed,

text: either

text: the

text: integral

text: over

text: z

text: or

text: z

text: can

text: be

text: carried

text: out

text: ﬁrst.

text: But

text: before

text: doing

text: that

text: it

text: is

text: worth

text: point-

text: b

text: −a

text: dz

text: dz

text: (λ

text: to

text: the

text: following

text: form:

text: in

text: the

text: expression

text: for

text: J

text: in

text: Eq.

text: of

text: Ref.

text: but

text: this

text: extra

text: term

text: can

text: be

text: shown

text: to

text: be

text: equal

text: to

text: zero.

text: (We

text: L

text: β

text: note

text: that

text: in

text: our

text: foregoing

text: derivation

text: this

text: term

text: did

text: not

text: arise

text: at

text: all.)

text: In

text: fact,

text: the

text: factor

text: of

text: λ

text: λ

text: changes

text: sign

text: but

text: that

text: of

text: L

text: L

text: [actually

text: each

text: of

text: L

text: and

text: L

text: upon

text: exchanging

text: as

text: Accordingly

text: with

text: some

text: further

text: algebra,

text: can

text: be

text: writ-

text: ten

text: as

text: above.

text: In

text: consequence,

text: b

text: b

text: dz

text: b

text: −a

text: dz

text: −a

text: dz

text: [α

text: R(z,

text: z

text: β

text: β

text: a

text: where

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 substituting Eqs. (B2), (B3), (B4) into Eq. (40), we where we have adopted (cid:12)the follow(cid:12)ing notation, obtain our ﬁnal result for J : (cid:12) (cid:12) 1 (cid:12) (cid:12) (cid:12)√ (cid:12) √C + 1 Y (z) = ln 0 − . (39) − | | C0 1 J1 = 16 + 4(5k + 4/k) ln b/a . (42) We then substitute Eqs. (28) and (38) into Eq. (27) and, with some further routine calculatio(cid:12)n, obta(cid:12)in (cid:12) (cid:12) (cid:12) (cid:12) The result is surprisingly elegant. (cid:12) (cid:12) IV. EVALUATION FOR J + J − 2 − k + 1 2 3 J = 20 + 16 ln k + (k2 1)2 ln − 1 k k 1 − − | | (k3 24k + 8) ln 2b Among the three expressions as shown in (13), (15), − − | | and (16), the expression in (13) for J is likely the one + (k3 24k 8) ln 2a 1 the most amenable to analytical computations. This is 1 − − − + [3η + 2(3 k2)η− (k2 1)2η− ], (40) in fact also the cause that it can be fully reduced to the k 1 1 3 explicit expression of (42). On the other hand, due to the logarithm forms in them, Eq. (15) and Eq. (16) where (cid:90) are much more diﬃcult. Further reduction of J2 and J , at least that of J , to one quadrature, had been 3 2 believed beyond barehanded eﬀort [24]. We ﬁnd indeed b that careless choices of procedures might easily make the η = k dzCn/2Y (z). (41) n − 0 task intractable. It is critical to search for optimal ones. a Instead of evaluating J and J separately, we choose 2 3 The explicit expressions for η , η− , and η− are given to combine them together and deal with them at one 1 1 3 in Eq. (B2), Eq. (B3), and Eq. (B4), respectively. By stroke. We write J = J +J , and accordingly have 23 2 3 (cid:90) (cid:90) (cid:8) (cid:9) b (cid:48) λ | (cid:48) − | − | − | | | J23 = 4 − dzdz z2z(cid:48)2 α2 ln [L1(α2) + λ λ + α2]/α2 β2 ln [a b + 2z + 2β/ β ]/β . (43) a We have also made the use of Eq. (21) in obtaining the above equation. We next perform some manipulation on Eq. (43) to bring it into the following form: (cid:90) (cid:90) b 1 | | b (cid:48) 1 − J23 = 16 − dz z λ ln 4λ − dz z(cid:48) 4N, (44) a a where (cid:90) (cid:90) (cid:112) b (cid:48) λ | (cid:48) − − (cid:48) | − | − − | | | N = dzdz (cid:48) [α2 ln α2 + λ λ 2 R(z, z ) β2 ln β(a b + 2z 2β/ β ) ]. (45) − z2z 2 a (cid:82) The ﬁrst term on the right hand side of Eq. (44) has vir- tually been reduced to a one-dimensional integral for the where b (cid:48) (cid:48) (cid:90) (cid:112) integration − dz 1/z is trivial. Hence we need concern a − ourselves only with the term of 4N. For this purpose, we ﬁrst rewrite N(cid:90) as the following: N¯1(z) = −b dz(cid:48) zα(cid:48)22 ln |α2 + λ(cid:48) − λ − 2 R(z, z(cid:48))|, (47) a b λ − N = dz [N¯ (z) N¯ (z)], (46) − z2 1 2 and a ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: substituting

text: Eqs.

text: into

text: Eq.

text: we

text: where

text: we

text: have

text: adopted

text: notation,

text: obtain

text: our

text: ﬁnal

text: result

text: for

text: J

text: √C

text: Y

text: (z)

text: ln

text: ln

text: b/a

text: We

text: then

text: substitute

text: Eqs.

text: and

text: into

text: Eq.

text: and,

text: with

text: some

text: further

text: routine

text: The

text: result

text: is

text: surprisingly

text: elegant.

text: IV.

label: section

text: EVALUATION

text: FOR

text: J

text: J

text: k

text: J

text: ln

text: k

text: ln

text: k

text: k

text: ln

text: Among

text: the

text: three

text: expressions

text: as

text: shown

text: in

text: and

text: the

text: expression

text: in

text: for

text: J

text: is

text: likely

text: the

text: one

text: ln

text: the

text: most

text: amenable

text: to

text: analytical

text: computations.

text: This

text: is

text: in

text: fact

text: also

text: the

text: cause

text: that

text: it

text: can

text: be

text: fully

text: reduced

text: to

text: the

text: k

text: explicit

text: expression

text: of

text: On

text: the

text: other

text: hand,

text: due

text: to

text: the

text: logarithm

text: forms

text: in

text: them,

text: Eq.

text: and

text: Eq.

text: where

text: are

text: much

text: more

text: diﬃcult.

text: Further

text: reduction

text: of

text: and

text: J

text: at

text: least

text: that

text: of

text: J

text: to

text: one

text: quadrature,

text: had

text: been

text: believed

text: beyond

text: barehanded

text: eﬀort

text: We

text: ﬁnd

text: indeed

text: b

text: that

text: careless

text: choices

text: of

text: procedures

text: might

text: easily

text: make

text: the

text: η

text: k

text: (z).

text: n

text: task

text: intractable.

text: It

text: is

text: critical

text: to

text: search

text: for

text: optimal

text: ones.

text: a

text: Instead

text: of

text: evaluating

text: J

text: and

text: J

text: separately,

text: we

text: choose

text: The

text: explicit

text: expressions

text: for

text: η

text: η−

text: and

text: η−

text: are

text: given

text: to

text: combine

text: them

text: together

text: and

text: deal

text: with

text: them

text: at

text: one

text: in

text: Eq.

text: Eq.

text: and

text: Eq.

text: respectively.

text: By

text: stroke.

text: We

text: write

text: J

text: J

text: +J

text: and

text: accordingly

text: have

text: b

text: λ

text: dzdz

text: α2

text: ln

text: λ

text: λ

text: β2

text: ln

text: [a

text: b

text: β

text: ]/β

text: a

text: We

text: have

text: also

text: made

text: the

text: use

text: of

text: Eq.

text: in

text: obtaining

text: the

text: above

text: equation.

text: We

text: next

text: perform

text: some

text: manipulation

text: on

text: Eq.

text: to

text: bring

text: it

text: into

text: the

text: following

text: form:

text: b

text: b

text: dz

text: z

text: λ

text: ln

text: 4λ

text: dz

text: a

text: a

text: where

text: b

text: λ

text: N

text: dzdz

text: ln

text: α2

text: λ

text: λ

text: R(z,

text: z

text: β2

text: ln

text: β(a

text: b

text: β

text: a

text: The

text: ﬁrst

text: term

text: on

text: the

text: right

text: hand

text: side

text: of

text: Eq.

text: has

text: vir-

text: tually

text: been

text: reduced

text: to

text: a

text: one-dimensional

text: integral

text: for

text: the

text: where

text: b

text: integration

text: dz

text: is

text: trivial.

text: Hence

text: we

text: need

text: concern

text: a

text: ourselves

text: only

text: with

text: the

text: term

text: of

text: For

text: this

text: purpose,

text: we

text: ﬁrst

text: rewrite

text: as

text: the

text: following:

text: −b

text: ln

text: λ

text: R(z,

text: a

text: b

text: λ

text: N

text: dz

text: [N¯

text: (z)

text: N¯

text: (z)],

text: and

text: a

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: 6 (cid:90) (cid:90) (cid:90) (cid:90) (cid:112) z (cid:48) β2 | − | b (cid:48) β2 | | b b (cid:48) 1 | (cid:48)| N¯2(z) = − dz z(cid:48)2 ln 2β(z b) + dz z(cid:48)2 ln 2β(z+a) . Φ3 = − dzλ − dz α R(z, z(cid:48)) ln z , (55) a z a a (48) (cid:48) and (cid:90) In the preceding expression for N¯ , the integral over z 2 − ≤ has been purposely separated into two regimes of a (cid:48) ≤ (cid:48) ≤ z < z and z z b. This is intended to take care b λ | | Φ = dz W (z) ln z . (56) of the subtle singular behavior of the logarithm factor 4 − z 1 | − − | | | a ln β(a b + 2z 2β/ β ) in Eq. (45). After performing partial integration on each of the integrals on the right Φ is clearly not our concern temporarily, since it is 4 hand sides of Eq. (4(cid:18)7) and Eq. (48(cid:12)(cid:12)), (cid:12)(cid:12)o(cid:19)ne obtains already in terms of one-dimensional integral. Fortunately (cid:12) (cid:12) it turns out that the two-dimensional integral in Φ can (cid:12) (cid:12) 1 be fully carried through. We ﬁrst attack it. To this end, (cid:90) − z2(cid:18) b | | (cid:19) N¯1(z) = 2 1 ab + z ln a ln 2λ we write (cid:90) b (cid:18) (cid:48) (cid:48) − z2 (cid:19) | (cid:48)| + −a dz(cid:112) z z(cid:48) + 2z ln z Φ1 = −b dz λz Φ¯1(z), (57) × 1 kz − a (cid:48) 1 , (49) α R(z, z ) where (cid:90) (cid:90) (cid:112) (cid:112) and (cid:18) (cid:12) (cid:12)(cid:19) (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) Φ¯1(z) = −b dz(cid:48) R(1z, z(cid:48)) −z −b dz(cid:48) z(cid:48) R1(z, z(cid:48)). (58) (cid:90) − z2(cid:18)− b | | (cid:19) | | a a N¯ (z) = 2 1 z ln ln 2λ + 2zW (z) ln z 2 ab a 1 (cid:48) Each of the integrals over z in Eq. (58) is routine. b (cid:48) (cid:48) − z2 − | (cid:48)| 1 After carrying out them, we obtain the following result + −a dz z z(cid:48) 2z ln z β . (50) for Φ¯1(z), (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Here we have introduced t(cid:12)he follo(cid:12)wing notation, √z aψ (z) √1 ψ (z) (cid:12) (cid:12) Φ¯ (z) = ln 1 + ln 3 , (59) (cid:12)(cid:12) (cid:12)(cid:12) 1 A0 bψ2(z) C0 ψ4(z) z + a W1(z) = ln z − b . (51) or, with the aid of Eqs. (36) and(cid:12)(cid:12)(37),(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Equation (46), together with Eqs. (49) and (50), is √2 4ab Φ¯ (z) = Y (z) + ln . (60) then substituted into Eq. (44) to get 1 C k2 0 − | | | | Substitution of Eq. (60) into Eq. (57), followed by J = 8(2 ln 2 1)(ab ln b/a + k) ln b/a 23 − − further algebra, yields +4( kΦ + 2Φ 2kΦ + 2Φ ), (52) 1 2 3 4 | | − − 1 1 Φ = k(ln 4ab 2 ln k) η + (1 + 2k2)η− .(61) where (cid:90) (cid:90) 1 k2 1 k2 1 (cid:112) The use of Eqs. (B2) and (B3) in the above equation − b λ b (cid:48) β 1 then yields Φ1 = − dz z − dz z(cid:48) R(z, z(cid:48)), (53) a a (cid:90) (cid:90) 1 − − | | Φ = [(8k3 + 9k2 + 4) ln b + (8k3 9k2 4) ln a 1 3k2 − | | − b λ b (cid:48) z(cid:48) | (cid:48)| 4k + 16k3 ln 2/k ]. (62) Φ = 2 dz dz ln z , (54) 2 − z − αβ a a This is another elegant result indeed. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: z

text: β2

text: b

text: β2

text: b

text: b

text: dz

text: ln

text: b)

text: dz

text: ln

text: Φ3

text: dzλ

text: dz

text: α

text: R(z,

text: ln

text: z

text: a

text: z

text: a

text: a

text: and

text: In

text: the

text: preceding

text: expression

text: for

text: N¯

text: the

text: integral

text: over

text: z

text: has

text: been

text: purposely

text: separated

text: into

text: two

text: regimes

text: of

text: a

text: z

text: z

text: and

text: z

text: z

text: b.

text: This

text: is

text: intended

text: to

text: take

text: care

text: b

text: λ

text: Φ

text: dz

text: W

text: (z)

text: ln

text: z

text: of

text: the

text: subtle

text: singular

text: behavior

text: of

text: the

text: logarithm

text: factor

text: z

text: a

text: ln

text: β(a

text: b

text: β

text: in

text: Eq.

text: After

text: performing

text: partial

text: integration

text: on

text: each

text: of

text: the

text: integrals

text: on

text: the

text: right

text: Φ

text: is

text: clearly

text: not

text: our

text: concern

text: temporarily,

text: since

text: it

text: is

text: hand

text: sides

text: of

text: Eq.

text: and

text: Eq.

text: obtains

text: already

text: in

text: terms

text: of

text: one-dimensional

text: integral.

text: Fortunately

text: it

text: turns

text: out

text: that

text: the

text: two-dimensional

text: integral

text: in

text: Φ

text: can

text: be

text: fully

text: carried

text: through.

text: We

text: ﬁrst

text: attack

text: it.

text: To

text: this

text: end,

text: b

text: ab

text: z

text: ln

text: a

text: ln

text: 2λ

text: we

text: write

text: b

text: −a

text: z

text: ln

text: z

text: Φ1

text: −b

text: dz

text: λz

text: kz

text: a

text: α

text: R(z,

text: z

text: where

text: and

text: −b

text: −z

text: −b

text: b

text: a

text: a

text: N¯

text: (z)

text: z

text: ln

text: ln

text: 2λ

text: (z)

text: ln

text: z

text: ab

text: a

text: Each

text: of

text: the

text: integrals

text: over

text: z

text: in

text: Eq.

text: is

text: routine.

text: b

text: After

text: carrying

text: out

text: them,

text: we

text: obtain

text: the

text: following

text: result

text: −a

text: dz

text: z

text: ln

text: z

text: β

text: for

text: Here

text: we

text: have

text: introduced

text: notation,

text: √z

text: aψ

text: (z)

text: √1

text: ψ

text: (z)

text: Φ¯

text: (z)

text: ln

text: ln

text: z

text: a

text: ln

text: z

text: b

text: or,

text: with

text: the

text: aid

text: of

text: Eqs.

text: Equation

text: together

text: with

text: Eqs.

text: and

text: is

text: √2

text: Φ¯

text: (z)

text: Y

text: (z)

text: ln

text: then

text: substituted

text: into

text: Eq.

text: to

text: get

text: C

text: Substitution

text: of

text: Eq.

text: into

text: Eq.

text: followed

text: by

text: J

text: ln

text: ln

text: b/a

text: k)

text: ln

text: b/a

text: further

text: algebra,

text: yields

text: kΦ

text: 2Φ

text: 2Φ

text: Φ

text: k(ln

text: ln

text: k)

text: η

text: where

text: The

text: use

text: of

text: Eqs.

text: and

text: in

text: the

text: above

text: equation

text: b

text: λ

text: b

text: β

text: then

text: yields

text: Φ1

text: dz

text: z

text: dz

text: R(z,

text: a

text: a

text: Φ

text: ln

text: b

text: ln

text: a

text: b

text: λ

text: b

text: ln

text: Φ

text: dz

text: dz

text: ln

text: z

text: z

text: αβ

text: a

text: a

text: This

text: is

text: another

text: elegant

text: result

text: indeed.

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text: ##LTLine##

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text: 8 − − It turns out that the integral in the expression for P J = 8[P + (k2 1)P + 2P kP ], (78) 3 23 0 1 2 3 can be further reﬁned. Indeed, it is shown in Appendix B that where 10 | | − − | | | | − P = (2k ln k/2 1/k) + 4(b ln2 b a ln2 a ) P = 8k2/3 ln k/2 4/3 3 3 0 − | | | | + [2 ln 2 1 + ln ab ]ab ln2 b/a + 4(3 ln |2/k| − 5 + 1/k2)(b ln b − a ln |a|) − − + 2[k ln 2 b(4k2 2k + 1)/3k] ln b 3 − − | | 1 | | 1 − 2[k ln 2 a(4k2 + 2k + 1)/3k] ln a , (79) + 3k (b ln b + a ln a ) + k (k2 1)P1. (83) (cid:90) Remarkably, it can be seen from Eq. (83) that the − − b 1 | | combination form of (k2 1)P1 kP3, the form in which P1 = − dz C ln z , (80) P1 and P3 enter the right hand side of Eq. (78), can be a 0 fully reduced to an explicit form with no more integral (cid:90) in it. In this light we substitute Eqs. (79) and (83) into Eq. (78) to obtain the ﬁnal expression for J as the 23 b 1 − − | | following: P = dz [λW (z) (b + z)(z a)W (z)] ln z , (81) 2 − z 1 2 a and (cid:90) J23 = 8[g(k) + 2P2], (84) b − | | P = dz(k + 2z)(2 + 3kz)C 3/2Y (z) ln z . (82) where 3 − 0 a | | | | | | − | | | | − g(k) = 2 + 4k2 ln 2/k + ab ln ab ln2 b/a + ab(2 ln 2 1) ln2 b/a + 4k(a ln2 a b ln2 b) − − +[2k2 + 9k/2 2/k 2k(k + 1) ln 2 + 4kb ln k] ln b − − − | | +[2k2 9k/2 + 2/k + 2k(1 k) ln 2 4ka ln k] ln a . (85) V. I(K) EXPRESSED IN TERMS OF ONE As a consequence, we have accomplished reducing I(k) QUADRATURE; AND FURTHER IN TERMS OF in terms of one quadrature. Equation (86) will be further (cid:20) (cid:12) (cid:12)(cid:18) (cid:12) (cid:12) (cid:19) SERIES reﬁned in the next sec(cid:12)tio(cid:12)n into t(cid:12)he(cid:12)following ﬁnal form: (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Further substitution of Eq. (42) and Eq. (84) into Eq. (cid:90)1 b 1(cid:12) (cid:12)(cid:18)b (cid:12) (cid:12) (cid:19)(cid:21) (18) then leads to I(k) = 2k2 b ln2 a 3a(cid:12)(cid:12) ln(cid:12)(cid:12)a + b (cid:12)(cid:12) (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) b2 1 x 1 x − − 1 + dx − ln ab ln k ,(88) I(k) = 2k2 [g0(k) + 2P2], (86) a2 x a2 a2 2 b2 (cid:20) (cid:12) (cid:12)(cid:18) (cid:12) (cid:12) (cid:19) or, (cid:12) (cid:12) (cid:12) (cid:12) where (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) g0(k) = − 2k(a ln2−|a| − |b l|n−2 b) −+ ab ln| |ab||ln2 |b/a| I(k) = (cid:90)2k12 a ln2 ab 13(cid:18)b ln ab +(cid:12)(cid:12)(cid:12)(cid:12) a (cid:12)(cid:12)(cid:12)(cid:12) (cid:19) (cid:21) 2(b ln b a l|n a | k k ln k/2 )2 a2/b2 1 1 a2 − + 2k2(1 + ln2 k/2 ). (87) + dx − ab ln k ln x . 1 x 2 b2x 1 (89) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: It

text: turns

text: out

text: that

text: the

text: integral

text: in

text: the

text: expression

text: for

text: P

text: J

text: kP

text: can

text: be

text: further

text: reﬁned.

text: Indeed,

text: it

text: is

text: shown

text: in

text: Appendix

text: B

text: that

text: where

text: P

text: ln

text: b

text: a

text: a

text: P

text: ln

text: ln

text: ln

text: ab

text: ]ab

text: b/a

text: ln

text: ln

text: b

text: a

text: ln

text: |a|)

text: ln

text: ln

text: b

text: ln

text: ln

text: a

text: (b

text: ln

text: b

text: a

text: ln

text: a

text: k

text: Remarkably,

text: it

text: can

text: be

text: seen

text: from

text: Eq.

text: that

text: the

text: b

text: combination

text: form

text: of

text: the

text: form

text: in

text: which

text: dz

text: C

text: ln

text: z

text: and

text: enter

text: the

text: right

text: hand

text: side

text: of

text: Eq.

text: can

text: be

text: a

text: fully

text: reduced

text: to

text: an

text: explicit

text: form

text: with

text: no

text: more

text: integral

text: in

text: it.

text: In

text: this

text: light

text: we

text: substitute

text: Eqs.

text: and

text: into

text: Eq.

text: to

text: obtain

text: the

text: ﬁnal

text: expression

text: for

text: J

text: as

text: the

text: b

text: following:

text: P

text: dz

text: [λW

text: (z)

text: (b

text: z)(z

text: a)W

text: (z)]

text: ln

text: z

text: z

text: a

text: and

text: b

text: P

text: dz(k

text: (z)

text: ln

text: z

text: where

text: a

text: g(k)

text: ln

text: ab

text: ln

text: ab

text: b/a

text: ln

text: b/a

text: a

text: b

text: b)

text: ln

text: ln

text: k]

text: ln

text: b

text: k)

text: ln

text: ln

text: k]

text: ln

text: a

text: V.

text: I(K)

text: EXPRESSED

text: IN

text: TERMS

text: OF

text: ONE

text: As

text: a

text: consequence,

text: we

text: have

text: accomplished

text: reducing

text: I(k)

text: QUADRATURE;

text: AND

text: FURTHER

text: IN

text: TERMS

text: OF

text: in

text: terms

text: of

text: one

text: quadrature.

text: Equation

text: will

text: be

text: further

text: SERIES

text: reﬁned

text: in

text: the

text: next

text: into

text: ﬁnal

text: form:

text: Further

text: substitution

text: of

text: Eq.

text: and

text: Eq.

text: into

text: Eq.

text: b

text: then

text: leads

text: to

text: I(k)

text: b

text: a

text: b

text: x

text: x

text: dx

text: ln

text: ab

text: ln

text: k

text: I(k)

text: x

text: or,

text: where

text: |b

text: b)

text: ab

text: ln|

text: |b/a|

text: I(k)

text: a

text: ab

text: ln

text: ab

text: a

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text: b

text: a

text: l|n

text: a

text: k

text: k

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text: k

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text: x

text: x

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: 11 [38] Ref. [37] in fact used the generating function rather than the CF. The former can be viewed as the latter that is evaluated on the imaginary axis. They are no essential diﬀerences if the nonequilibrium process is ﬁnished within a ﬁnite time interval. This notion will be used again in the following discussions. [39] A´. Rivas, A. D. K. Plato, S. F. Huelga, and M. B. Plenio, New J. Phys. 12, 113032 (2010). [40] H. Carmichael, An open systems approach to Quantum Optics: lectures presented at the Universit´e Libre de Bruxelles, October 28 to November 4, 1991, Vol. 18 (Springer, 1993). [41] H. M. Wiseman and G. J. Milburn, Quantum measurement and control (Cambridge University Press, 2010). [42] H. H. Risken, The Fokker-Planck equation : methods of solution and applications (Springer-Verlag, Berlin, New York, 1984). [43] F. Liu, H. Tong, R. Ma, and Z.-c. Ou-Yang, J. Phys. A: Math. Theor. 43, 495003 (2010). [44] W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev. Lett. 56, 2797 (1986). [45] K. Murch, S. Weber, C. Macklin, and I. Siddiqi, Nature 502, 211 (2013). . ∗ [1] G. Bochkov and Y. E. Kuzovlev, Zh. Eksp. Teor. Fiz 72, 238 (1977). [2] J. Kurchan, arXiv preprint cond-mat/0007360 (2000). [3] H. Tasaki, arXiv preprint cond-mat/0009244 (2000). [4] S. Yukawa, J. Phys. Soc. Jpn. 69, 2367 (2000). [5] S. Mukamel, Phys. Rev. Lett. 90, 170604 (2003). [6] W. De Roeck and C. Maes, Phys. Rev. E 69, 026115 (2004). [7] A. E. Allahverdyan and T. M. Nieuwenhuizen, Phys. Rev. E 71, 066102 (2005). [8] P. Talkner, E. Lutz, and P. Ha¨nggi, Phys. Rev. E 75, 050102 (2007). [9] W. De Roeck, C. R. Phys. 8, 674 (2007). [10] D. Andrieux and P. Gaspard, Phys. Rev. Lett. 100, 230404 (2008). [11] P. Talkner, M. Campisi, and P. Ha¨nggi, J. Stat. Mech.: Theor. Exp. 2009, P02025 (2009). [12] M. Esposito, U. Harbola, and S. Mukamel, Rev. Mod. Phys. 81, 1665 (2009). [13] M. Campisi, P. H¨anggi, and P. Talkner, Rev. Mod. Phys. 83, 771 (2011). [14] Y. Suba¸sı and B. L. Hu, Phys. Rev. E 85, 011112 (2012). [15] J. M. Horowitz and J. M. Parrondo, New J. Phys. 15, 085028 (2013). [16] M. Esposito and S. Mukamel, Phys. Rev. E 73, 046129 (2006). [17] G. E. Crooks, Phys. Rev. A 77, 034101 (2008). [18] J. M. Horowitz, Phys. Rev. E 85, 031110 (2012). [19] F. Liu, Phys. Rev. E 86, 010103 (2012). [20] R. Chetrite and K. Mallick, J. Stat. Phys. 148, 480 (2012). [21] F. W. J. Hekking and J. P. Pekola, Phys. Rev. Lett. 111, 093602 (2013). [22] B. Leggio, A. Napoli, A. Messina, and H.-P. Breuer, Phys. Rev. A 88, 042111 (2013). [23] F. Liu, Phys. Rev. E 89, 042122 (2014). [24] F. Liu, Phys. Rev. E 90, 032121 (2014). [25] M. Silaev, T. T. Heikkila¨, and P. Virtanen, Phys. Rev. E 90, 022103 (2014). [26] S. Suomela, P. Solinas, J. P. Pekola, J. Ankerhold, and T. Ala-Nissila, Phys. Rev. B 90, 094304 (2014). [27] S. Suomela, J. Salmilehto, I. G. Savenko, T. Ala-Nissila, and M. Mo¨tto¨nen, Phys. Rev. E 91, 022126 (2015). [28] T. B. Batalha˜o, A. M. Souza, L. Mazzola, R. Auccaise, R. S. Sarthour, I. S. Oliveira, J. Goold, G. De Chiara, M. Paternostro, and R. M. Serra, Phys. Rev. Lett. 113, 140601 (2014). [29] S.-M. An, J.-N. Zhang, M. Um, D.-S. Lv, Y. Lu, J.-H. Zhang, Z.-Q. Yi, H.-T. Quan, and K. Kim, Nat. Phys. 11, 193 (2015). [30] C. Jarzynski, H. Quan, and S. Rahav, Phys. Rev. X 5, 031038 (2015). [31] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997). [32] C. Jarzynski, C. R. Phys. 8, 495 (2007). [33] A. Allahverdyan, Phys. Rev. E 90, 032137 (2014). [34] F. Liu and Z.-C. Ouyang, Chin. Phys. B 23, 070512 (2014). [35] H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford university press, 2002). [36] J. Derezin´ski, W. De Roeck, and C. Maes, J. Stat. Phys. 131, 341 (2008). [37] J. P. Garrahan and I. Lesanovsky, Phys. Rev. Lett. 104, 160601 (2010). [38] Ref. [37] in fact used the generating function rather than the CF. The former can be viewed as the latter that is evaluated on the imaginary axis. They are no essential diﬀerences if the nonequilibrium process is ﬁnished within a ﬁnite time interval. This notion will be used again in the following discussions. [39] A´. Rivas, A. D. K. Plato, S. F. Huelga, and M. B. Plenio, New J. Phys. 12, 113032 (2010). [40] H. Carmichael, An open systems approach to Quantum Optics: lectures presented at the Universit´e Libre de Bruxelles, October 28 to November 4, 1991, Vol. 18 (Springer, 1993). ##LTLine## ##LTLine## ##LTLine##

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text: P.

text: Garrahan

text: and

text: I.

text: Lesanovsky,

text: Phys.

text: Rev.

text: Lett.

text: Ref.

text: in

text: fact

text: used

text: the

text: generating

text: function

text: rather

text: than

text: the

text: CF.

text: The

text: former

text: can

text: be

text: viewed

text: as

text: the

text: latter

text: that

text: is

text: evaluated

text: on

text: the

text: imaginary

text: axis.

text: They

text: are

text: no

text: essential

text: diﬀerences

text: if

text: the

text: nonequilibrium

text: process

text: is

text: ﬁnished

text: within

text: a

text: ﬁnite

text: time

text: interval.

text: This

text: notion

text: will

text: be

text: used

text: again

text: in

text: the

text: following

text: discussions.

text: A´.

text: Rivas,

text: A.

text: D.

text: K.

text: Plato,

text: S.

text: F.

text: Huelga,

text: and

text: M.

text: B.

text: Plenio,

text: New

text: J.

text: Phys.

text: H.

text: Carmichael,

text: An

text: open

text: systems

text: approach

text: to

text: Quantum

text: Optics:

text: lectures

text: presented

text: at

text: the

text: Universit´e

text: Libre

text: de

text: Bruxelles,

text: October

text: to

text: November

text: Vol.

text: (Springer,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 over the reservoir) could correspond to the previous K (0; µ). In the following discussion, these two conjectures are 0 veriﬁed. (cid:48) − (cid:48) Let us introduce the time evolution operator U (t )=exp[ i(H +H )t ] and rewrite Eq. (19) using the interaction 0r 0 r picture, (cid:48) (cid:48) (cid:48) (cid:48) − (cid:48) ∂ (cid:48)K (t ; µ) + i[HI(t ), K (t ; µ)] + iK (t ; µ)eiµH0[e iµH0, HI(t )] −t I (cid:48) (cid:48) 1− I(cid:48) (cid:48) I 1 = i[V (t )K (t ; µ) K (t ; µ)V (t )], (20) I I I Iµ where the operator subscripts I denote that these are the interaction picture operators, and (cid:48) † (cid:48) − (cid:48) V (t ) = U (t )eiµ(H0+Hr)V e iµ(H0+Hr)U (t ). (21) Iµ 0r 0r Note that I have moved all terms that do not involve the interaction term, V , to the left-hand side (LHS) of the equation. This is in preparation for the perturbation calculation below. Eq. (20) has an integral form: (cid:90) (cid:48) (cid:90) tf − (cid:90) K (t ; µ) = I+i dτ[V (τ)K (τ; µ) K (τ; µ)V (τ)] I I I I Iµ (cid:48) t tf tf − +i dτ[HI(τ), K (τ; µ)] + i dτK (τ; µ)eiµH0[e iµH0, HI(τ)]. (22) (cid:48) 1 I (cid:48) I 1 t t I substitute Eq. (22) into the right-hand side (RHS) of Eq. (20) and obtain (cid:90) (cid:48) (cid:48) (cid:48) (cid:48) − (cid:48) ∂ (cid:48)K (t ; µ) + i[HI(t ), K (t ; µ)] + iK (t ; µ)eiµH0[e iµH0, HI(t )] t I 1 I I 1 tf(cid:90) (cid:48) − = dτV (t )[V (τ)K (τ; µ) K (τ; µ)V (τ)] I I I I Iµ (cid:48) t − (cid:90) tf − (cid:90) (cid:48) − (cid:48) − (cid:48) dτ[V (τ)K (τ; µ) K (τ; µ)V (τ)]V (t ) i[V (t ) V (t )] I I I Iµ Iµ I Iµ (cid:48) t (cid:90) tf (cid:48) − tf (cid:48) + dτV (t )[HI(τ), K (τ; µ)] dτ[HI(τ), K (τ; µ)]V (t ) (cid:48) I 1 I (cid:48) 1 I Iµ t t (cid:90) tf (cid:48) − + dτV (t )K (τ; µ)eiµH0[e iµH0, HI(τ)] (cid:48) I I 1 t − tf − (cid:48) dτK (τ; µ)eiµH0[e iµH0, HI(τ)]V (t ). (23) (cid:48) I 1 Iµ t Multiplying both sides by ρ and taking a trace over the reservoir, I transform the LHS of the above equation into r (cid:48) (cid:48) (cid:48) (cid:48) − (cid:48) LHS = ∂ (cid:48)K (t ; µ) + i[HI(t ), K (t ; µ)] + iK (t ; µ)eiµH0[e iµH0, HI(t )], (24) t 0I 1 0I 0I 1 (cid:48) (cid:48) (cid:48) where K (t ; µ)=Tr [K(t ; µ)ρ ]. Note that HI(t ) is now the interaction picture operator for the system due to 0I (cid:48) r r 1 the fact that [H (t ), H ]=0. So far, all these derivations are exact. In order to attend to the complicated RHS of 1 r Eq. (23), however, I resort to certain approximations. Following the standard concepts pertaining to the dynamics ≈ ⊗ of open systems [35], I make an important assumption, that K (τ; µ) K (τ; µ) I for all K -terms on the RHS. I 0I r I This is justiﬁed if the ﬁeld term, H (t), and the coupling term, V , are so very weak that the reservoir is almost 1 (cid:48) not aﬀected by these interactions. By imposing the conventional condition Tr[R (t )ρ ]=0 and performing a Markov I r approximation, I immediately obtain the following expression:

text: over

text: the

text: reservoir)

text: could

text: correspond

text: to

text: the

text: previous

text: K

text: µ).

text: In

text: the

text: following

text: discussion,

text: these

text: two

text: conjectures

text: are

text: veriﬁed.

text: Let

text: us

text: introduce

text: the

text: time

text: evolution

text: operator

text: U

text: (t

text: )=exp[

text: i(H

text: +H

text: )t

text: and

text: rewrite

text: Eq.

text: using

text: the

text: interaction

text: r

text: picture,

text: (t

text: µ)

text: i[HI(t

text: K

text: (t

text: µ)]

text: iK

text: (t

text: HI(t

text: −t

text: I

text: 1−

text: I

text: i[V

text: (t

text: )K

text: (t

text: µ)

text: K

text: (t

text: µ)V

text: (t

text: I

text: I

text: I

text: Iµ

text: where

text: the

text: operator

text: subscripts

text: I

text: denote

text: that

text: these

text: are

text: the

text: interaction

text: picture

text: operators,

text: and

text: V

text: (t

text: U

text: (t

text: e

text: (t

text: Iµ

text: Note

text: that

text: I

text: have

text: moved

text: all

text: terms

text: that

text: do

text: not

text: involve

text: the

text: interaction

text: term,

text: V

text: to

text: the

text: left-hand

text: side

text: (LHS)

text: of

text: the

text: equation.

text: This

text: is

text: in

text: preparation

text: for

text: the

text: perturbation

text: calculation

text: below.

text: Eq.

text: has

text: an

text: integral

text: form:

text: tf

text: K

text: (t

text: µ)

text: I+i

text: dτ[V

text: (τ)K

text: (τ;

text: µ)

text: K

text: (τ;

text: µ)V

text: (τ)]

text: I

text: I

text: I

text: I

text: Iµ

text: t

text: tf

text: tf

text: +i

text: dτ[HI(τ),

text: K

text: (τ;

text: µ)]

text: i

text: dτK

text: (τ;

text: HI(τ)].

text: I

text: I

text: t

text: t

text: I

text: substitute

text: Eq.

text: into

text: the

text: right-hand

text: side

text: (RHS)

text: of

text: Eq.

text: and

text: obtain

text: (t

text: µ)

text: i[HI(t

text: K

text: (t

text: µ)]

text: iK

text: (t

text: HI(t

text: t

text: I

text: I

text: I

text: dτV

text: (t

text: )[V

text: (τ)K

text: (τ;

text: µ)

text: K

text: (τ;

text: µ)V

text: (τ)]

text: I

text: I

text: I

text: I

text: Iµ

text: t

text: tf

text: dτ[V

text: (τ)K

text: (τ;

text: µ)

text: K

text: (τ;

text: µ)V

text: (τ)]V

text: (t

text: i[V

text: (t

text: V

text: (t

text: I

text: I

text: I

text: Iµ

text: Iµ

text: I

text: Iµ

text: t

text: tf

text: tf

text: dτV

text: (t

text: )[HI(τ),

text: K

text: (τ;

text: µ)]

text: dτ[HI(τ),

text: K

text: (τ;

text: µ)]V

text: (t

text: I

text: I

text: I

text: Iµ

text: t

text: t

text: tf

text: dτV

text: (t

text: )K

text: (τ;

text: HI(τ)]

text: I

text: I

text: t

text: tf

text: dτK

text: (τ;

text: HI(τ)]V

text: (t

text: I

text: Iµ

text: t

text: Multiplying

text: both

text: sides

text: by

text: ρ

text: and

text: taking

text: a

text: trace

text: over

text: the

text: reservoir,

text: I

text: transform

text: the

text: LHS

text: of

text: the

text: above

text: equation

text: into

text: r

text: LHS

text: (t

text: µ)

text: i[HI(t

text: K

text: (t

text: µ)]

text: iK

text: (t

text: HI(t

text: t

text: where

text: K

text: (t

text: µ)=Tr

text: [K(t

text: µ)ρ

text: Note

text: that

text: HI(t

text: is

text: now

text: the

text: interaction

text: picture

text: operator

text: for

text: the

text: system

text: due

text: to

text: r

text: r

text: the

text: fact

text: that

text: [H

text: (t

text: H

text: So

text: far,

text: all

text: these

text: derivations

text: are

text: exact.

text: In

text: order

text: to

text: attend

text: to

text: the

text: complicated

text: RHS

text: of

text: r

text: Eq.

text: however,

text: I

text: resort

text: to

text: certain

text: approximations.

text: Following

text: the

text: standard

text: concepts

text: pertaining

text: to

text: the

text: dynamics

text: of

text: open

text: systems

text: I

text: make

text: an

text: important

text: assumption,

text: that

text: K

text: (τ;

text: µ)

text: K

text: (τ;

text: µ)

text: I

text: for

text: all

text: K

text: -terms

text: on

text: the

text: RHS.

text: I

text: r

text: I

text: This

text: is

text: justiﬁed

text: if

text: the

text: ﬁeld

text: term,

text: H

text: (t),

text: and

text: the

text: coupling

text: term,

text: V

text: are

text: so

text: very

text: weak

text: that

text: the

text: reservoir

text: is

text: almost

text: not

text: aﬀected

text: by

text: these

text: interactions.

text: By

text: imposing

text: the

text: conventional

text: condition

text: Tr[R

text: (t

text: )ρ

text: and

text: performing

text: a

text: Markov

text: I

text: r

text: approximation,

text: I

text: immediately

text: obtain

text: the

text: following

text: expression:


text: 7 (cid:90) (cid:90)tf (cid:48) (cid:48) (cid:104) (cid:48) (cid:105) RHS = dτS (t )S (τ)K (t ; µ) R (t )R (τ) I I 0I I I r (cid:48) t − (cid:90) tf (cid:48) (cid:48) (cid:104) (cid:48) (cid:105) dτS (t )K (t ; µ)S (τ) R (t )R (τ) I 0I Iµ I Iµ r (cid:48) t − (cid:90) tf (cid:48) (cid:48) (cid:104) (cid:48) (cid:105) dτS (τ)K (t ; µ)S (t ) R (τ)R (t ) I 0I Iµ I Iµ r (cid:48) t tf (cid:48) (cid:48) (cid:104) (cid:48) (cid:105) + dτK (t ; µ)S (τ)S (t ) R (τ)R (t ) , (25) 0I Iµ Iµ Iµ Iµ r (cid:48) t (cid:104)· · · (cid:105) · · · where =Tr [ ρ ] are the reservoir correlation functions. If I further assume the RWA to be valid, the r r r integrals in the above equation may be eliminated, and I arrive at a ﬁnal form: − (cid:48) − (cid:48) RHS = i[H , K (t ; µ)] D(cid:63)[K (t ; µ)], (26) LS 0I 0I where H represents the Lamb shift [35]. Considering that this is a standard procedure, I only highlight the key LS steps in Appendix II. If I transform Eqs. (24) and (26) back into Schr¨odinger’s theoretical framework and neglect the smaller Lamb shift, I reproduce Eq. (6). The backward equation of the heat. As discussed earlier, the PDFs of the heat Q and Q are the same in the TEM QJT master equation (1). Hence, deﬁning one CF for the heat is adequate, e.g., Ξ(µ). Previous results [9, 12, 25, 36, 37] have shown that this function can be evaluated by Ξ(µ) = E[eiµQQJT ] = Tr0[ρˆ(tf; µ)], (27) where ρˆ in the latter equation satisﬁes the same Eq. (9) but with a diﬀerent initial condition of ρ . In comparison (cid:48) ≤ (cid:48) ≤ 0 with the case of the work, one may expect that there exists an operator F (t ; µ) (0 t t ) that leads to the 0 f relation Ξ(µ) = Tr[F (0; µ)ρ ]. (28) 0 0 It is not diﬃcult to ﬁnd the backward equation of the operator if one follows either of the two methods related to the work. Here I only present the ﬁnal result: (cid:48) −L (cid:48) ∂ (cid:48)F (t ; µ) = ˘(cid:63)(cid:48)(µ)F (t ; µ) (29) t 0 t 0 with a terminal condition F (t ; µ) = I. The current discussion also highlights the fact that, if the initial density 0 f matrix of the system is a completely random ensemble, ρ , such as in the N-level system, ρ =I/N, the PDF of the C C heat must obey an exact equality, − EC[e βQQJT ] = 1, (30) where I used the subscript C to indicate the fact that all the QJTs star from this speciﬁc initial condition. The proof of this is reserved for Appendix III. Note that there exists a classical version of the equality in the classical stochastic process [43]. I would like to assert that, since the use of the backward and forward equations related to the work (heat) always leads to the same CFs, from the viewpoint of computing, there are no speciﬁc reasons to favor one more than the other.

text: RHS

text: dτS

text: (t

text: )S

text: (τ)K

text: (t

text: µ)

text: R

text: (t

text: )R

text: (τ)

text: I

text: I

text: I

text: I

text: r

text: t

text: tf

text: dτS

text: (t

text: )K

text: (t

text: µ)S

text: (τ)

text: R

text: (t

text: )R

text: (τ)

text: I

text: Iµ

text: I

text: Iµ

text: r

text: t

text: tf

text: dτS

text: (τ)K

text: (t

text: µ)S

text: (t

text: R

text: (τ)R

text: (t

text: I

text: Iµ

text: I

text: Iµ

text: r

text: t

text: tf

text: dτK

text: (t

text: µ)S

text: (τ)S

text: (t

text: R

text: (τ)R

text: (t

text: Iµ

text: Iµ

text: Iµ

text: Iµ

text: r

text: t

text: where

text: =Tr

text: ρ

text: are

text: the

text: reservoir

text: correlation

text: functions.

text: If

text: I

text: further

text: assume

text: the

text: RWA

text: to

text: be

text: valid,

text: the

text: r

text: r

text: r

text: integrals

text: in

text: the

text: above

text: equation

text: may

text: be

text: eliminated,

text: and

text: I

text: arrive

text: at

text: a

text: ﬁnal

text: form:

text: RHS

text: i[H

text: K

text: (t

text: µ)]

text: (t

text: µ)],

text: LS

text: where

text: H

text: represents

text: the

text: Lamb

text: shift

text: Considering

text: that

text: this

text: is

text: a

text: standard

text: procedure,

text: I

text: only

text: highlight

text: the

text: key

text: LS

text: steps

text: in

text: Appendix

text: II.

text: If

text: I

text: transform

text: Eqs.

text: and

text: back

text: into

text: Schr¨odinger’s

text: theoretical

text: framework

text: and

text: neglect

text: the

text: smaller

text: Lamb

text: shift,

text: I

text: reproduce

text: Eq.

text: The

text: backward

text: equation

text: of

text: the

text: heat.

text: As

text: discussed

text: earlier,

text: the

text: PDFs

text: of

text: the

text: heat

text: Q

text: and

text: Q

text: are

text: the

text: same

text: in

text: the

text: TEM

text: QJT

text: master

text: equation

text: Hence,

text: deﬁning

text: one

text: CF

text: for

text: the

text: heat

text: is

text: adequate,

text: e.g.,

text: Ξ(µ).

text: Previous

text: results

text: have

text: shown

text: that

text: this

text: function

text: can

text: be

text: evaluated

text: by

text: Ξ(µ)

text: E[eiµQQJT

text: µ)],

text: where

text: ρˆ

text: in

text: the

text: latter

text: equation

text: satisﬁes

text: the

text: same

text: Eq.

text: but

text: with

text: a

text: diﬀerent

text: initial

text: condition

text: of

text: ρ

text: In

text: comparison

text: with

text: the

text: case

text: of

text: the

text: work,

text: one

text: may

text: expect

text: that

text: there

text: exists

text: an

text: operator

text: F

text: (t

text: µ)

text: t

text: t

text: that

text: leads

text: to

text: the

text: f

text: relation

text: Ξ(µ)

text: Tr[F

text: µ)ρ

text: It

text: is

text: not

text: diﬃcult

text: to

text: ﬁnd

text: the

text: backward

text: equation

text: of

text: the

text: operator

text: if

text: one

text: follows

text: either

text: of

text: the

text: two

text: methods

text: related

text: to

text: the

text: work.

text: Here

text: I

text: only

text: present

text: the

text: ﬁnal

text: result:

text: −L

text: (t

text: µ)

text: (t

text: µ)

text: t

text: t

text: with

text: a

text: terminal

text: condition

text: F

text: (t

text: µ)

text: I.

text: The

text: current

text: discussion

text: also

text: highlights

text: the

text: fact

text: that,

text: if

text: the

text: initial

text: density

text: f

text: matrix

text: of

text: the

text: system

text: is

text: a

text: completely

text: random

text: ensemble,

text: ρ

text: such

text: as

text: in

text: the

text: N-level

text: system,

text: ρ

text: =I/N,

text: the

text: PDF

text: of

text: the

text: C

text: C

text: heat

text: must

text: obey

text: an

text: exact

text: equality,

text: EC[e

text: βQQJT

text: where

text: I

text: used

text: the

text: subscript

text: C

text: to

text: indicate

text: the

text: fact

text: that

text: all

text: the

text: QJTs

text: star

text: from

text: this

text: speciﬁc

text: initial

text: condition.

text: The

text: proof

text: of

text: this

text: is

text: reserved

text: for

text: Appendix

text: III.

text: Note

text: that

text: there

text: exists

text: a

text: classical

text: version

text: of

text: the

text: equality

text: in

text: the

text: classical

text: stochastic

text: process

text: I

text: would

text: like

text: to

text: assert

text: that,

text: since

text: the

text: use

text: of

text: the

text: backward

text: and

text: forward

text: equations

text: related

text: to

text: the

text: work

text: (heat)

text: always

text: leads

text: to

text: the

text: same

text: CFs,

text: from

text: the

text: viewpoint

text: of

text: computing,

text: there

text: are

text: no

text: speciﬁc

text: reasons

text: to

text: favor

text: one

text: more

text: than

text: the

text: other.


text: where (cid:90) (cid:114) η ν2 − ≡ 1 Ψ = dτ s k2c2, ν2 + η2U , (2.22) s η∗ τ2 s s 4 s For the tensor perturbations, the power spectrum given by the second of (2.9) can be approximated as −2k3 η P = e2Ψt, (2.23) t π2 a2ν t where (cid:90) (cid:114) η ν2 − ≡ 1 Ψ = dτ t k2, ν2 + η2U . (2.24) t η∗ τ2 t 4 t → − Note that the limit η 0 is implicit in the expressions for P and P above. These s t two expressions for the power spectra are our working equations to leading order in the UA. 3 Series Expansion of the Conformal time and the Hubble and Sound Flow Functions 3.1 Conformal Time With the working equation for P and P given in the previous section, the possibility s t of calculating power spectra to any order in the Hubble and sound ﬂow parameters mainly relies on the convenience provided by the recursive deﬁnitions of (cid:15) and δ given by (2.10) n n and (2.11) and the ability to express the conformal time in terms of (cid:15) up to any order with n respect to (cid:15) . It is often the case that recursive integration is used to expressed η up to second n or third order in (cid:15) with the possibility of extending it indeﬁnitely. In this subsection, we n give a short and simple derivation of a series expansion for the conformal time by indeﬁnitely extending the recursive integration by parts usually used in the literature. ≡ −∞ We start with the deﬁnition dη dt/a. With η in the range ( , 0], this deﬁnition implies that (cid:90) (cid:90) (cid:0) (cid:1) da − 1 dt d − η = = + H 1 . (3.1) a2H aH a dt The quantity dt/a = da/(a2H) appears again in the integrand in the right hand side so following the ﬁrst one, we perform another round(cid:90)of integ(cid:18)ration by parts(cid:19). (cid:0) (cid:1) (cid:0) (cid:1) − 1 − 1 − d − dt d − d − η = H 1 H 1 + H 1 H 1 (3.2) aH a dt a dt dt – 10 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: η

text: ν2

text: Ψ

text: dτ

text: s

text: ν2

text: s

text: η∗

text: τ2

text: s

text: s

text: s

text: For

text: the

text: tensor

text: perturbations,

text: the

text: power

text: spectrum

text: given

text: by

text: the

text: second

text: of

text: can

text: be

text: approximated

text: as

text: η

text: P

text: t

text: π2

text: t

text: where

text: η

text: ν2

text: Ψ

text: dτ

text: t

text: ν2

text: t

text: η∗

text: τ2

text: t

text: t

text: Note

text: that

text: the

text: limit

text: η

text: is

text: implicit

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