text: (cid:48) where ρ, ρ are obtained from (56) by substituting t for T , and the last term on the RHS should be interprete(cid:110)d as zero when ρ is ze(cid:111)ro. (cid:0) (cid:1) It thus follows from (57), (59), (60), and the above interpretation that (cid:110) (cid:111) ≥ h f min h (X − (cid:48) , . . . , X ) α Z ≤ (cid:48)≤ − α n ρ +1 n 1 ρ n 1 (cid:26)(cid:22) (cid:23) (cid:27) ∧ + 0 min h (X , . . . , X ) ≤ ≤ − α 1 ρ 1 ρ n 1 − m n + t + min h (f ) . (61) ≤ ≤ − α n 0 t n 1 n −∞ The ﬁrst two terms do not depend on m and are greater than when- (cid:0) (cid:1) ever (47) holds. Dividing (61) by m and letting m tend to inﬁnity (with n held ﬁxed), establishes (48). To establish (50) we need an upper bound on h f . Such a bound α Z can be obtained from Lemma 8. The exact form of the bound depends on whether α exceeds 1 or not. But either form leads to (50) upon dividing by m and letting it tend to inﬁnity. To conclude the proof we note that (51) follows from (50) and (48). · ∞ Proof of Theorem 2. Since h(cid:63)( ) is continuous on the ray (Γ , ), and since · 0 Γ > Γ by the theorem’s hypotheses, h(cid:63)( ) is continuous at Γ. Consequently, 0 (cid:48) we can ﬁnd some Γ for which (cid:48) Γ < Γ (62a) (cid:48) − h(cid:63)(Γ ) > h(cid:63)(Γ) ε˜. (62b) These inequalities imply that we can ﬁnd some δ > 0 small enough so that (cid:48) Γ + δ < Γ (63a) (cid:48) − − h(cid:63)(Γ ) δ > h(cid:63)(Γ) ε˜. (63b) 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: ρ,

text: ρ

text: are

text: obtained

text: from

text: by

text: substituting

text: t

text: for

text: T

text: and

text: the

text: last

text: term

text: on

text: the

text: RHS

text: should

text: be

text: as

text: zero

text: when

text: ρ

text: is

text: It

text: thus

text: follows

text: from

text: and

text: the

text: above

text: interpretation

text: that

text: h

text: f

text: min

text: h

text: (X

text: X

text: α

text: Z

text: α

text: n

text: ρ

text: n

text: ρ

text: n

text: min

text: h

text: (X

text: X

text: α

text: ρ

text: ρ

text: n

text: m

text: n

text: t

text: min

text: h

text: (f

text: α

text: n

text: t

text: n

text: n

text: The

text: ﬁrst

text: two

text: terms

text: do

text: not

text: depend

text: on

text: m

text: and

text: are

text: greater

text: than

text: when-

text: ever

text: holds.

text: Dividing

text: by

text: m

text: and

text: letting

text: m

text: tend

text: to

text: inﬁnity

text: (with

text: n

text: held

text: ﬁxed),

text: establishes

text: To

text: establish

text: we

text: need

text: an

text: upper

text: bound

text: on

text: h

text: f

text: Such

text: a

text: bound

text: α

text: Z

text: can

text: be

text: obtained

text: from

text: Lemma

text: The

text: exact

text: form

text: of

text: the

text: bound

text: depends

text: on

text: whether

text: α

text: exceeds

text: or

text: not.

text: But

text: either

text: form

text: leads

text: to

text: upon

text: dividing

text: by

text: m

text: and

text: letting

text: it

text: tend

text: to

text: inﬁnity.

text: To

text: conclude

text: the

text: proof

text: we

text: note

text: that

text: follows

text: from

text: and

text: Proof

text: of

text: Theorem

text: Since

text: is

text: continuous

text: on

text: the

text: ray

text: (Γ

text: and

text: since

text: Γ

text: Γ

text: by

text: the

text: theorem’s

text: hypotheses,

text: is

text: continuous

text: at

text: Γ.

text: Consequently,

text: we

text: can

text: ﬁnd

text: some

text: Γ

text: for

text: which

text: Γ

text: Γ

text: ε˜.

text: These

text: inequalities

text: imply

text: that

text: we

text: can

text: ﬁnd

text: some

text: δ

text: small

text: enough

text: so

text: that

text: Γ

text: δ

text: Γ

text: δ

text: ε˜.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: When a random variable (RV) X is of density f we sometimes write X h (X) instead of h (f ). The R´enyi entropy of some multivariate densities α α X are computed in [8]. (cid:16) (cid:17) S If the support of f is contained in , then ≤ |S| (cid:54) h (f) log , α > 0, α = 1 , (6) α |A| A where denotes the Lebesgue measure of the set , and where we |S| ∞ |S| interpret log as + when is inﬁnite. (Throughout this paper we ∞ ∞ −∞ deﬁne log = and log 0 = (cid:90) .) The R´enyi entropy is closely related to the Shannon entropy: ∞ − h(f) = f(x) log f(x) dx. (7) −∞ (The integral on the RHS of (7) need not exist. If it does not, then we say that h(f) does not exist.) Depending on whether α is smaller or larger than one, the R´enyi entropy can be larger or smaller than the Shannon entropy. ∞ Indeed, if f is of Shannon entropy h(f) (possibly + ), then by [9, Lemma 5.1 (iv)]: ≤ h (f) h(f), for α > 1; (8) α ≥ h (f) h(f), for 0 < α < 1. (9) α Moreover, under some mild technical conditions [9, Lemma 5.1 (ii)]: lim h (f) = h(f). (10) → α α 1 { } { } The order-α R´enyi rate h ( X ) of a stochastic process (SP) X is α k k (cid:0) (cid:1) deﬁned as { } 1 h ( X ) = lim h Xn (11) α k n→∞ n α 1 whenever the limit exists.2 Here Xj denotes the tuple (X , . . . , X ). i i j ∞ 2We say that the limit exists and is equal to + if for every M > 0 there exists some n such that for all n > n the R´enyi entropy h (X , . . . , X ) exceeds nM, possibly by 0 ∞ 0 α 1 n being + . 3 ##LTLine## ##LTLine##

text: When

label: paragraph

text: a

text: random

text: variable

text: (RV)

text: X

text: is

text: of

text: density

text: f

text: we

text: sometimes

text: write

text: X

text: h

text: (X)

text: instead

text: of

text: h

text: (f

text: The

text: R´enyi

text: entropy

text: of

text: some

text: multivariate

text: densities

text: α

text: α

text: X

text: are

text: computed

text: in

label: equation

text: S

text: If

text: the

text: support

text: of

text: f

text: is

text: contained

text: in

text: then

text: |S|

font: default

text: h

text: (f)

text: log

text: α

text: α

text: α

text: |A|

text: A

text: where

text: denotes

text: the

text: Lebesgue

text: measure

text: of

text: the

text: set

text: and

text: where

text: we

text: |S|

text: |S|

text: interpret

text: log

text: as

text: when

text: is

text: inﬁnite.

text: (Throughout

text: this

text: paper

text: we

text: deﬁne

text: log

text: and

text: log

text: The

text: R´enyi

text: entropy

text: is

text: closely

text: related

text: to

text: the

text: Shannon

text: entropy:

text: h(f)

text: f(x)

text: log

text: f(x)

text: dx.

text: (The

text: integral

text: on

text: the

text: RHS

text: of

text: need

text: not

text: exist.

text: If

text: it

text: does

text: not,

text: then

text: we

text: say

text: that

text: h(f)

text: does

text: not

text: exist.)

text: Depending

text: on

text: whether

text: α

text: is

text: smaller

text: or

text: larger

text: than

text: one,

text: the

text: R´enyi

text: entropy

text: can

text: be

text: larger

text: or

text: smaller

text: than

text: the

text: Shannon

text: entropy.

text: Indeed,

text: if

text: f

text: is

text: of

text: Shannon

text: entropy

text: h(f)

text: (possibly

text: then

text: by

text: Lemma

text: (iv)]:

text: h

text: (f)

text: h(f),

text: for

text: α

text: α

text: h

text: (f)

text: h(f),

text: for

text: α

text: α

text: Moreover,

text: under

text: some

text: mild

text: technical

text: conditions

text: Lemma

text: (ii)]:

text: lim

text: h

text: (f)

text: h(f).

text: α

text: α

text: The

text: order-α

text: R´enyi

text: rate

text: h

text: X

text: of

text: a

text: stochastic

text: process

text: (SP)

text: X

text: is

text: α

text: k

text: k

text: deﬁned

text: as

text: h

text: X

text: lim

text: h

text: Xn

text: α

text: k

text: n→∞

text: n

text: α

text: whenever

text: the

text: limit

text: Here

text: Xj

text: denotes

text: the

text: tuple

text: (X

text: X

text: i

text: i

text: j

text: say

text: that

text: the

text: limit

text: exists

text: and

text: is

text: equal

text: to

text: if

text: for

text: every

text: M

text: there

text: exists

text: some

text: n

text: such

text: that

text: for

text: all

text: n

text: n

text: the

text: R´enyi

text: entropy

text: h

text: (X

text: X

text: exceeds

text: nM,

text: possibly

text: by

text: α

text: n

text: being

text: ##LTLine##

text: ##LTLine##


text: (cid:90) ∧ (cid:0) (cid:1) ≤ By (111), f(x) M = f(x) whenever f(x) 1, so (cid:90) ∧ 1 f(x) M log ∧ dx ≤ f(x) M x: f(x) 1 1 = f(x) log dx. (118) ≤ f(x) x: f(x) 1 − ∧ Since ξ log ξ 1 is decreasing for ξ > 1, and since f(x) > 1 implies f(x) M > 1 (cid:16) (cid:17) (cid:0) (cid:1) (by (112c)), ∧ 1 ≥ 1 f(x) M log ∧ f(x) log , f(x) > 1 f(x) M f(x) (cid:90) (cid:0) (cid:1) and hence (cid:90) ∧ 1 f(x) M log ∧ dx f(x) M x: f(x)>1 ≥ 1 f(x) log dx. (119) f(x) x: f(x)>1 (cid:90) (cid:0) (cid:1) Summing (118) and (119) we obtain ∧ 1 ≥ f(x) M log ∧ dx h(f). (120) f(x) M Using this, (117), and (114) we conclude that ∞ ˜ h(f) = h(f), whenever h(f) = and ≥ − − ε | | h(f˜) log(1 ε) + h(f) − h(f) , 1 ε | | ∞ whenever h(f) < . (121) ≥ −∞ ˜ And obviously h(f) h(f) whenever h(f) = . The result now follows by choosing ε small enough to guarantee that the RHS of (116) does not exceed Γ + δ and—if h(f) is ﬁnite—that the RHS of − (121) exceeds h(f) δ. The following lemma addresses the case where (107) does not hold. 31 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: By

text: f(x)

text: M

text: f(x)

text: whenever

text: f(x)

text: so

text: f(x)

text: M

text: log

text: dx

text: f(x)

text: M

text: x:

text: f(x)

text: f(x)

text: log

text: dx.

text: f(x)

text: x:

text: f(x)

text: Since

text: ξ

text: log

text: ξ

text: is

text: decreasing

text: for

text: ξ

text: and

text: since

text: f(x)

text: implies

text: f(x)

text: M

text: (by

text: f(x)

text: M

text: log

text: f(x)

text: log

text: f(x)

text: f(x)

text: M

text: f(x)

text: and

text: hence

text: f(x)

text: M

text: log

text: dx

text: f(x)

text: M

text: x:

text: f(x)

text: log

text: dx.

text: f(x)

text: x:

text: Summing

text: and

text: we

text: obtain

text: f(x)

text: M

text: log

text: dx

text: h(f).

text: f(x)

text: M

text: Using

text: this,

text: and

text: we

text: conclude

text: that

text: h(f)

text: h(f),

text: whenever

text: h(f)

text: and

text: ε

text: h(f˜)

text: ε)

text: h(f)

text: h(f)

text: ε

text: whenever

text: h(f)

text: And

text: obviously

text: h(f)

text: h(f)

text: whenever

text: h(f)

text: The

text: result

text: now

text: follows

text: by

text: choosing

text: ε

text: small

text: enough

text: to

text: guarantee

text: that

text: the

text: RHS

text: of

text: does

text: not

text: exceed

text: Γ

text: δ

text: and—if

text: h(f)

text: is

text: ﬁnite—that

text: the

text: RHS

text: of

text: exceeds

text: h(f)

text: δ.

text: The

text: following

text: lemma

text: addresses

text: the

text: case

text: where

text: does

text: not

text: hold.

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: Note that since f is a density, ≤ β 1 k and (by the MCT) ↑ β 1. (130) k Consequently, ≤ 0 < β 1, k large. (131) k For every such suﬃciently large k, deﬁne the density (cid:44) − { ∈ D } f˜ (x) β 1f(x) I x . k k k (cid:90) S ˜ It is supported by , and its entropy h(f ) can be expressed as k (cid:90) 1 ˜ ˜ h(f ) = f (x) log dx k k ˜ f (x) k (cid:90) 1 ˜ = f (x) log dx D k f˜ (x) k (cid:90) k 1 β k = f(x) log dx D β f(x) k k 1 1 = log β + f(x) log dx. k β D f(x) k k From this, (129), and (130) we obtain ˜ lim h(f ) = h(f). (132) →∞ k k (cid:90) ˜ And as to the expectation of r(x) under f : k (cid:90) ˜ f (x) r(x) dx k (cid:90) (cid:90) 1 = f(x) r(x) dx β D k k 1 − 1 − = f(x) r+(x) dx f(x) r (x) dx. β D β D k k k k 34 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Note

text: that

text: since

text: f

text: is

text: a

text: density,

text: β

text: k

text: and

text: (by

text: the

text: MCT)

text: β

text: k

text: Consequently,

text: β

text: k

text: large.

text: k

text: For

text: every

text: such

text: suﬃciently

text: large

text: k,

text: deﬁne

text: the

text: density

text: D

text: f˜

text: (x)

text: β

text: I

text: x

text: k

text: k

text: k

text: S

text: It

text: is

text: supported

text: by

text: and

text: its

text: entropy

text: h(f

text: can

text: be

text: expressed

text: as

text: k

text: h(f

text: f

text: (x)

text: log

text: dx

text: k

text: k

text: f

text: (x)

text: k

text: f

text: (x)

text: log

text: dx

text: D

text: k

text: f˜

text: (x)

text: k

text: k

text: β

text: k

text: f(x)

text: log

text: dx

text: D

text: β

text: f(x)

text: k

text: k

text: log

text: β

text: f(x)

text: log

text: dx.

text: k

text: β

text: D

text: f(x)

text: k

text: k

text: From

text: this,

text: and

text: we

text: obtain

text: lim

text: h(f

text: h(f).

text: k

text: k

text: And

text: as

text: to

text: the

text: expectation

text: of

text: r(x)

text: under

text: f

text: k

text: f

text: (x)

text: r(x)

text: dx

text: k

text: f(x)

text: r(x)

text: dx

text: β

text: D

text: k

text: k

text: f(x)

text: r+(x)

text: dx

text: f(x)

text: r

text: (x)

text: dx.

text: β

text: D

text: β

text: D

text: k

text: k

text: k

text: k

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: The ﬁrst term on the LHS is(cid:90)ﬁnite by (131) and (124b). The second tends −∞ to by (130) and (127). Hence, −∞ ˜ lim f (x) r(x) dx = . (133) →∞ k k (cid:90) Moreover, (cid:90) (cid:90) | | ˜ f (x) r(x) dx k (cid:90) 1 1 − = f(x) r+(x) dx + f(x) r (x) dx β D β D k k k k ≤ 1 f(x) r+(x) dx + k β ∞k < , (134) where the ﬁrst inequality follows from the nonnegativity of r+ and from the D deﬁnition of the set (125), and the second inequality follows from (124b) k and (131). The lemma now follows from (134), (132), and (133). Proof of Proposition 10. Since Γ exceeds Γ , it follows from (17) that 0 −∞ ∞ < h(cid:63)(Γ) < . (135) Let the den(cid:90)sity f nearly achieve h(cid:63)(Γ) in the sense that it is supported S by and that ≤ − δ f(x) r(x) dx Γ, and h(f) > h(cid:63)(Γ) . (136) 2 By (135), (136), and the deﬁnition of h(cid:63)(Γ), −∞ ∞ (cid:82) < h(f) < . (137) (cid:82) | | If f(x) r(x) dx is ﬁnite, then the result follows directly from Lemma 13. It remains to prove the result when this integral is inﬁnite. In this case −∞ ∞ f(x) r(x) dx = by (136) (because Γ < ). Using this, the ﬁniteness 35 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: ﬁrst

text: term

text: on

text: the

text: LHS

text: by

text: and

text: The

text: second

text: tends

text: to

text: by

text: and

text: Hence,

text: lim

text: f

text: (x)

text: r(x)

text: dx

text: k

text: k

text: Moreover,

text: f

text: (x)

text: r(x)

text: dx

text: k

text: f(x)

text: r+(x)

text: dx

text: f(x)

text: r

text: (x)

text: dx

text: β

text: D

text: β

text: D

text: k

text: k

text: k

text: k

text: f(x)

text: r+(x)

text: dx

text: k

text: β

text: ∞k

text: where

text: the

text: ﬁrst

text: inequality

text: follows

text: from

text: the

text: nonnegativity

text: of

text: r+

text: and

text: from

text: the

text: D

text: deﬁnition

text: of

text: the

text: set

text: and

text: the

text: second

text: inequality

text: follows

text: from

text: k

text: and

text: The

text: lemma

text: now

text: follows

text: from

text: and

text: Proof

text: of

text: Proposition

text: Since

text: Γ

text: exceeds

text: Γ

text: it

text: follows

text: from

text: that

text: Let

text: the

text: f

text: nearly

text: achieve

text: in

text: the

text: sense

text: that

text: it

text: is

text: supported

text: S

text: by

text: and

text: that

text: δ

text: f(x)

text: r(x)

text: dx

text: Γ,

text: and

text: h(f)

text: By

text: and

text: the

text: deﬁnition

text: of

text: h(f)

text: If

text: f(x)

text: r(x)

text: dx

text: is

text: ﬁnite,

text: then

text: the

text: result

text: follows

text: directly

text: from

text: Lemma

text: It

text: remains

text: to

text: prove

text: the

text: result

text: when

text: this

text: integral

text: is

text: inﬁnite.

text: In

text: this

text: case

text: f(x)

text: r(x)

text: dx

text: by

text: (because

text: Γ

text: Using

text: this,

text: the

text: ﬁniteness

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ˜ of h(f) (137), and Lemma(cid:90) 14, we infer the existence of a density f that S supported by and for which | | ∞ ˜ f(x) r(x) dx < , (138a) − δ ˜ h(f) > h(f) , (138b) (cid:90) 2 ˜ f(x) r(x) dx < Γ. (138c) ˜ Applying Lemma 13 to the density(cid:90) f, we conclude that there exists a S bounded density f(cid:63) that is supported by and that satisﬁes − δ ≤ h(f(cid:63)) > h(f˜) and f(cid:63)(x) r(x) dx Γ + δ (139) 2 (cid:90) and hence, in view of (138) and (136), − ≤ h(f(cid:63)) > h(cid:63)(Γ) δ and f(cid:63)(x) r(x) dx Γ + δ. (140) (cid:82) The existence of f(cid:63) concludes the proof of the proposition for the case | | where f(x) r(x) dx is inﬁnite. Acknowledgment Discussions with Stefan M. Moser and Igal Sason are gratefully acknowl- edged. References [1] C. Bunte and A. Lapidoth, “R´enyi entropy and quantization for densi- ties,” in Proc. Information Theory Workshop, Nov. 2014, pp. 258–262. 36 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: of

text: h(f)

text: and

text: we

text: infer

text: the

text: existence

text: of

text: a

text: density

text: f

text: that

text: S

text: supported

text: by

text: and

text: for

text: which

text: f(x)

text: r(x)

text: dx

text: δ

text: h(f)

text: h(f)

text: f(x)

text: r(x)

text: dx

text: Γ.

text: Applying

text: Lemma

text: to

text: the

text: f,

text: we

text: conclude

text: that

text: there

text: exists

text: a

text: S

text: bounded

text: density

text: that

text: is

text: supported

text: by

text: and

text: that

text: satisﬁes

text: δ

text: h(f˜)

text: and

text: r(x)

text: dx

text: Γ

text: δ

text: and

text: hence,

text: in

text: view

text: of

text: and

text: δ

text: and

text: r(x)

text: dx

text: Γ

text: δ.

text: The

text: existence

text: of

text: concludes

text: the

text: proof

text: of

text: the

text: proposition

text: for

text: the

text: case

text: where

text: f(x)

text: r(x)

text: dx

text: is

text: inﬁnite.

text: Acknowledgment

text: Discussions

text: with

text: Stefan

text: M.

text: Moser

text: and

text: Igal

text: Sason

text: are

text: gratefully

text: acknowl-

text: edged.

text: References

text: C.

text: Bunte

text: and

text: A.

text: Lapidoth,

text: “R´enyi

text: entropy

text: and

text: quantization

text: for

text: densi-

text: ties,”

text: in

text: Proc.

text: Information

text: Theory

text: Workshop,

text: Nov.

text: pp.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [2] J. P. Burg, “Maximum entropy spectral analysis,” in Proc. 37th Meet. Society of Exploration Geophysicists, 1967. Reprinted in Modern Spec- trum Analysis, D. G. Childers, Ed. New York: IEEE Press, 1978 pp. 34–41, 1967. [3] T. Cover and J. Thomas, Elements of Information Theory, 2nd ed. Hoboken, NJ: John Wiley & Sons, 2006. [4] C. Bunte and A. Lapidoth, “Maximizing R´enyi entropy rate,” in Proc. of the 2014 IEEE 28-th Convention of Electrical and Electronics Engineers in Israel, Eilat, Israel, December 3–5 2014. [5] M. A. Kumar and R. Sundaresan, “Minimization problems based on a parametric family of relative entropies I: Forward projection.” arXiv preprint arXiv:1410.2346, 2014. [6] E. Lutwak, D. Yang, and G. Zhang, “Moment-entropy inequalities,” Ann. Probab, vol. 32, no. 1B, pp. 757–774, 2004. [7] J. Costa, A. Hero, and C. Vignat, “On solutions to multivariate max- imum α-entropy problems,” in Energy Minimization Methods in Com- puter Vision and Pattern Recognition. Springer, 2003, pp. 211–226. [8] K. Zografos and S. Nadarajah, “Expressions for R´enyi and Shannon en- tropies for multivariate distributions,” Statistics and Probability Letters, no. 71, pp. 71–84, 2005. [9] L. Wang and M. Madiman, “Beyond the entropy power inequality, via rearrangements,” IEEE Trans. Inf. Theory, vol. 60, no. 9, pp. 5116–5137, Sept. 2014. [10] Z. Rached, F. Alajaji, and L. Campbell, “R´enyi’s divergence and entropy rates for ﬁnite alphabet markov sources,” IEEE Trans. Inf. Theory, vol. 47, no. 4, pp. 1553–1561, May 2001. [11] L. Golshani, E. Pasha, and G. Yari, “Some properties of R´enyi entropy and R´enyi entropy rate,” Information Sciences, vol. 179, no. 14, pp. 2426–2433, 2009. [12] L. Golshani and E. Pasha, “R´enyi entropy rate for Gaussian processes,” Information Sciences, vol. 180, no. 8, pp. 1486–1491, 2010. 37

label: reference

text: J.

text: P.

text: Burg,

text: “Maximum

text: entropy

text: spectral

text: analysis,”

text: in

text: Proc.

text: Meet.

text: Society

text: of

text: Exploration

text: Geophysicists,

text: Reprinted

text: in

text: Modern

text: Spec-

text: trum

text: Analysis,

text: D.

text: G.

text: Childers,

text: Ed.

text: New

text: York:

text: IEEE

text: Press,

text: pp.

text: T.

text: Cover

text: and

text: J.

text: Thomas,

text: Elements

text: of

text: Information

text: Theory,

text: ed.

text: Hoboken,

text: NJ:

text: John

text: Wiley

text: Sons,

text: C.

text: Bunte

text: and

text: A.

text: Lapidoth,

text: “Maximizing

text: R´enyi

text: entropy

text: rate,”

text: in

text: Proc.

text: of

text: the

text: IEEE

text: Convention

text: of

text: Electrical

text: and

text: Electronics

text: Engineers

text: in

text: Israel,

text: Eilat,

text: Israel,

text: December

text: M.

text: A.

text: Kumar

text: and

text: R.

text: Sundaresan,

text: “Minimization

text: problems

text: based

text: on

text: a

text: parametric

text: family

text: of

text: relative

text: entropies

text: I:

text: Forward

text: projection.”

text: arXiv

text: preprint

text: E.

text: Lutwak,

text: D.

text: Yang,

text: and

text: G.

text: Zhang,

text: “Moment-entropy

text: inequalities,”

text: Ann.

text: Probab,

text: vol.

text: no.

text: pp.

text: J.

text: Costa,

text: A.

text: Hero,

text: and

text: C.

text: Vignat,

text: “On

text: solutions

text: to

text: multivariate

text: max-

text: imum

text: α-entropy

text: problems,”

text: in

text: Energy

text: Minimization

text: Methods

text: in

text: Com-

text: puter

text: Vision

text: and

text: Pattern

text: Recognition.

text: Springer,

text: pp.

text: K.

text: Zografos

text: and

text: S.

text: Nadarajah,

text: “Expressions

text: for

text: R´enyi

text: and

text: Shannon

text: en-

text: tropies

text: for

text: multivariate

text: distributions,”

text: Statistics

text: and

text: Probability

text: Letters,

text: no.

text: pp.

text: L.

text: Wang

text: and

text: M.

text: Madiman,

text: “Beyond

text: the

text: entropy

text: power

text: inequality,

text: via

text: rearrangements,”

text: IEEE

text: Trans.

text: Inf.

text: Theory,

text: vol.

text: no.

text: pp.

text: Sept.

text: Z.

text: Rached,

text: F.

text: Alajaji,

text: and

text: L.

text: Campbell,

text: “R´enyi’s

text: divergence

text: and

text: entropy

text: rates

text: for

text: ﬁnite

text: alphabet

text: markov

text: sources,”

text: IEEE

text: Trans.

text: Inf.

text: Theory,

text: vol.

text: no.

text: pp.

text: May

text: L.

text: Golshani,

text: E.

text: Pasha,

text: and

text: G.

text: Yari,

text: “Some

text: properties

text: of

text: R´enyi

text: entropy

text: and

text: R´enyi

text: entropy

text: rate,”

text: Information

text: Sciences,

text: vol.

text: no.

text: pp.

text: L.

text: Golshani

text: and

text: E.

text: Pasha,

text: “R´enyi

text: entropy

text: rate

text: for

text: Gaussian

text: processes,”

text: Information

text: Sciences,

text: vol.

text: no.

text: pp.


text: because when the RHS is ﬁnite it can be achieve by a uniform distribution { ∈ S ≤ } on the set x : r(x) Γ , a distribution under which (16) clearly holds, and when it is inﬁnite, it can be approached by uniform distributions on ever- increasing compact subsets of this set. We next note that, by the Monotone Convergence Theorem (MCT), |{ ∈ S ≤ }| |S| lim x : r(x) Γ = . (22) →∞ Γ Combining (21) and (22) establishes (20) and hence completes the proof of (18). For α > 1 we note that (11), (14), and the deﬁnition of h(cid:63)(Γ) imply that { } for every SP Z satisfying (1) k { } ≤ h ( Z ) h(cid:63)(Γ), α > 1, (23) α k and consequently, { } ≤ sup h ( Z ) h(cid:63)(Γ), α > 1, (24) α k where the supremum is over all SPs satisfying (1). Perhaps surprisingly, this bound is tight: Theorem 2 (Max R´enyi Rate for α > 1). Suppose that α > 1, and that Γ > Γ , where Γ satisﬁes (17). Then for every ε˜ > 0 there exists a stationary { 0 } 0 − SP Z satisfying (1) whose R´enyi rate is deﬁned and exceeds h(cid:63)(Γ) ε˜. k For 0 < α < 1 we can use (12) to obtain for the same supremum { } ≤ |S| sup h ( Z ) log , 0 < α < 1. (25) α k This seemingly crude bound is tight: Theorem 3 (Max R´enyi Rate for 0 < α < 1). Suppose that 0 < α < 1 and that Γ > Γ , where Γ satisﬁes (17). 0 0 • |S| ∞ ∈ R { } If = , then for every M there exists a stationary SP Z k satisfying (1) whose R´enyi rate is deﬁned and exceeds M. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: because

text: when

text: the

text: RHS

text: is

text: ﬁnite

text: it

text: can

text: be

text: achieve

text: by

text: a

text: uniform

text: distribution

text: S

text: on

text: the

text: set

text: x

text: r(x)

text: Γ

text: a

text: distribution

text: under

text: which

text: clearly

text: holds,

text: and

text: when

text: it

text: is

text: inﬁnite,

text: it

text: can

text: be

text: approached

text: by

text: uniform

text: distributions

text: on

text: ever-

text: increasing

text: compact

text: subsets

text: of

text: this

text: set.

text: We

text: next

text: note

text: that,

text: by

text: the

text: Monotone

text: Convergence

text: Theorem

text: (MCT),

text: S

text: |S|

text: lim

text: x

text: r(x)

text: Γ

text: Γ

text: Combining

text: and

text: establishes

text: and

text: hence

text: completes

text: the

text: proof

text: of

text: For

text: α

text: we

text: note

text: that

text: and

text: the

text: deﬁnition

text: of

text: imply

text: that

text: for

text: every

text: SP

text: Z

text: satisfying

text: k

text: h

text: Z

text: α

text: α

text: k

text: and

text: consequently,

text: sup

text: h

text: Z

text: α

text: α

text: k

text: where

text: the

text: supremum

text: is

text: over

text: all

text: SPs

text: satisfying

text: Perhaps

text: surprisingly,

text: this

text: bound

text: is

text: tight:

text: Theorem

text: (Max

text: R´enyi

text: Rate

text: for

text: α

text: Suppose

text: that

text: α

text: and

text: that

text: Γ

text: Γ

text: where

text: Γ

text: satisﬁes

text: Then

text: for

text: every

text: ε˜

text: there

text: exists

text: a

text: stationary

text: SP

text: Z

text: satisfying

text: whose

text: R´enyi

text: rate

text: is

text: deﬁned

text: and

text: exceeds

text: ε˜.

text: k

text: For

text: α

text: we

text: can

text: use

text: to

text: obtain

text: for

text: the

text: same

text: supremum

text: |S|

text: sup

text: h

text: Z

text: log

text: α

text: α

text: k

text: This

text: seemingly

text: crude

text: bound

text: is

text: tight:

text: Theorem

text: (Max

text: R´enyi

text: Rate

text: for

text: α

text: Suppose

text: that

text: α

text: and

text: that

text: Γ

text: Γ

text: where

text: Γ

text: satisﬁes

text: |S|

text: R

text: If

text: then

text: for

text: every

text: M

text: there

text: exists

text: a

text: stationary

text: SP

text: Z

text: k

text: satisfying

text: whose

text: R´enyi

text: rate

text: is

text: deﬁned

text: and

text: exceeds

text: M.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 15 (cid:48) where γ = 0.5772 is the Euler constant. In Ref. [29], with V ((cid:126)k;(cid:126)k ). The spherical harmonics decomposition s (cid:48) a further analysis based on Gor’kov, Melik-Barkhudarov of V ((cid:126)k;(cid:126)k ) can be done in a similar way to Eq. 81, and s (GM) approach95 is performed, which takes into account the resultant V (cid:48) takes the same form as Eq. 82 but (cid:48) ll ;m the media polarization eﬀect due to the virtual process with l, l only taking even integer values. However, the of creating particle-hole excitations. This GM approach Hartree interaction does not exist in the pairing channel, (cid:48) shows≈that the energy cutoﬀ eﬀect can be approximated and thus, for l = l = m = 0, V00;0 = 0 for the case of by ω¯ 0.42(cid:15)f. the purely dipolar interaction. C. Competition between the triplet and singlet The analysis of pairing eigenvalues in the triplet sector pairings in the two-component dipolar Fermi gases is the same as that in the single component case. Thus, the leading pairing symmetry still lies in the p -channel. z The new ingredient of the two-component dipolar The pairing eigenvalue and eigenvector are the same as fermions is that both the spin singlet and triplet pair- Eq. 83, and T is still approximately determined by Eq. c ings are allowed50–53. We de(cid:104)ﬁne the pair(cid:105)ing operators in 84. We can express the triplet pairing in terms of the these two sectors as d-vector as Ps((cid:126)k) = √1 tr(cid:104)P((cid:126)k)(−iσy) , (cid:105) ∆t,µ((cid:126)k) = ∆teiγφz(Ωk)dˆµ, (90) 2 Pµ((cid:126)k) = √1 tr P((cid:126)k)(−iσyσµ) , (85) where γ is the U(1) phase and ∆t is the pairing am- t plitude. This spin-triplet pairing breaks the U(1) gauge 2 symmetry in the charge channel and the SU(2) symmetry with µ = x, y, z, respectively, where P ((cid:126)k) = in the spin channel, and thus, there are two diﬀerent low ψα((cid:126)k)ψβ(−(cid:126)k). (cid:88)Th(cid:110)en the pairing(cid:88)Hamiltoniαanβ is ex- epnaeirrignyg eisxcsittialltiionnvsa:ripahntonuonndearnda csopminb-winaevde Zmo-sdyems.mTethriys pressed as → − → 2 of dˆ dˆ and γ γ + π96,97, and thus, it supports µ µ H = 1 V ((cid:126)k;(cid:126)k(cid:48)) [ (cid:111) P†,µ((cid:126)k)Pµ((cid:126)k(cid:48)) ] two diﬀerent classes of vortices: the usual integer vor- pair 2V t t t tex of superﬂuidity, and the half-integer quantum vortex 0 (cid:48) k,k , µ=x,y,z of superﬂuidity combined with a π-disclination of the d- (cid:48) † (cid:48) + V ((cid:126)k;(cid:126)k )P ((cid:126)k)P ((cid:126)k ) , (86) vector. s s s In the singlet channel, the lowest eigenvalue and the (cid:48) − (cid:48) ∓ (cid:48) corresponding eigenvector are where V ((cid:126)k;(cid:126)k ) = V ((cid:126)k (cid:126)k ) V ((cid:126)k+(cid:126)k ) are pairing inter- (cid:113) t,s d d − ≈ − actions in the singlet and triplet channels, respectively. w0 = 1.93λ, ψs+d(Ω ) 0.6Y 0.8Y , (91) The Bogoliubov quasiparticle spectra become E ((cid:126)k) = s k 00 20 i − respectively. The eigenvector mixes the s and d − - ((cid:15)k µ)2 + λ2i ((cid:126)k), and λ21,2((cid:126)k) are the eigenvalues of channels. All other negative eigenvalues are smak2ll 3aknz2d † the positive-deﬁ(cid:90)nite Hermitian matrix ∆ ((cid:126)k)∆((cid:126)k). negligible. Although for the purely dipolar interac- The gap equation takes the matrix form as tion, the singlet channel pairing instability is signiﬁcantly (cid:48) weaker than that of the triplet channel. Nevertheless, the − d3k − (cid:48) (cid:104)| − |(cid:105) ∆αβ((cid:126)k) = (2π)3 Vd((cid:126)k (cid:126)k ) ψα((cid:126)k)ψβ( (cid:126)k) , (87) mfroamtritxheelesmhoerntt-rVa0n0g;0ems-awyarveecsecivaettaedridnigtioinntaelrcaocntitornib,uwthioicnhs where (cid:104)||(cid:105) means the thermal ensemble average. ∆αβ((cid:126)k) iVn00p;0r/inλci≈pl−e 3is.1t5u,ntahbelesitnhgrloetugahndFetrsihpblaetchchraensonnealnincestsa. bAilt- can be decomposed into the singlet and triplet channel ities become degenerate. pairings as Considering the competition between the spin triplet and singlet channel pairings, we generally expect two ∆ ((cid:126)k) = ∆ ((cid:126)k)iσy + ∆ ((cid:126)k)(iσµσy) , (88) αβ s αβ t,µ αβ pairing superﬂuid transitions. The ﬁrst transition is trig- in which ∆ and ∆ (cid:88)satis(cid:90)fy gered by the stronger pairing channel, say, the triplet pz- s t,µ channel, which determines T . In the case that the sin- c1 (cid:48) glet channel pairing is weaker but nearly degenerate with −1 d3k (cid:48) (cid:48) ∆ ((cid:126)k) = V ((cid:126)k;(cid:126)k )[K ((cid:126)k ) the triplet one, a second transition may occur at T as t(s),µ 2 (2π)3 t(s) i c2 i further lowering temperature. The mixing between the − 1 (cid:48) single and triplet pairings breaks parity, and thus, the ]∆ ((cid:126)k ), (89) 2(cid:15) t(s),µ second transition is also a genuine phase transition. The k coupling between these two pairing channels can be cap- where K ((cid:126)k) = tanh[β E ((cid:126)k)]/[2E ((cid:126)k)]. After linearizing tured by the following Ginzburg-Landau (GL) free energy i 2 i i the gap equation around T , we perform the eigenvalue as50 c analysis for the pairing problem. The eigen-equation of ∗ · | | { ∗ · ∗ } ∆F = γ (∆(cid:126) ∆(cid:126) ) ∆ 2 + γ ∆(cid:126) ∆(cid:126) ∆ ∆ + cc the triplet sector is the same as Eq. 80, while that of 1 t t s 2 t t s s (cid:48) | ∗ × | the singlet sector can be obtained by replacing V ((cid:126)k;(cid:126)k ) + γ ∆(cid:126) ∆(cid:126) 2, (92) t 3 t t ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: γ

text: is

text: the

text: Euler

text: constant.

text: In

text: Ref.

text: with

text: V

text: The

text: spherical

text: harmonics

text: decomposition

text: s

text: a

text: further

text: analysis

text: based

text: on

text: Gor’kov,

text: Melik-Barkhudarov

text: of

text: V

text: can

text: be

text: done

text: in

text: a

text: similar

text: way

text: to

text: Eq.

text: and

text: s

text: (GM)

text: is

text: performed,

text: which

text: takes

text: into

text: account

text: the

text: resultant

text: V

text: takes

text: the

text: same

text: form

text: as

text: Eq.

text: but

text: ll

text: ;m

text: the

text: media

text: polarization

text: eﬀect

text: due

text: to

text: the

text: virtual

text: process

text: with

text: l,

text: l

text: only

text: taking

text: even

text: integer

text: values.

text: However,

text: the

text: of

text: creating

text: particle-hole

text: excitations.

text: This

text: GM

text: approach

text: Hartree

text: interaction

text: does

text: not

text: exist

text: in

text: the

text: pairing

text: channel,

text: shows≈that

text: the

text: energy

text: cutoﬀ

text: eﬀect

text: can

text: be

text: approximated

text: and

text: thus,

text: for

text: l

text: l

text: m

text: for

text: the

text: case

text: of

text: by

text: ω¯

text: the

text: purely

text: dipolar

text: interaction.

text: C.

text: Competition

text: between

text: the

text: triplet

text: and

text: singlet

text: The

text: analysis

text: of

text: pairing

text: eigenvalues

text: in

text: the

text: triplet

text: sector

text: pairings

text: in

text: the

text: two-component

text: dipolar

text: Fermi

text: gases

text: is

text: the

text: same

text: as

text: that

text: in

text: the

text: single

text: component

text: case.

text: Thus,

text: the

text: leading

text: pairing

text: symmetry

text: still

text: lies

text: in

text: the

text: p

text: -channel.

text: z

text: The

text: new

text: ingredient

text: of

text: the

text: two-component

text: dipolar

text: The

text: pairing

text: eigenvalue

text: and

text: eigenvector

text: are

text: the

text: same

text: as

text: fermions

text: is

text: that

text: both

text: the

text: spin

text: singlet

text: and

text: triplet

text: pair-

text: Eq.

text: and

text: T

text: is

text: still

text: approximately

text: determined

text: by

text: Eq.

text: c

text: ings

text: are

text: We

text: the

text: operators

text: in

text: We

text: can

text: express

text: the

text: triplet

text: pairing

text: in

text: terms

text: of

text: the

text: these

text: two

text: sectors

text: as

text: d-vector

text: as

text: √1

text: ∆teiγφz(Ωk)dˆµ,

text: √1

text: tr

text: where

text: γ

text: is

text: the

text: phase

text: and

text: ∆t

text: is

text: the

text: pairing

text: am-

text: t

text: plitude.

text: This

text: spin-triplet

text: pairing

text: breaks

text: the

text: gauge

text: symmetry

text: in

text: the

text: charge

text: channel

text: and

text: the

text: symmetry

text: with

text: µ

text: x,

text: y,

text: z,

text: respectively,

text: where

text: P

text: in

text: the

text: spin

text: channel,

text: and

text: thus,

text: there

text: are

text: two

text: diﬀerent

text: low

text: the

text: is

text: ex-

text: epnaeirrignyg

text: csopminb-winaevde

text: Zmo-sdyems.mTethriys

text: pressed

text: as

text: of

text: dˆ

text: dˆ

text: and

text: γ

text: γ

text: and

text: thus,

text: it

text: supports

text: µ

text: µ

text: H

text: V

text: two

text: diﬀerent

text: classes

text: of

text: vortices:

text: the

text: usual

text: integer

text: vor-

text: pair

text: t

text: t

text: t

text: tex

text: of

text: superﬂuidity,

text: and

text: the

text: half-integer

text: quantum

text: vortex

text: k,k

text: µ=x,y,z

text: of

text: superﬂuidity

text: combined

text: with

text: a

text: π-disclination

text: of

text: the

text: d-

text: V

text: )P

text: vector.

text: s

text: s

text: s

text: In

text: the

text: singlet

text: channel,

text: the

text: lowest

text: eigenvalue

text: and

text: the

text: corresponding

text: eigenvector

text: are

text: where

text: V

text: V

text: V

text: are

text: pairing

text: inter-

text: t,s

text: d

text: d

text: actions

text: in

text: the

text: singlet

text: and

text: triplet

text: channels,

text: respectively.

text: ψs+d(Ω

text: The

text: Bogoliubov

text: quasiparticle

text: spectra

text: become

text: E

text: s

text: k

text: i

text: respectively.

text: The

text: eigenvector

text: mixes

text: the

text: s

text: and

text: d

text: and

text: are

text: the

text: eigenvalues

text: of

text: channels.

text: All

text: other

text: negative

text: eigenvalues

text: are

text: the

text: Hermitian

text: matrix

text: negligible.

text: Although

text: for

text: the

text: purely

text: dipolar

text: interac-

text: The

text: gap

text: equation

text: takes

text: the

text: matrix

text: form

text: as

text: tion,

text: the

text: singlet

text: channel

text: pairing

text: instability

text: is

text: signiﬁcantly

text: weaker

text: than

text: that

text: of

text: the

text: triplet

text: channel.

text: Nevertheless,

text: the

text: where

text: means

text: the

text: thermal

text: ensemble

text: average.

text: bAilt-

text: can

text: be

text: decomposed

text: into

text: the

text: singlet

text: and

text: triplet

text: channel

text: ities

text: become

text: degenerate.

text: pairings

text: as

text: Considering

text: the

text: competition

text: between

text: the

text: spin

text: triplet

text: and

text: singlet

text: channel

text: pairings,

text: we

text: generally

text: expect

text: two

text: αβ

text: s

text: αβ

text: t,µ

text: αβ

text: pairing

text: superﬂuid

text: transitions.

text: The

text: ﬁrst

text: transition

text: is

text: trig-

text: in

text: which

text: and

text: gered

text: by

text: the

text: stronger

text: pairing

text: channel,

text: say,

text: the

text: triplet

text: pz-

text: s

text: t,µ

text: channel,

text: which

text: determines

text: T

text: In

text: the

text: case

text: that

text: the

text: sin-

text: glet

text: channel

text: pairing

text: is

text: weaker

text: but

text: nearly

text: degenerate

text: with

text: −1

text: V

text: )[K

text: the

text: triplet

text: one,

text: a

text: second

text: transition

text: may

text: occur

text: at

text: T

text: as

text: t(s),µ

text: t(s)

text: i

text: i

text: further

text: lowering

text: temperature.

text: The

text: mixing

text: between

text: the

text: single

text: and

text: triplet

text: pairings

text: breaks

text: parity,

text: and

text: thus,

text: the

text: t(s),µ

text: second

text: transition

text: is

text: also

text: a

text: genuine

text: phase

text: transition.

text: The

text: k

text: coupling

text: between

text: these

text: two

text: pairing

text: channels

text: can

text: be

text: cap-

text: where

text: K

text: tanh[β

text: E

text: After

text: linearizing

text: tured

text: by

text: the

text: following

text: Ginzburg-Landau

text: (GL)

text: free

text: energy

text: i

text: i

text: i

text: the

text: gap

text: equation

text: around

text: T

text: we

text: perform

text: the

text: eigenvalue

text: c

text: analysis

text: for

text: the

text: pairing

text: problem.

text: The

text: eigen-equation

text: of

text: ∆F

text: γ

text: γ

text: cc

text: the

text: triplet

text: sector

text: is

text: the

text: same

text: as

text: Eq.

text: while

text: that

text: of

text: t

text: t

text: s

text: t

text: t

text: s

text: s

text: the

text: singlet

text: sector

text: can

text: be

text: obtained

text: by

text: replacing

text: V

text: γ

text: t

text: t

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 18 (cid:48) (cid:48) ± (cid:48) where L = L , or, L = L 2, and L, L are odd in- tegers. The expressions of the dimensionless matrix el- ements V (cid:48) (cid:48) (cid:48) are the same as those in Eq. 56 LMSz;L M Sz except an overall minus sign and a numeric factor. (cid:90) The free ene(cid:88)rgy can be calc(cid:2)ulated as (cid:3) F = − 2(cid:88) (cid:90) (cid:90) d3k ln 2 co(cid:8)sh βE(cid:126)k,i β (2π)3 2 i=1,2 (cid:48) − 1 d3k d3k ∗ − (cid:48) (cid:9) ∆ ((cid:126)k)V 1 (cid:48) ((cid:126)k;(cid:126)k ) | | 2 (cid:48) (2π)3 (2π)3 Sz SzSz FIG. 8: The angular distribution of the gap function ∆((cid:126)k) 2 × Sz,Sz v.s. cos θk in the helical polar pairing state (the red line) and ∆ (cid:48) ((cid:126)k) , (104) the axial pairing state (the black lines). From Ref. [60]. S z − (cid:48) wher(cid:90)e V 1 (cid:48) ((cid:126)k;(cid:126)k ) is the inverse of the interaction matrix d(cid:88)eﬁned SazsSz free energy can be linearized, and these two instabilities are degenerate, while this degeneracy is lifted below T (cid:48) c (cid:48) (d23πk)3 VSz,Sz(cid:48) ((cid:126)k;(cid:126)k(cid:48))VS−z(cid:48)1,Sz(cid:48)(cid:48)((cid:126)k(cid:48);(cid:126)k(cid:48)(cid:48)) = δ(cid:126)k,(cid:126)k(cid:48)(cid:48)δSz,Sz(cid:48)(cid:48). derugey.to the non-linearity of the Ginzburg-Landau free en- S z In quantum mechanics, if a system possesses rotation symmetry, of course, the eigenstates of its energy in the In order to analyze the pairing eigenvalues, the gap J = 1 sector and all of their superpositions are degener- equation is linearized around T for states around the c ate because of the linearity of quantum mechanics. How- Fermi surface. The total angular momentum quantum ever, the index J labeling the pairing order parameter is number J is employed to classify the eigen-gap functions denoted as φa,JJz((cid:126)k), in which the index a is used to not the angular momentum of the many-body eigenstate. Sz The description of a many-body system in terms of or- distinguish diﬀerent channels with the same value of J. (cid:90) der parameters is a great simpliﬁcation by only keeping The eigen-equation for φa,JJz((cid:126)k) is Sz a very limit but essential amount of degrees of freedom. A price to pay is that the description in terms of or- dΩ (cid:48) (cid:48) (cid:48) N0 4πk VSzSz(cid:48) ((cid:126)k;(cid:126)k )φaS;z(cid:48)JJz((cid:126)k ) = wJaφaS;zJJz((cid:126)k),(105) dneorn-plianreaamreetveerns, tshaoy,utghheqLuaanndtuaum-Gmineczhbaunrgicsfrereemenaeinrgsyl,inis- ± (cid:48) ear. This is the reason why the J = 0 and J = 1 where wa are dimensionless eigenvalues; (cid:126)k,(cid:126)k are at the z z J in principle are non-equivalent and the superposition law Fermi surface. Employing the spherical harmonics de- does not hold for Cooper pairing symmetries. The two composition of Eq. 103, the most negative eigenvalue is − pairing patterns with J = 1 and J = 1 are equivalent calculated lying the channel of J = L = 1 as wJ=1 = z z − to each other which can be connected either by a rotation 2πλ . All other negative eigenvalues are signiﬁcantly m or by the TR transformation. smaller. Thus, the dominant pairing channel remains 1. Helical polar pairing in the J-triplet channel in the weak coupling theory in agreement with the real space analysis. The pairing symmetry with J = 0 is also called the For later convenience, the pairing matrix in the J = 1 z J-polar pairing, characterized by the following pairing sector is represented as | | (cid:20) − (cid:21) matrix ∆pl = 1 ∆ [k σ k σ )iσ ] as αβ 2 y 1 x 2 2 αβ ∆ − ∆µ ((cid:126)k) = (cid:15) (k σ k σ ) , (106) | | − αβ 2 µνλ ν λ λ ν αβ ∆ kˆ ikˆ 0 ∆pl = x y , (107) αβ 2 0 kˆ + ikˆ for µ = x, y and z. It represents a pairing symmetry x y whose angular momentum projection along the direction · which is a unitary pairing and preserves TR symmetry. of eˆ is zero, i.e., it is an eigenstate of eˆ J(cid:126) with the − µ µ The spin-up fermions are paired in the p ip symme- zero eigenvalue. x y try, while, the spin-down fermions are paired with the C. Helical polar pairing and chiral axial pairing p +ip symmetry. Thus, it is equivalent to a helical ver- x y sion of the 3He-A phase. In the 3He-A phase, the pairing In the sector of J = 1, there are still two non-equivalent ± symmetry deﬁnes an orbital angular momentum direc- pairing possibilities: J = 0, or J = 1, whose pairing z { z ± } tion represented by the l-vector. Here, the z-axis plays a matrices are ∆z ((cid:126)k) and √1 ∆x ((cid:126)k) i∆y ((cid:126)k) , respec- similar role, but due to the TR symmetry, it represents αβ 2 αβ αβ tively. Based on the GL analysis up to the quartic or- the bi-direction of the polar axis, and thus it is no long a der of the pairing order parameter, it can be proved that vector but a director. Thus if we rotation the polar pair- ◦ these two are the only non-equivalent pairing symmetries ing around any axis in the xy-plane for 180 , then the under 3D rotations. Right at T , the Ginzburg-Landau system returns to itself up to a global phase diﬀerence. c ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: L

text: L

text: or,

text: L

text: L

text: and

text: L,

text: L

text: are

text: odd

text: in-

text: tegers.

text: The

text: expressions

text: of

text: the

text: dimensionless

text: matrix

text: el-

text: ements

text: V

text: are

text: the

text: same

text: as

text: those

text: in

text: Eq.

text: LMSz;L

text: M

text: Sz

text: except

text: an

text: overall

text: minus

text: sign

text: and

text: a

text: numeric

text: factor.

text: The

text: free

text: can

text: be

text: as

text: F

text: ln

text: β

label: caption

text: Sz

text: SzSz

text: FIG.

text: The

text: angular

text: distribution

text: of

text: the

text: gap

text: function

text: Sz,Sz

text: v.s.

text: cos

text: θk

text: in

text: the

text: helical

text: polar

text: pairing

text: state

text: (the

text: red

text: line)

text: and

text: the

text: axial

text: pairing

text: state

text: (the

text: black

text: lines).

text: From

text: Ref.

text: S

text: z

text: V

text: is

text: the

text: inverse

text: of

text: the

text: interaction

text: matrix

text: SazsSz

text: free

text: energy

text: can

text: be

text: linearized,

text: and

text: these

text: two

text: instabilities

text: are

text: degenerate,

text: while

text: this

text: degeneracy

text: is

text: lifted

text: below

text: T

text: c

text: derugey.to

text: the

text: non-linearity

text: of

text: the

text: Ginzburg-Landau

text: free

text: en-

text: S

text: z

text: In

text: quantum

text: mechanics,

text: if

text: a

text: system

text: possesses

text: rotation

text: symmetry,

text: of

text: course,

text: the

text: eigenstates

text: of

text: its

text: energy

text: in

text: the

text: In

text: order

text: to

text: analyze

text: the

text: pairing

text: eigenvalues,

text: the

text: gap

text: J

text: sector

text: and

text: all

text: of

text: their

text: superpositions

text: are

text: degener-

text: equation

text: is

text: linearized

text: around

text: T

text: for

text: states

text: around

text: the

text: c

text: ate

text: because

text: of

text: the

text: linearity

text: of

text: quantum

text: mechanics.

text: How-

text: Fermi

text: surface.

text: The

text: total

text: angular

text: momentum

text: quantum

text: ever,

text: the

text: index

text: J

text: labeling

text: the

text: pairing

text: order

text: parameter

text: is

text: number

text: J

text: is

text: employed

text: to

text: classify

text: the

text: eigen-gap

text: functions

text: denoted

text: as

text: in

text: which

text: the

text: index

text: a

text: is

text: used

text: to

text: not

text: the

text: angular

text: momentum

text: of

text: the

text: many-body

text: eigenstate.

text: Sz

text: The

text: description

text: of

text: a

text: many-body

text: system

text: in

text: terms

text: of

text: or-

text: distinguish

text: diﬀerent

text: channels

text: with

text: the

text: same

text: value

text: of

text: J.

text: der

text: parameters

text: is

text: a

text: great

text: simpliﬁcation

text: by

text: only

text: keeping

text: The

text: eigen-equation

text: for

text: is

text: Sz

text: a

text: very

text: limit

text: but

text: essential

text: amount

text: of

text: degrees

text: of

text: freedom.

text: A

text: price

text: to

text: pay

text: is

text: that

text: the

text: description

text: in

text: terms

text: of

text: or-

text: dΩ

text: dneorn-plianreaamreetveerns,

text: ear.

text: This

text: is

text: the

text: reason

text: why

text: the

text: J

text: and

text: J

text: where

text: wa

text: are

text: dimensionless

text: eigenvalues;

text: are

text: at

text: the

text: z

text: z

text: J

text: in

text: principle

text: are

text: non-equivalent

text: and

text: the

text: superposition

text: law

text: Fermi

text: surface.

text: Employing

text: the

text: spherical

text: harmonics

text: de-

text: does

text: not

text: hold

text: for

text: Cooper

text: pairing

text: symmetries.

text: The

text: two

text: composition

text: of

text: Eq.

text: the

text: most

text: negative

text: eigenvalue

text: is

text: pairing

text: patterns

text: with

text: J

text: and

text: J

text: are

text: equivalent

text: calculated

text: lying

text: the

text: channel

text: of

text: J

text: L

text: as

text: z

text: z

text: to

text: each

text: other

text: which

text: can

text: be

text: connected

text: either

text: by

text: a

text: rotation

text: 2πλ

text: All

text: other

text: negative

text: eigenvalues

text: are

text: signiﬁcantly

text: m

text: or

text: by

text: the

text: TR

text: transformation.

text: smaller.

text: Thus,

text: the

text: dominant

text: pairing

text: channel

text: remains

text: Helical

text: polar

text: pairing

text: in

text: the

text: J-triplet

text: channel

text: in

text: the

text: weak

text: coupling

text: theory

text: in

text: agreement

text: with

text: the

text: real

text: space

text: analysis.

text: The

text: pairing

text: symmetry

text: with

text: J

text: is

text: also

text: called

text: the

text: For

text: later

text: convenience,

text: the

text: pairing

text: matrix

text: in

text: the

text: J

text: z

text: J-polar

text: pairing,

text: characterized

text: by

text: the

text: following

text: pairing

text: sector

text: is

text: represented

text: as

text: matrix

text: ∆pl

text: [k

text: σ

text: k

text: σ

text: )iσ

text: as

text: αβ

text: y

text: x

text: αβ

text: ∆µ

text: (k

text: σ

text: k

text: σ

text: αβ

text: µνλ

text: ν

text: λ

text: λ

text: ν

text: αβ

text: kˆ

text: ikˆ

text: ∆pl

text: x

text: y

text: αβ

text: kˆ

text: ikˆ

text: for

text: µ

text: x,

text: y

text: and

text: z.

text: It

text: represents

text: a

text: pairing

text: symmetry

text: x

text: y

text: whose

text: angular

text: momentum

text: projection

text: along

text: the

text: direction

text: which

text: is

text: a

text: unitary

text: pairing

text: and

text: preserves

text: TR

text: symmetry.

text: of

text: eˆ

text: is

text: zero,

text: i.e.,

text: it

text: is

text: an

text: eigenstate

text: of

text: eˆ

text: with

text: the

text: µ

text: µ

text: The

text: spin-up

text: fermions

text: are

text: paired

text: in

text: the

text: p

text: ip

text: symme-

text: zero

text: eigenvalue.

text: x

text: y

text: try,

text: while,

text: the

text: spin-down

text: fermions

text: are

text: paired

text: with

text: the

text: C.

text: Helical

text: polar

text: pairing

text: and

text: chiral

text: axial

text: pairing

text: p

text: +ip

text: symmetry.

text: Thus,

text: it

text: is

text: equivalent

text: to

text: a

text: helical

text: ver-

text: x

text: y

text: sion

text: of

text: the

text: phase.

text: In

text: the

text: phase,

text: the

text: pairing

text: In

text: the

text: sector

text: of

text: J

text: there

text: are

text: still

text: two

text: non-equivalent

text: symmetry

text: deﬁnes

text: an

text: orbital

text: angular

text: momentum

text: direc-

text: pairing

text: possibilities:

text: J

text: or

text: J

text: whose

text: pairing

text: z

text: z

text: tion

text: represented

text: by

text: the

text: l-vector.

text: Here,

text: the

text: z-axis

text: plays

text: a

text: matrices

text: are

text: ∆z

text: and

text: √1

text: ∆x

text: i∆y

text: respec-

text: similar

text: role,

text: but

text: due

text: to

text: the

text: TR

text: symmetry,

text: it

text: represents

text: αβ

text: αβ

text: αβ

text: tively.

text: Based

text: on

text: the

text: GL

text: analysis

text: up

text: to

text: the

text: quartic

text: or-

text: the

text: bi-direction

text: of

text: the

text: polar

text: axis,

text: and

text: thus

text: it

text: is

text: no

text: long

text: a

text: der

text: of

text: the

text: pairing

text: order

text: parameter,

text: it

text: can

text: be

text: proved

text: that

text: vector

text: but

text: a

text: director.

text: Thus

text: if

text: we

text: rotation

text: the

text: polar

text: pair-

text: these

text: two

text: are

text: the

text: only

text: non-equivalent

text: pairing

text: symmetries

text: ing

text: around

text: any

text: axis

text: in

text: the

text: xy-plane

text: for

text: then

text: the

text: under

text: rotations.

text: Right

text: at

text: T

text: the

text: Ginzburg-Landau

text: system

text: returns

text: to

text: itself

text: up

text: to

text: a

text: global

text: phase

text: diﬀerence.

text: c

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: ##LTLine##

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: ygrenE sbbiG 6 The aqueous phase is then gently ﬁlled in the cuvette through Free Surfactants a syringe needle pierced through the cuvette cap. The two phases—water and ethanol—are held in stable contact. The Disc aggregates mixing occurs solely due to molecular diﬀusion of one phase into the other. Adequate care is taken not to introduce con- vective mixing in the two phases near the interface during the 1.0 process. c. Dynamic light scattering (DLS)/Photon correlation 0.75 Intrinsic Size spectroscopy (PCS) The liposomes were characterized by DLS or PCS (Zetasizer nano ZS, Malvern, UK) for size and 0.5 size distribution (polydispersity), without any further pro- cessing, except aqueous dilution, when required, for reliable 0.25 measurements. The laser used internally is He-Ne laser with Unilamellar Vesicles 633 nm wavelength and 4 mW power and the scattered light is detected at a single angle of 173◦. The size and size dis- tribution was calculated by a software (DTS 6.1, supplied Reaction Coordinate by Malvern). The polydispersity and average diameter were FIG. 5: Schematic representation of the free-energy for a evaluated using mono-modal or cumulant analysis method suspension of vesicles. The highest energy state is that of the and the intensity-weighted distribution of diameter was eval- free surfactants subject to sudden change in the environment uated using multimodal/distribution analysis. Usual care is that leads to a spontaneous aggregation to form discs. The taken to prevent dust and other contaminants in the system. discs, beyond a critical diameter, spontaneously close up to d. TEM using Negative staining High resolution trans- form vesicles. Vesicles of larger diameter are formed by mission electron microscopy (Jeol, JEM-2100F) was used to fusion of smaller vesicles reducing the free energy further. visualize the liposome lamellarity. Negative staining method But this process has an energy barrier of the order of 10 k T. was used to observe the liposomes under the HR-TEM. An electron dense material—phosphotungstic acid (PTA) 2%w/v—is used as the negative staining agent. PTA is acidic fusion to conserve area, which is expected to be valid only for in nature and needs to be neutralized; the pH is adjusted by a dilute mono-disperse unilamellar suspensions, in the initial dilute NaOH solution. A drop of liposome solution (10 µl) is times. placed carefully on a carbon coated TEM grid with the help of Supplementary Material B: Materials and methods a micropipette and kept for air drying for 10 minutes, which is followed by a placing drop of PTA. The sample was air dried before observing. The operating voltage is kept at 120 kV. a. Preparation of phospholipid solution. The phospho- lipids used to synthesize the liposomes are: 1,2-dimyristoyl- sn-glycero-3-phosphocholine (14:0-DMPC); 1,2-dipalmitoyl- sn-glycero-3-phosphocholine (16:0-DPPC); and L-α-phosphatidylchSoulipnpelementary Material C: Stages of Liposome formation in (Soy PC, a mixture of various lipids: 18:2, 16:0, 18:1, 18:3, SPI 18:0 phosphatidylcholine), all purchased from Avanti Polar Lipids, was used without further puriﬁcation. 1,2-dioleoyl- The phospholipid dissolved in ethanol is taken in an in- sn-glycero-3-phosphocholine (18:1-DOPC) was purchased verted tube and is in contact with an aqueous phase (cuvette) from Sigma Aldrich. Ethanol (AR grade) was purchased from as shown in Figure 6. As soon as the interface contact is es- Merck and MilliQ water was used for the aqueous medium. tablished, the progress of liposome formation is clearly visi- All the lipids from Avanti Polar Lipids were in solution form ble in the form of a turbid front that moves upward through the in chloroform (20 mg/ml). An appropriate volume of lipid ethanol section. Time lapse photographs of the visual changes solution is taken out in a round bottom ﬂask (RBF) and the observed are shown in Figure 7. chloroform is evaporated from the lipid solution using a rotary A copy of the time-lapse video of the formation of liposome evaporator. This forms a thin ﬁlm on the glass surface of the suspension is available from: RBF. The RBF is then kept in a vacuum desiccator overnight http://goo.gl/KOW3VN to remove all the traces of chloroform. An appropriate volume of ethanol is added to the RBF to make various concentrations The video has been made from photographs taken in 10 min of phospholipid in ethanol (5, 10, 20, 40, 80 mg/ml). time intervals (the timestamp is labelled on the top of the cu- b. Device setup The experimental setup is shown in Fig- vette, in the format DD:HH:MM). Three distinct phenomena ure 6. The key components of the device are: a container for are observed—(a) Initially, both the lipid and aqueous phases aqueous phase (square cuvette 10×40 mm) and organic phase are clear, and soon after, a turbid layer is observed inside the container (cut syringe tube 4.8 mm inner diameter). As shown inverted tube which moves upwards, indicating that liposomes in the setup, a hole is made in the cuvette cap and the cut tube are being formed. This is completed by about 6 hours (b) As is inserted through it. The phospholipid solution in ethanol is the white region moves vertically, some amount of liposome ﬁlled in the tube (held by the vacuum of the syringe piston). suspension also comes out in the aqueous phase (cuvette) and ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ygrenE

text: sbbiG

text: The

text: aqueous

text: phase

text: is

text: then

text: gently

text: ﬁlled

text: in

text: the

text: cuvette

text: through

text: Free

text: Surfactants

text: a

text: syringe

text: needle

text: pierced

text: through

text: the

text: cuvette

text: cap.

text: The

text: two

text: phases—water

text: and

text: ethanol—are

text: held

text: in

text: stable

text: contact.

text: The

text: Disc

text: aggregates

text: mixing

text: occurs

text: solely

text: due

text: to

text: molecular

text: diﬀusion

text: of

text: one

text: phase

text: into

text: the

text: other.

text: Adequate

text: care

text: is

text: taken

text: not

text: to

text: introduce

text: con-

text: vective

text: mixing

text: in

text: the

text: two

text: phases

text: near

text: the

text: interface

text: during

text: the

text: process.

text: c.

text: Dynamic

text: light

text: scattering

text: (DLS)/Photon

text: correlation

text: Intrinsic

text: Size

text: spectroscopy

text: (PCS)

text: The

text: liposomes

text: were

text: characterized

text: by

text: DLS

text: or

text: PCS

text: (Zetasizer

text: nano

text: ZS,

text: Malvern,

text: UK)

text: for

text: size

text: and

text: size

text: distribution

text: (polydispersity),

text: without

text: any

text: further

text: pro-

text: cessing,

text: except

text: aqueous

text: dilution,

text: when

text: required,

text: for

text: reliable

text: measurements.

text: The

text: laser

text: used

text: internally

text: is

text: He-Ne

text: laser

text: with

text: Unilamellar

text: Vesicles

text: nm

text: wavelength

text: and

text: mW

text: power

text: and

text: the

text: scattered

text: light

text: is

text: detected

text: at

text: a

text: single

text: angle

text: of

text: The

text: size

text: and

text: size

text: dis-

text: tribution

text: was

text: calculated

text: by

text: a

text: software

text: (DTS

text: supplied

text: Reaction

text: Coordinate

text: by

text: Malvern).

text: The

text: polydispersity

text: and

text: average

text: diameter

text: were

text: FIG.

text: Schematic

text: representation

text: of

text: the

text: free-energy

text: for

text: a

text: evaluated

text: using

text: mono-modal

text: or

text: cumulant

text: analysis

text: method

text: suspension

text: of

text: vesicles.

text: The

text: highest

text: energy

text: state

text: is

text: that

text: of

text: the

text: and

text: the

text: intensity-weighted

text: distribution

text: of

text: diameter

text: was

text: eval-

text: free

text: surfactants

text: subject

text: to

text: sudden

text: change

text: in

text: the

text: environment

text: uated

text: using

text: multimodal/distribution

text: analysis.

text: Usual

text: care

text: is

text: that

text: leads

text: to

text: a

text: spontaneous

text: aggregation

text: to

text: form

text: discs.

text: The

text: taken

text: to

text: prevent

text: dust

text: and

text: other

text: contaminants

text: in

text: the

text: system.

text: discs,

text: beyond

text: a

text: critical

text: diameter,

text: spontaneously

text: close

text: up

text: to

text: d.

text: TEM

text: using

text: Negative

text: staining

text: High

text: resolution

text: trans-

text: form

text: vesicles.

text: Vesicles

text: of

text: larger

text: diameter

text: are

text: formed

text: by

text: mission

text: electron

text: microscopy

text: (Jeol,

text: was

text: used

text: to

text: fusion

text: of

text: smaller

text: vesicles

text: reducing

text: the

text: free

text: energy

text: further.

text: visualize

text: the

text: liposome

text: lamellarity.

text: Negative

text: staining

text: method

text: But

text: this

text: process

text: has

text: an

text: energy

text: barrier

text: of

text: the

text: order

text: of

text: k

text: T.

text: was

text: used

text: to

text: observe

text: the

text: liposomes

text: under

text: the

text: HR-TEM.

text: An

text: electron

text: dense

text: material—phosphotungstic

text: acid

text: (PTA)

text: used

text: as

text: the

text: negative

text: staining

text: agent.

text: PTA

text: is

text: acidic

text: fusion

text: to

text: conserve

text: area,

text: which

text: is

text: expected

text: to

text: be

text: valid

text: only

text: for

text: in

text: nature

text: and

text: needs

text: to

text: be

text: neutralized;

text: the

text: pH

text: is

text: adjusted

text: by

text: a

text: dilute

text: mono-disperse

text: unilamellar

text: suspensions,

text: in

text: the

text: initial

text: dilute

text: NaOH

text: solution.

text: A

text: drop

text: of

text: liposome

text: solution

text: µl)

text: is

text: times.

text: placed

text: carefully

text: on

text: a

text: carbon

text: coated

text: TEM

text: grid

text: with

text: the

text: help

text: of

text: Supplementary

text: Material

text: B:

text: Materials

text: and

text: methods

text: a

text: micropipette

text: and

text: kept

text: for

text: air

text: drying

text: for

text: minutes,

text: which

text: is

text: followed

text: by

text: a

text: placing

text: drop

text: of

text: PTA.

text: The

text: sample

text: was

text: air

text: dried

text: before

text: observing.

text: The

text: operating

text: voltage

text: is

text: kept

text: at

text: kV.

text: a.

text: Preparation

text: of

text: phospholipid

text: solution.

text: The

text: phospho-

text: lipids

text: used

text: to

text: synthesize

text: the

text: liposomes

text: are:

text: and

text: L-α-phosphatidylchSoulipnpelementary

text: Material

text: C:

text: Stages

text: of

text: Liposome

text: formation

text: in

text: (Soy

text: PC,

text: a

text: mixture

text: of

text: various

text: lipids:

text: SPI

text: phosphatidylcholine),

text: all

text: purchased

text: from

text: Avanti

text: Polar

text: Lipids,

text: was

text: used

text: without

text: further

text: puriﬁcation.

text: The

text: phospholipid

text: dissolved

text: in

text: ethanol

text: is

text: taken

text: in

text: an

text: in-

text: was

text: purchased

text: verted

text: tube

text: and

text: is

text: in

text: contact

text: with

text: an

text: aqueous

text: phase

text: (cuvette)

text: from

text: Sigma

text: Aldrich.

text: Ethanol

text: (AR

text: grade)

text: was

text: purchased

text: from

text: as

text: shown

text: in

text: Figure

text: As

text: soon

text: as

text: the

text: interface

text: contact

text: is

text: es-

text: Merck

text: and

text: MilliQ

text: water

text: was

text: used

text: for

text: the

text: aqueous

text: medium.

text: tablished,

text: the

text: progress

text: of

text: liposome

text: formation

text: is

text: clearly

text: visi-

text: All

text: the

text: lipids

text: from

text: Avanti

text: Polar

text: Lipids

text: were

text: in

text: solution

text: form

text: ble

text: in

text: the

text: form

text: of

text: a

text: turbid

text: front

text: that

text: moves

text: upward

text: through

text: the

text: in

text: chloroform

text: mg/ml).

text: An

text: appropriate

text: volume

text: of

text: lipid

text: ethanol

text: section.

text: Time

text: lapse

text: photographs

text: of

text: the

text: visual

text: changes

text: solution

text: is

text: taken

text: out

text: in

text: a

text: round

text: bottom

text: ﬂask

text: (RBF)

text: and

text: the

text: observed

text: are

text: shown

text: in

text: Figure

text: chloroform

text: is

text: evaporated

text: from

text: the

text: lipid

text: solution

text: using

text: a

text: rotary

text: A

text: copy

text: of

text: the

text: time-lapse

text: video

text: of

text: the

text: formation

text: of

text: liposome

text: evaporator.

text: This

text: forms

text: a

text: thin

text: ﬁlm

text: on

text: the

text: glass

text: surface

text: of

text: the

text: suspension

text: is

text: available

text: from:

text: RBF.

text: The

text: RBF

text: is

text: then

text: kept

text: in

text: a

text: vacuum

text: desiccator

text: overnight

text: to

text: remove

text: all

text: the

text: traces

text: of

text: chloroform.

text: An

text: appropriate

text: volume

text: of

text: ethanol

text: is

text: added

text: to

text: the

text: RBF

text: to

text: make

text: various

text: concentrations

text: The

text: video

text: has

text: been

text: made

text: from

text: photographs

text: taken

text: in

text: min

text: of

text: phospholipid

text: in

text: ethanol

text: mg/ml).

text: time

text: intervals

text: (the

text: timestamp

text: is

text: labelled

text: on

text: the

text: top

text: of

text: the

text: cu-

text: b.

text: Device

text: setup

text: The

text: experimental

text: setup

text: is

text: shown

text: in

text: Fig-

text: vette,

text: in

text: the

text: format

text: DD:HH:MM).

text: Three

text: distinct

text: phenomena

text: ure

text: The

text: key

text: components

text: of

text: the

text: device

text: are:

text: a

text: container

text: for

text: are

text: observed—(a)

text: Initially,

text: both

text: the

text: lipid

text: and

text: aqueous

text: phases

text: aqueous

text: phase

text: (square

text: cuvette

text: mm)

text: and

text: organic

text: phase

text: are

text: clear,

text: and

text: soon

text: after,

text: a

text: turbid

text: layer

text: is

text: observed

text: inside

text: the

text: container

text: (cut

text: syringe

text: tube

text: mm

text: inner

text: diameter).

text: As

text: shown

text: inverted

text: tube

text: which

text: moves

text: upwards,

text: indicating

text: that

text: liposomes

text: in

text: the

text: setup,

text: a

text: hole

text: is

text: made

text: in

text: the

text: cuvette

text: cap

text: and

text: the

text: cut

text: tube

text: are

text: being

text: formed.

text: This

text: is

text: completed

text: by

text: about

text: hours

text: (b)

text: As

text: is

text: inserted

text: through

text: it.

text: The

text: phospholipid

text: solution

text: in

text: ethanol

text: is

text: the

text: white

text: region

text: moves

text: vertically,

text: some

text: amount

text: of

text: liposome

text: ﬁlled

text: in

text: the

text: tube

text: (held

text: by

text: the

text: vacuum

text: of

text: the

text: syringe

text: piston).

text: suspension

text: also

text: comes

text: out

text: in

text: the

text: aqueous

text: phase

text: (cuvette)

text: and

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text: 2 Yuan Liu and Xiaobo Li The geometry of BN11 is shown in Figure 3. The half- ◦ opening angle of the torus is ﬁxed at 60 . The inner radius of the torus is assumed to be zero in BN11’s model. The comparison between the BN11’s model and our results is shown in Figure 4 (Γ=1.8). The total spectra are shown, i.e. direct+ scattered components, since BN11’s model only provides the total spectra. The ﬂuores- cent lines (including the Compton shoulders) in the spectra of our simulations are not shown for clarity, because it is not easy to directly compare the strength of such a quasi-δ Figure 1. Geometry of MY09. The red and blue arrows indicate function in the ﬁgure. the possible trajectories of low-energy scattered photons that can Although the continua of the face-on directions are con- escape to face-on and edge-on directions, respectively. The yellow sistent with each other, the spectra of the edge-on directions, arrows indicate the trajectories of low-energy scattered photons especially for low-energy band, are signiﬁcantly diﬀerent. It that will be absorbed by the near side of the torus for edge-on directions. seems there is an “additional” component at the low-energy band in the simulations of BN11. We have further compared EW(Fe Kα) of our simula- Liu & Li (2014). We have included several physical pro- tions with that in Figure 3 of BN11. The photon index Γ is cesses in our code, e.g. photoelectric eﬀect, Compton scat- ◦ 2.0 in these simulations. For the face-on direction (θ = 60 tering, Rayleigh scattering, γ conversion, ﬂuorescent lines − ◦ tor and θ = 0 37 ), EW(Fe Kα) of BN11 is lower than our in and Auger process. Similar physical processes have been con- simulation results by 30% and 35% for N = 1024 and 1025 sidered in the simulations of MY09 and BN11. Thus, it is − H ◦ cm 2, respectively. For the edge-on direction (θ = 60 and convenient to modify our code to simulate their cases. Our − ◦ tor θ = 78 90 ), their EW(Fe Kα) is higher than our results code and the simulations by MY09 and BN11 have included in − by 60% for N = 1024 cm 2. Due to the large deviation of multi-scatterings, e.g. Figure 7 in Liu & Li (2014). We then H − the continuum of the edge-on direction for N = 1025 cm 2, H use the same assumptions adopted by BN11 and MY09 re- we have not compared the EW(Fe Kα) in this case. spectively to try to reproduce their results(cid:63). The incident × We explore these discrepancies in detail in the next sec- ﬂux is 5 108 photons (1-500 keV) for all simulations pre- tion. sented in this paper. 3 DISCUSSION 2.1 Results under the assumptions of MY09 Under the N discussed in Figure 2 and 4, the direct The geometry of MY09 is shown in Figure 1. The half- H ◦ component is highly absorbed in the low-energy band. Thus, opening angle of the torus is 60 . the low-energy spectra are dominated by the reﬂection from Using the same geometry, cross sections, element abun- the visible inner wall of the torus for a given inclination dances and incident spectra (a single power law with the angle, which is referred to as the ‘reﬂection component 2’ photon index Γ = 1.8), we can well reproduce MY09’s con- − in IK09 (the Figure 2 in IK09 illustrates its geometry and tinua, for both N = 1024 and 1025 cm 2 (Figure 2, where H origin). θ is the inclination angle relative to an observer). The di- in The strength of this component depends on the geom- rect component is not shown for clarity. Since the shape of etry of the torus, i.e. the location and shape of the surface, the scattered component is determined by both the scatter- and should also depend on the inclination angles. Therefore, ing and absorption processes (e.g. Fe K absorption edge at to judge whether the results of BN11’s model in Figure 4 are 7 keV), these results verify the accuracy of MY09’s simula- reasonable, we investigate the variation of this component tions. with inclination angles in diﬀerent models. We have further compared the equivalent width (EW) Because r /r = 0.01 adopted in the simulations of of the Fe Kα line of our simulations with that in Figure in out IK09 is small, IK09’s geometry is actually very similar to 8 of Murphy & Yaqoob (2011), which is the erratum of that of BN11, i.e. the N distribution is almost constant for MY09. The photon index Γ is 1.9 in these simulations. For H N = 1024 and 1025 cm−2, the diﬀerence of EWs between diﬀerent inclination angles. We plot the distribution of NH H using equation (3) in IK09 to show this (Figure 5). our simulations and MY09’s model is smaller than 1% for ∼ As shown in the Figure 9 in IK09, the ‘reﬂection com- most of the directions. The maximum deviation is 2% for N = 1024 cm−2 and cos θ = 0.3−0.4. This small deviation ponent 2’ signiﬁcantly decreases with increasing inclination H in angles, i.e. it is very weak in the edge-on directions. could be due to the diﬀerent simulation method adopted by MY09. The accuracy of this model is suﬃcient, since the A similar trend is also observed in MY09’s model (Fig- statistical error of EW(Fe Kα) in observed X-ray spectra ure 6), though the variation is not as dramatic as that in the is larger than 10% in most cases (Liu & Wang 2010; Shu, Figure 9 in IK09. We will explain this ◦later. The spectrum Yaqoob, & Wang 2011). at 1 keV with◦inclination angle θin = 65 is higher than that 2.2 Results under the assumptions of BN11 with θin = 85 by more than one order. However, the strength of this low-energy reﬂection com- ponent in BN11’s model only weakly depends on the inclina- ◦ tion angles (Figure 7). The spectrum at 1 keV with θ = 65 (cid:63) The model ﬁle of MY09 is downloaded from ◦ in is only higher than that with θ = 85 by a factor of two. http://mytorus.com/model-ﬁles-mytorus-downloads.html in The model ﬁle of BN11 is downloaded from The strength of this low-energy component depends on ∼ http://www.mpe.mpg.de/ mbright/data/torus1006.ﬁts the visibility of the inner wall of the torus at diﬀerent in- (cid:13) c 2002 RAS, MNRAS 000, 1–5 ##LTFigure##

text: Yuan

text: Liu

text: and

text: Xiaobo

text: Li

text: The

text: geometry

text: of

text: is

text: shown

text: in

text: Figure

text: The

text: half-

text: opening

text: angle

text: of

text: the

text: torus

text: is

text: ﬁxed

text: at

text: The

text: inner

text: radius

text: of

text: the

text: torus

text: is

text: assumed

text: to

text: be

text: zero

text: in

text: model.

text: The

text: comparison

text: between

text: the

text: model

text: and

text: our

text: results

text: is

text: shown

text: in

text: Figure

text: The

text: total

text: spectra

text: are

text: shown,

text: i.e.

text: direct+

text: scattered

text: components,

text: since

text: model

text: only

text: provides

text: the

text: total

text: spectra.

text: The

text: ﬂuores-

text: cent

text: lines

text: (including

text: the

text: Compton

text: shoulders)

text: in

text: the

text: spectra

text: of

text: our

text: simulations

text: are

text: not

text: shown

text: for

text: clarity,

text: because

text: it

text: is

text: not

text: easy

text: to

text: directly

text: compare

text: the

text: strength

text: of

text: such

text: a

text: quasi-δ

text: Figure

text: Geometry

text: of

text: The

text: red

text: and

text: blue

text: arrows

text: indicate

text: function

text: in

text: the

text: ﬁgure.

text: the

text: possible

text: trajectories

text: of

text: low-energy

text: scattered

text: photons

text: that

text: can

text: Although

text: the

text: continua

text: of

text: the

text: face-on

text: directions

text: are

text: con-

text: escape

text: to

text: face-on

text: and

text: edge-on

text: directions,

text: respectively.

text: The

text: yellow

text: sistent

text: with

text: each

text: other,

text: the

text: spectra

text: of

text: the

text: edge-on

text: directions,

text: arrows

text: indicate

text: the

text: trajectories

text: of

text: low-energy

text: scattered

text: photons

text: especially

text: for

text: low-energy

text: band,

text: are

text: signiﬁcantly

text: diﬀerent.

text: It

text: that

text: will

text: be

text: absorbed

text: by

text: the

text: near

text: side

text: of

text: the

text: torus

text: for

text: edge-on

text: directions.

text: seems

text: there

text: is

text: an

text: “additional”

text: component

text: at

text: the

text: low-energy

text: band

text: in

text: the

text: simulations

text: of

text: We

text: have

text: further

text: compared

text: EW(Fe

text: Kα)

text: of

text: our

text: simula-

text: Liu

text: Li

text: We

text: have

text: included

text: several

text: physical

text: pro-

text: tions

text: with

text: that

text: in

text: Figure

text: of

text: The

text: photon

text: index

text: Γ

text: is

text: cesses

text: in

text: our

text: code,

text: e.g.

text: photoelectric

text: eﬀect,

text: Compton

text: scat-

text: in

text: these

text: simulations.

text: For

text: the

text: face-on

text: direction

text: (θ

text: tering,

text: Rayleigh

text: scattering,

text: γ

text: conversion,

text: ﬂuorescent

text: lines

text: tor

text: and

text: θ

text: EW(Fe

text: Kα)

text: of

text: is

text: lower

text: than

text: our

text: in

text: and

text: Auger

text: process.

text: Similar

text: physical

text: processes

text: have

text: been

text: con-

text: simulation

text: results

text: by

text: and

text: for

text: N

text: and

text: sidered

text: in

text: the

text: simulations

text: of

text: and

text: Thus,

text: it

text: is

text: H

text: cm

text: respectively.

text: For

text: the

text: edge-on

text: direction

text: (θ

text: and

text: convenient

text: to

text: modify

text: our

text: code

text: to

text: simulate

text: their

text: cases.

text: Our

text: tor

text: θ

text: their

text: EW(Fe

text: Kα)

text: is

text: higher

text: than

text: our

text: results

text: code

text: and

text: the

text: simulations

text: by

text: and

text: have

text: included

text: in

text: by

text: for

text: N

text: cm

text: Due

text: to

text: the

text: large

text: deviation

text: of

text: multi-scatterings,

text: e.g.

text: Figure

text: in

text: Liu

text: Li

text: We

text: then

text: H

text: the

text: continuum

text: of

text: the

text: edge-on

text: direction

text: for

text: N

text: cm

text: H

text: use

text: the

text: same

text: assumptions

text: adopted

text: by

text: and

text: re-

text: we

text: have

text: not

text: compared

text: the

text: EW(Fe

text: Kα)

text: in

text: this

text: case.

text: spectively

text: to

text: try

text: to

text: reproduce

text: their

text: The

text: incident

text: We

text: explore

text: these

text: discrepancies

text: in

text: detail

text: in

text: the

text: next

text: sec-

text: ﬂux

text: is

text: photons

text: keV)

text: for

text: all

text: simulations

text: pre-

text: tion.

text: sented

text: in

text: this

text: paper.

text: DISCUSSION

text: Results

text: under

text: the

text: assumptions

text: of

text: Under

text: the

text: N

text: discussed

text: in

text: Figure

text: and

text: the

text: direct

text: The

text: geometry

text: of

text: is

text: shown

text: in

text: Figure

text: The

text: half-

text: H

text: component

text: is

text: highly

text: absorbed

text: in

text: the

text: low-energy

text: band.

text: Thus,

text: opening

text: angle

text: of

text: the

text: torus

text: is

text: the

text: low-energy

text: spectra

text: are

text: dominated

text: by

text: the

text: reﬂection

text: from

text: Using

text: the

text: same

text: geometry,

text: cross

text: sections,

text: element

text: abun-

text: the

text: visible

text: inner

text: wall

text: of

text: the

text: torus

text: for

text: a

text: given

text: inclination

text: dances

text: and

text: incident

text: spectra

text: (a

text: single

text: power

text: law

text: with

text: the

text: angle,

text: which

text: is

text: referred

text: to

text: as

text: the

text: ‘reﬂection

text: component

text: 2’

text: photon

text: index

text: Γ

text: we

text: can

text: well

text: reproduce

text: con-

text: in

text: (the

text: Figure

text: in

text: illustrates

text: its

text: geometry

text: and

text: tinua,

text: for

text: both

text: N

text: and

text: cm

text: (Figure

text: where

text: H

text: origin).

text: θ

text: is

text: the

text: inclination

text: angle

text: relative

text: to

text: an

text: observer).

text: The

text: di-

text: in

text: The

text: strength

text: of

text: this

text: component

text: depends

text: on

text: the

text: geom-

text: rect

text: component

text: is

text: not

text: shown

text: for

text: clarity.

text: Since

text: the

text: shape

text: of

text: etry

text: of

text: the

text: torus,

text: i.e.

text: the

text: location

text: and

text: shape

text: of

text: the

text: surface,

text: the

text: scattered

text: component

text: is

text: determined

text: by

text: both

text: the

text: scatter-

text: and

text: should

text: also

text: depend

text: on

text: the

text: inclination

text: angles.

text: Therefore,

text: ing

text: and

text: absorption

text: processes

text: (e.g.

text: Fe

text: K

text: absorption

text: edge

text: at

text: to

text: judge

text: whether

text: the

text: results

text: of

text: model

text: in

text: Figure

text: are

text: keV),

text: these

text: results

text: verify

text: the

text: accuracy

text: of

text: simula-

text: reasonable,

text: we

text: investigate

text: the

text: variation

text: of

text: this

text: component

text: tions.

text: with

text: inclination

text: angles

text: in

text: diﬀerent

text: models.

text: We

text: have

text: further

text: compared

text: the

text: equivalent

text: width

text: (EW)

text: Because

text: r

text: /r

text: adopted

text: in

text: the

text: simulations

text: of

text: of

text: the

text: Fe

text: Kα

text: line

text: of

text: our

text: simulations

text: with

text: that

text: in

text: Figure

text: in

text: out

text: is

text: small,

text: geometry

text: is

text: actually

text: very

text: similar

text: to

text: of

text: Murphy

text: Yaqoob

text: which

text: is

text: the

text: erratum

text: of

text: that

text: of

text: i.e.

text: the

text: N

text: distribution

text: is

text: almost

text: constant

text: for

text: The

text: photon

text: index

text: Γ

text: is

text: in

text: these

text: simulations.

text: For

text: H

text: N

text: and

text: the

text: diﬀerence

text: of

text: EWs

text: between

text: diﬀerent

text: inclination

text: angles.

text: We

text: plot

text: the

text: distribution

text: of

text: NH

text: H

text: using

text: equation

text: in

text: to

text: show

text: this

text: (Figure

text: our

text: simulations

text: and

text: model

text: is

text: smaller

text: than

text: for

text: As

text: shown

text: in

text: the

text: Figure

text: in

text: the

text: ‘reﬂection

text: com-

text: most

text: of

text: the

text: directions.

text: The

text: maximum

text: deviation

text: is

text: for

text: N

text: and

text: cos

text: θ

text: This

text: small

text: deviation

text: ponent

text: 2’

text: signiﬁcantly

text: decreases

text: with

text: increasing

text: inclination

text: H

text: in

text: angles,

text: i.e.

text: it

text: is

text: very

text: weak

text: in

text: the

text: edge-on

text: directions.

text: could

text: be

text: due

text: to

text: the

text: diﬀerent

text: simulation

text: method

text: adopted

text: by

text: The

text: accuracy

text: of

text: this

text: model

text: is

text: suﬃcient,

text: since

text: the

text: A

text: similar

text: trend

text: is

text: also

text: observed

text: in

text: model

text: (Fig-

text: statistical

text: error

text: of

text: EW(Fe

text: Kα)

text: in

text: observed

text: X-ray

text: spectra

text: ure

text: though

text: the

text: variation

text: is

text: not

text: as

text: dramatic

text: as

text: that

text: in

text: the

text: is

text: larger

text: than

text: in

text: most

text: cases

text: (Liu

text: Wang

text: Shu,

text: Figure

text: in

text: We

text: will

text: explain

text: this

text: ◦later.

text: The

text: spectrum

text: Yaqoob,

text: Wang

text: at

text: keV

text: with◦inclination

text: angle

text: θin

text: is

text: higher

text: than

text: that

text: Results

text: under

text: the

text: assumptions

text: of

text: with

text: θin

text: by

text: more

text: than

text: one

text: order.

text: However,

text: the

text: strength

text: of

text: this

text: low-energy

text: reﬂection

text: com-

text: ponent

text: in

text: model

text: only

text: weakly

text: depends

text: on

text: the

text: inclina-

text: tion

text: angles

text: (Figure

text: The

text: spectrum

text: at

text: keV

text: with

text: θ

text: The

text: model

text: ﬁle

text: of

text: is

text: downloaded

text: from

text: in

text: is

text: only

text: higher

text: than

text: that

text: with

text: θ

text: by

text: a

text: factor

text: of

text: two.

text: in

text: The

text: model

text: ﬁle

text: of

text: is

text: downloaded

text: from

text: The

text: strength

text: of

text: this

text: low-energy

text: component

text: depends

text: on

text: the

text: visibility

text: of

text: the

text: inner

text: wall

text: of

text: the

text: torus

text: at

text: diﬀerent

text: in-

text: c

text: RAS,

text: MNRAS

text: ##LTFigure##