text: (cid:2) (cid:3) − − − − T { − }→{ } ∂ r = ∂ ((1 ρ)(1 + αr)r) + (γ γ )∂ ((1 ρ)br) r (t(cid:0) ) = r (t ) + i 1,j i,j r − (cid:1) t x 1 2 y i,j Tk+{1 }→i,{j k} r T { − }→{i 1},j − h − − − +− Tr i{,j+}1→{ i,j }ri,j+T1{+ }r→i{,j −1 } i,jT r{i,j−}→1 { } (2a) 2 ∂x2(r(1 ρ)(1 + αr)) 2∂x((1 ρ)∂xr) i,j i+1,j + i,j i,j 1 + i,j i,j+1 r . h − r r r i,j + [(γ + γ )∂ ((1 ρ)∂ (rb) + br∂ ρ) (5a) 2 1 2 y y y − + 2γ ∂ ((1 ρ)∂ r + r∂ ρ) 0 y− y − y Hence the probability to ﬁnd a red particle at location + 2(γ1 γ2)∂y ((1 ρ)r∂xb)] , (xi, yj) corresponds to the probability that a particle located at (cid:2) − − − − (cid:3) (xi−1, yj) jumps forwards (ﬁrst term), particles located above ∂tb = ∂x ((1 ρ)(1 + αb)b) (γ1 γ2)∂y ((1 ρ)br) or below, i.e. at (x , y ± ) jump up or down (second line), − h − − − i j 1 ∂2(b(1 ρ)(1 + αb)) 2∂ ((1 ρ)∂ b) minus the probability that a particle located at (xi, yj) moves 2 x x x forward or steps aside (third line). The evolution of the blue h − particles can be formulated analogously: + 2 [(γ1 + γ2)∂y ((1 ρ)∂y(rb) + br∂yρ) (5b) − + 2γ ∂ ((1 ρ)∂ b + b∂ ρ) 0 −y y− y +2(γ γ )∂ ((1 ρ)b∂ r)] . 1 2 y x T { }→{ } bi,j(t(cid:0)k+1) = bi,j(tk) + b i+1,j i,j bi+1,j (cid:1) T { − }→{ } T { }→{ } + b i,j 1 i,j bi,j−1 + b i,j+1 i,j bi,j+1 (2b) The ﬁrst terms on the right-hand side of (5a) and (5b) − T {i,j}→{i−1,j} + T {i,j}→{i,j+1} + T {i,j}→{i,j−1} b . result from the ﬁrst-order terms in the Taylor expansion. They b b b i,j correspond to the movement of the reds and blues to the right and left respectively as well as the preference of either stepping − to the right or left (depending on the difference γ γ ). 1 2 B. Derivation of the macroscopic model The second line corresponds to the second-order terms in x- direction, the last three lines to the second-order terms due to In the next step we formally derive the limiting mean-ﬁeld side-stepping. Note that Equations (5a) and (5b) can be written −∇ · −∇ · equations as the grid size h and the time steps ∆t tend to in a conservative form, i.e. ∂ r = F and ∂ b = F t r t b zero. Hence we c→onsider the formal hyperbolic limit as h = for some matrices Fr and Fb. These so-called continuity ∆t = ∆x = ∆y 0 in Equations (2). We ﬁrst substitute the equations are always useful as they describe the transport of a transition rates (1) in Equation (2a) and obtain conserved quantity, in our case mass conservation. III. ALGORITHMIC DERIVATION OF THE MEAN-FIELD PDES − − − ri,j(tk+1) ri,j(tk) = (1 bi,j −ri,j)(−1 + αri+1,j)ri−1,j Symbolic computation, the ﬁeld of mathematics that is +((cid:0)γ + γ b )(1 b r )r concerned with computer-implemented exact manipulation 0 1 i+1,j+1 − i,j − i,j i,j+1 +(γ + γ b − )(1 b r )r − of mathematical expressions involving variables/symbols, is 0 2 i+1,j 1 i,j i,j i,j 1 − − − meanwhile a well-established area of research and has nu- (1 b r )(1 + αr ) (cid:1) (3) i+1,j i+−1,j − i+2,j merous applications. Unfortunately, it is not as widely known +(γ0 + γ1bi+1,j)(1 bi,j−1 ri,j−1) as it should be. One reason may be that some applications − − +(γ + γ b )(1 b r ) r , are not straightforward and require at least some insight or 0 2 i+1,j i,j+1 i,j+1 i,j programming skills. But to those who get moderately familiar with symbolic computation software, it becomes an indis- pensable tool. There are plenty of general-purpose computer and a similar equation for the evolution of the blue parti- algebra systems available, the most well-known being probably cles. Next we employ Taylor expansions up to second order of Mathematica, Maple, and Sage. For our implementation we all the occurring probabilities. For example, the probability to have chosen Mathematica. ﬁnd a red particle at location (x , y ) can be expanded as i+1 j In this section we demonstrate how the transition from the discrete lattice-based model to a macroscopic PDE-based formulation is achieved using techniques from symbolic com- O r = r + h∂ r + 1h2∂2r + (h3). (4) putation. i+1,j i,j x i,j 2 x i,j A. Expansion Recall that the lattice sites are given by (x , y ) = (ih, jh) ∈ Z → i j After expanding all probability densities we keep the terms for i, j ; in the limit h 0 one obtains the problem up to second order and consider the formal limit as ∆t = formulation for the macroscopic model. Let r = r(x, y) and → ∆x = ∆y 0. This leads to the following system of PDEs for b = b(x, y) denote the densities of red and blue particles in the densities of the red and blue particles, which can be either the macroscopic model. In order to perform the transition from obtained by tedious hand calculations, or by the computer- partial difference equations for r and b to partial differen- i,j i,j algebra methods described in Section 2: tial equations for r(x, y) and b(x, y), we employ formal Taylor ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: T

text: r

text: αr)r)

text: (γ

text: γ

text: ρ)br)

text: r

text: r

text: (t

text: i

text: i,j

text: r

text: t

text: x

text: y

text: i,j

text: }→i,{j

text: k}

text: r

text: T

text: }→{i

text: h

text: Tr

text: i,j

text: }r→i{,j

text: −1

text: i,jT

text: αr))

text: ρ)∂xr)

text: i,j

text: i,j

text: i,j

text: i,j

text: r

text: h

text: r

text: r

text: r

text: i,j

text: [(γ

text: γ

text: ρ)∂

text: (rb)

text: br∂

text: ρ)

text: y

text: y

text: y

text: 2γ

text: ρ)∂

text: r

text: r∂

text: ρ)

text: y−

text: y

text: y

text: Hence

text: the

text: probability

text: to

text: ﬁnd

text: a

text: red

text: particle

text: at

text: location

text: ρ)r∂xb)]

text: (xi,

text: yj)

text: corresponds

text: to

text: the

text: probability

text: that

text: a

text: particle

text: located

text: at

text: yj)

text: jumps

text: forwards

text: (ﬁrst

text: term),

text: particles

text: located

text: above

text: ∂tb

text: ∂x

text: αb)b)

text: ρ)br)

text: or

text: below,

text: i.e.

text: at

text: (x

text: y

text: jump

text: up

text: or

text: down

text: (second

text: line),

text: h

text: i

text: j

text: αb))

text: 2∂

text: ρ)∂

text: b)

text: minus

text: the

text: probability

text: that

text: a

text: particle

text: located

text: at

text: (xi,

text: yj)

text: moves

text: x

text: x

text: x

text: forward

text: or

text: steps

text: aside

text: (third

text: line).

text: The

text: evolution

text: of

text: the

text: blue

text: h

text: particles

text: can

text: be

text: formulated

text: analogously:

text: ρ)∂y(rb)

text: br∂yρ)

text: 2γ

text: ρ)∂

text: b

text: b∂

text: ρ)

text: −y

text: y−

text: y

text: γ

text: ρ)b∂

text: r)]

text: y

text: x

text: T

text: bi,j(tk)

text: b

text: i,j

text: T

text: T

text: b

text: i,j

text: i,j

text: b

text: i,j

text: The

text: ﬁrst

text: terms

text: on

text: the

text: right-hand

text: side

text: of

text: and

text: T

text: T

text: T

text: b

text: result

text: from

text: the

text: ﬁrst-order

text: terms

text: in

text: the

text: Taylor

text: expansion.

text: They

text: b

text: b

text: b

text: i,j

text: correspond

text: to

text: the

text: movement

text: of

text: the

text: reds

text: and

text: blues

text: to

text: the

text: right

text: and

text: left

text: respectively

text: as

text: well

text: as

text: the

text: preference

text: of

text: either

text: stepping

text: to

text: the

text: right

text: or

text: left

text: (depending

text: on

text: the

text: difference

text: γ

text: γ

text: B.

text: Derivation

text: of

text: the

text: macroscopic

text: model

text: The

text: second

text: line

text: corresponds

text: to

text: the

text: second-order

text: terms

text: in

text: x-

text: direction,

text: the

text: last

text: three

text: lines

text: to

text: the

text: second-order

text: terms

text: due

text: to

text: In

text: the

text: next

text: step

text: we

text: formally

text: derive

text: the

text: limiting

text: mean-ﬁeld

text: side-stepping.

text: Note

text: that

text: Equations

text: and

text: can

text: be

text: written

text: equations

text: as

text: the

text: grid

text: size

text: h

text: and

text: the

text: time

text: steps

text: ∆t

text: tend

text: to

text: in

text: a

text: conservative

text: form,

text: i.e.

text: r

text: F

text: and

text: b

text: F

text: t

text: r

text: t

text: b

text: zero.

text: Hence

text: we

text: c→onsider

text: the

text: formal

text: hyperbolic

text: limit

text: as

text: h

text: for

text: some

text: matrices

text: Fr

text: and

text: Fb.

text: These

text: so-called

text: continuity

text: ∆t

text: ∆x

text: ∆y

text: in

text: Equations

text: We

text: ﬁrst

text: substitute

text: the

text: equations

text: are

text: always

text: useful

text: as

text: they

text: describe

text: the

text: transport

text: of

text: a

text: transition

text: rates

text: in

text: Equation

text: and

text: obtain

text: conserved

text: quantity,

text: in

text: our

text: case

text: mass

text: conservation.

text: III.

text: ALGORITHMIC

text: DERIVATION

text: OF

text: THE

text: MEAN-FIELD

text: PDES

text: ri,j(tk)

text: bi,j

text: Symbolic

text: computation,

text: the

text: ﬁeld

text: of

text: mathematics

text: that

text: is

text: γ

text: b

text: b

text: r

text: )r

text: concerned

text: with

text: computer-implemented

text: exact

text: manipulation

text: i,j

text: i,j

text: +(γ

text: γ

text: b

text: b

text: r

text: )r

text: of

text: mathematical

text: expressions

text: involving

text: variables/symbols,

text: is

text: i,j

text: i,j

text: i,j

text: meanwhile

text: a

text: well-established

text: area

text: of

text: research

text: and

text: has

text: nu-

text: b

text: r

text: αr

text: merous

text: applications.

text: Unfortunately,

text: it

text: is

text: not

text: as

text: widely

text: known

text: as

text: it

text: should

text: be.

text: One

text: reason

text: may

text: be

text: that

text: some

text: applications

text: +(γ

text: γ

text: b

text: b

text: r

text: r

text: are

text: not

text: straightforward

text: and

text: require

text: at

text: least

text: some

text: insight

text: or

text: i,j

text: programming

text: skills.

text: But

text: to

text: those

text: who

text: get

text: moderately

text: familiar

text: with

text: symbolic

text: computation

text: software,

text: it

text: becomes

text: an

text: indis-

text: pensable

text: tool.

text: There

text: are

text: plenty

text: of

text: general-purpose

text: computer

text: and

text: a

text: similar

text: equation

text: for

text: the

text: evolution

text: of

text: the

text: blue

text: parti-

text: algebra

text: systems

text: available,

text: the

text: most

text: well-known

text: being

text: probably

text: cles.

text: Next

text: we

text: employ

text: Taylor

text: expansions

text: up

text: to

text: second

text: order

text: of

text: Mathematica,

text: Maple,

text: and

text: Sage.

text: For

text: our

text: implementation

text: we

text: all

text: the

text: occurring

text: probabilities.

text: For

text: example,

text: the

text: probability

text: to

text: have

text: chosen

text: Mathematica.

text: ﬁnd

text: a

text: red

text: particle

text: at

text: location

text: (x

text: y

text: can

text: be

text: expanded

text: as

text: j

text: In

text: this

text: section

text: we

text: demonstrate

text: how

text: the

text: transition

text: from

text: the

text: discrete

text: lattice-based

text: model

text: to

text: a

text: macroscopic

text: PDE-based

text: formulation

text: is

text: achieved

text: using

text: techniques

text: from

text: symbolic

text: com-

text: O

text: r

text: r

text: h∂

text: r

text: putation.

text: i,j

text: x

text: i,j

text: x

text: i,j

text: A.

text: Expansion

text: Recall

text: that

text: the

text: lattice

text: sites

text: are

text: given

text: by

text: (x

text: y

text: (ih,

text: jh)

text: Z

text: i

text: j

text: After

text: expanding

text: all

text: probability

text: densities

text: we

text: keep

text: the

text: terms

text: for

text: i,

text: j

text: in

text: the

text: limit

text: h

text: one

text: obtains

text: the

text: problem

text: up

text: to

text: second

text: order

text: and

text: consider

text: the

text: formal

text: limit

text: as

text: ∆t

text: formulation

text: for

text: the

text: macroscopic

text: model.

text: Let

text: r

text: r(x,

text: y)

text: and

text: ∆x

text: ∆y

text: This

text: leads

text: to

text: the

text: following

text: system

text: of

text: PDEs

text: for

text: b

text: b(x,

text: y)

text: denote

text: the

text: densities

text: of

text: red

text: and

text: blue

text: particles

text: in

text: the

text: densities

text: of

text: the

text: red

text: and

text: blue

text: particles,

text: which

text: can

text: be

text: either

text: the

text: macroscopic

text: model.

text: In

text: order

text: to

text: perform

text: the

text: transition

text: from

text: obtained

text: by

text: tedious

text: hand

text: calculations,

text: or

text: by

text: the

text: computer-

text: partial

text: difference

text: equations

text: for

text: r

text: and

text: b

text: to

text: partial

text: differen-

text: i,j

text: i,j

text: algebra

text: methods

text: described

text: in

text: Section

text: tial

text: equations

text: for

text: r(x,

text: y)

text: and

text: b(x,

text: y),

text: we

text: employ

text: formal

text: Taylor

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: has an antiderivative, i.e., E = ∂ I for some polynomial For each integration variable, say x, we proceed as follows: x expression I, then it is easy to see that ∂nf occurs linearly we determine the highest derivative with respect to x that x in E, i.e., E is quasi-linear. Hence, if ∂nf does not occur occurs in the input, no matter which function is involved. We x linearly, then the corresponding monomials are put into the say that the highest x-derivative is of order n if ∂nf occurs ≤ ≤ x i remainder, as they cannot be integrated. Now assume that E for some 1 i k, but there is no index i such that ∂n+mf − ≥ ≤ ≤x i is linear in ∂nf. Let m be the highest power of ∂n 1f and for some m 1 occurs. Then for each f , 1 i k, (in x − x i denote by u the coefﬁcient of (∂n 1f)m(∂nf) in E, which is the order as speciﬁed by the user) the terms involving ∂nf x − x x i itself a polynomial in f, ∂ f, . . . , ∂n 2f. Then integration by are treated. Note that in this step derivatives with order n + 1 x x parts(cid:0)yields (cid:1) can be produced, as can be seen in (11). In order to avoid that the algorithm runs into an inﬁnite loop, we keep these terms, − · −(cid:16) (cid:0) (cid:1) (cid:17) (cid:0) (cid:1) and continue by considering derivatives of order n 1. This u ∂n 1f m(∂nf) = algorithm is described in detail in the following pseudo-code: x x u − − ∂ u − (10) ∂ ∂n 1f m+1 x ∂n 1f m+1. x m + 1 x m + 1 x (cid:0) (cid:1) Algorithm PartialIntegrate Hence the ﬁrst term on the right-hand side of (10) goes into Input: E: differential polynomial expression the integrable part, while the second term is used to replace u· ∂xn−1f m(∂xnf) in E. After performing this step repeatedly fx1,,y.,....,.f,kz:: uinntsepgercaitﬁioend vfuarnicatbiolenss (at most−m times), E involves only derivatives of f up to d: depth order n 1. This shows that the algorithm terminates. { } ∅ 1: if E = 0 or x, . . . , z = or d = 0 then We have seen that in the case of a single unspeciﬁed function, there is a canonical choice which term to integrate in 2: return E each step of the algorithm. In contrast when several unspeciﬁed 3: end if ← functions are involved, the situation is less clear, as the 4: R 0 (cid:0) (cid:1) ← following example shows: 5: I 0 (cid:0) (cid:1) ← (cid:0) (cid:1) 6: n highest x-derivative that appears in E for some f − i (∂xf)(∂xg) = ∂x f(∂xg) − f(∂x2g) (11) 7: if n = 0 then = ∂ (∂ f)g (∂2f)g. 8: return PartialIntegrate E, (f , . . . , f ), (y, . . . , z), d x x x 1 k 9: end if Hence one has to specify an order in which the terms are processed, and which at the same time doesn’t lead to inﬁnite loops. The same kinds of problems are faced when the un- speciﬁed functions depend on several variables. The following example demonstrates the ambiguity of the decomposition in the case of a singl(cid:0)e unspeciﬁed(cid:1) funct(cid:0)ion(cid:1)f(x, y): (∂ f)(∂ f(cid:0)) +(cid:1)∂ f +(cid:0) ∂ f (cid:1) x y x y · − · = ∂ f ∂ f + f + ∂ f f ∂ ∂ f x y y x y · − · = ∂ f + ∂ f ∂ f + f f ∂ ∂ f. x y x x y In our application we have to deal with several unspeciﬁed functions f , . . . , f in several variables, say x, y, . . . , z. So 1 k the question is in which order we should treat the terms of the input expression to obtain the desired result. One natural choice is to consider the variables in a ﬁxed order as the main loop of the algorithm. This means that we ﬁrst decompose the input with respect to the ﬁrst variable, say ∂ I + R; then the x remainder R is decomposed with respect to the next variable, and so on, yielding a result of the form · · · ∂ I + ∂ I + + ∂ I + R. x x y y z z Additionally, one can also decompose I further, yielding a nested decomposition of the following form (we show only the case of a single variable): · · · · · · ∂ (∂ ( (∂ (I) + R ) + + R ) + R ) + R . x x x d 2 1 0 In our description of the algorithm we use the parameter d to specify the desired maximal integration depth of the output expression. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: has

label: paragraph

text: an

text: antiderivative,

text: i.e.,

text: E

text: I

text: for

text: some

text: polynomial

text: For

text: each

text: integration

text: variable,

text: say

text: x,

text: we

text: proceed

text: as

text: follows:

text: x

text: expression

text: I,

text: then

text: it

text: is

text: easy

text: to

text: see

text: that

text: ∂nf

text: occurs

text: linearly

text: we

text: determine

text: the

text: highest

text: derivative

text: with

text: respect

text: to

text: x

text: that

text: x

text: in

text: E,

text: i.e.,

text: E

text: is

text: quasi-linear.

text: Hence,

text: if

text: ∂nf

text: does

text: not

text: occur

text: occurs

text: in

text: the

text: input,

text: no

text: matter

text: which

text: function

text: is

text: involved.

text: We

text: x

text: linearly,

text: then

text: the

text: corresponding

text: monomials

text: are

text: put

text: into

text: the

text: say

text: that

text: the

text: highest

text: x-derivative

text: is

text: of

text: order

text: n

text: if

text: ∂nf

text: occurs

text: x

text: i

text: remainder,

text: as

text: they

text: cannot

text: be

text: integrated.

text: Now

text: assume

text: that

text: E

text: for

text: some

text: i

text: k,

text: but

text: there

text: is

text: no

text: index

text: i

text: such

text: that

text: ∂n+mf

text: ≤x

text: i

text: is

text: linear

text: in

text: ∂nf.

text: Let

text: m

text: be

text: the

text: highest

text: power

text: of

text: ∂n

text: and

text: for

text: some

text: m

text: occurs.

text: Then

text: for

text: each

text: f

text: i

text: k,

text: (in

text: x

text: x

text: i

text: denote

text: by

text: u

text: the

text: coefﬁcient

text: of

text: (∂n

text: in

text: E,

text: which

text: is

text: the

text: order

text: as

text: speciﬁed

text: by

text: the

text: user)

text: the

text: terms

text: involving

text: ∂nf

text: x

text: x

text: x

text: i

text: itself

text: a

text: polynomial

text: in

text: f,

text: f,

text: ∂n

text: Then

text: integration

text: by

text: are

text: treated.

text: Note

text: that

text: in

text: this

text: step

text: derivatives

text: with

text: order

text: n

text: x

text: x

text: can

text: be

text: produced,

text: as

text: can

text: be

text: seen

text: in

text: In

text: order

text: to

text: avoid

text: that

text: the

text: algorithm

text: runs

text: into

text: an

text: inﬁnite

text: loop,

text: we

text: keep

text: these

text: terms,

text: and

text: continue

text: by

text: considering

text: derivatives

text: of

text: order

text: n

text: This

text: u

text: ∂n

text: m(∂nf)

text: algorithm

text: is

text: described

text: in

text: detail

text: in

text: the

text: following

text: pseudo-code:

text: x

text: x

text: u

text: u

text: ∂n

text: x

text: ∂n

text: x

text: m

text: x

text: m

text: x

text: Algorithm

text: PartialIntegrate

text: Hence

text: the

text: ﬁrst

text: term

text: on

text: the

text: right-hand

text: side

text: of

text: goes

text: into

text: Input:

text: E:

text: differential

text: polynomial

text: expression

text: the

text: integrable

text: part,

text: while

text: the

text: second

text: term

text: is

text: used

text: to

text: replace

text: u·

text: m(∂xnf)

text: in

text: E.

text: After

text: performing

text: this

text: step

text: repeatedly

text: uinntsepgercaitﬁioend

text: vfuarnicatbiolenss

text: (at

text: most−m

text: times),

text: E

text: involves

text: only

text: derivatives

text: of

text: f

text: up

text: to

text: d:

text: depth

text: order

text: n

text: This

text: shows

text: that

text: the

text: algorithm

text: terminates.

text: if

text: E

text: or

text: x,

text: z

text: or

text: d

text: then

text: We

text: have

text: seen

text: that

text: in

text: the

text: case

text: of

text: a

text: single

text: unspeciﬁed

text: function,

text: there

text: is

text: a

text: canonical

text: choice

text: which

text: term

text: to

text: integrate

text: in

text: return

text: E

text: each

text: step

text: of

text: the

text: algorithm.

text: In

text: contrast

text: when

text: several

text: unspeciﬁed

text: end

text: if

text: functions

text: are

text: involved,

text: the

text: situation

text: is

text: less

text: clear,

text: as

text: the

text: R

text: following

text: example

text: shows:

text: I

text: n

text: highest

text: x-derivative

text: that

text: appears

text: in

text: E

text: for

text: some

text: f

text: i

text: (∂xf)(∂xg)

text: ∂x

text: f(∂xg)

text: if

text: n

text: then

text: f)g

text: return

text: PartialIntegrate

text: E,

text: (f

text: f

text: (y,

text: z),

text: d

text: x

text: x

text: x

text: k

text: end

text: if

text: Hence

text: one

text: has

text: to

text: specify

text: an

text: order

text: in

text: which

text: the

text: terms

text: are

text: processed,

text: and

text: which

text: at

text: the

text: same

text: time

text: doesn’t

text: lead

text: to

text: inﬁnite

text: loops.

text: The

text: same

text: kinds

text: of

text: problems

text: are

text: faced

text: when

text: the

text: un-

text: speciﬁed

text: functions

text: depend

text: on

text: several

text: variables.

text: The

text: following

text: example

text: demonstrates

text: the

text: ambiguity

text: of

text: the

text: decomposition

text: in

text: the

text: case

text: of

text: a

text: y):

text: f)(∂

text: f

text: f

text: x

text: y

text: x

text: y

text: f

text: f

text: f

text: f

text: f

text: f

text: x

text: y

text: y

text: x

text: y

text: f

text: f

text: f

text: f

text: f

text: f.

text: x

text: y

text: x

text: x

text: y

text: In

text: our

text: application

text: we

text: have

text: to

text: deal

text: with

text: several

text: unspeciﬁed

text: functions

text: f

text: f

text: in

text: several

text: variables,

text: say

text: x,

text: y,

text: z.

text: So

text: k

text: the

text: question

text: is

text: in

text: which

text: order

text: we

text: should

text: treat

text: the

text: terms

text: of

text: the

text: input

text: expression

text: to

text: obtain

text: the

text: desired

text: result.

text: One

text: natural

text: choice

text: is

text: to

text: consider

text: the

text: variables

text: in

text: a

text: ﬁxed

text: order

text: as

text: the

text: main

text: loop

text: of

text: the

text: algorithm.

text: This

text: means

text: that

text: we

text: ﬁrst

text: decompose

text: the

text: input

text: with

text: respect

text: to

text: the

text: ﬁrst

text: variable,

text: say

text: I

text: R;

text: then

text: the

text: x

text: remainder

text: R

text: is

text: decomposed

text: with

text: respect

text: to

text: the

text: next

text: variable,

text: and

text: so

text: on,

text: yielding

text: a

text: result

text: of

text: the

text: form

text: I

text: I

text: I

text: R.

text: x

text: x

text: y

text: y

text: z

text: z

text: Additionally,

text: one

text: can

text: also

text: decompose

text: I

text: further,

text: yielding

text: a

text: nested

text: decomposition

text: of

text: the

text: following

text: form

text: (we

text: show

text: only

text: the

text: case

text: of

text: a

text: single

text: variable):

text: (I)

text: R

text: R

text: R

text: R

text: x

text: x

text: x

text: d

text: In

text: our

text: description

text: of

text: the

text: algorithm

text: we

text: use

text: the

text: parameter

text: d

text: to

text: specify

text: the

text: desired

text: maximal

text: integration

text: depth

text: of

text: the

text: output

text: expression.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − − with a diffusivity of the form D(c) = 3α(c 2)2 + 1 4α, 3 3 see [28]. Again in [1] several other transition rates were proposed, which lead to different nonlinear diffusitivies, see Table 1 there. Using our implementation, the entries in that table can be generated automatically. For example, we have tried one of their most complicated models [1, Equation (13)], which combines contact-forming or contact-breaking interac- tions with contact-maintaining interactions. Our implementa- tion correctly derives the diffusivity given in [1, Equation (14)], where we have chosen the two-dimensional square lattice with Moore interacting neighborhoods. (a) Density of reds at time T = 5 ACKNOWLEDGMENT CK was supported by the Austrian Science Fund (FWF): W1214. HR and MTW acknowledge support by the Austrian Academy of Sciences O¨ AW via the New Frontiers project NST-0001. GR was supported by the Austrian Science Fund (FWF): P27229. REFERENCES [1] A. E. Fernando, K. A. Landman, and M. J. Simpson, “Nonlinear diffusion and exclusion processes with contact interactions,” Physical (b) Density of blues at time T = 5 Review E, vol. 81, no. 011903, 2010. [2] M. J. Simpson, R. E. Baker, and S. W. McCue, “Models of collective Fig. 2. Solution to system (5) with cohesion and aversion. cell spreading with variable cell aspect ratio: a motivation for degenerate diffusion models,” Physical Review E, vol. 83, no. 021901, 2011. [3] M. Burger, M. Di Francesco, J.-F. Pietschmann, and B. Schlake, example we observe lane formation. Since γ > γ individuals “Nonlinear cross-diffusion with size exclusion,” SIAM Journal on 1 2 have a tendency to step to the right. This tendency can also Mathematical Analysis, vol. 42, no. 6, pp. 2842–2871, 2010. be observed in the formation of the directional lanes. The red [4] M. Burger, B. Schlake, and M. Wolfram, “Nonlinear Poisson–Nernst– Planck equations for ion ﬂux through conﬁned geometries,” Nonlinear- individuals concentrate on the bottom of the corridor, whereas ity, vol. 25, no. 961, 2012. the blue individuals move to the top. [5] M. J. Lighthill and G. B. Whitham, “On kinematic waves. II. a theory of trafﬁc ﬂow on long crowded roads,” in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 229, no. 1178. The Royal Society, 1955, pp. 317–345. V. CONCLUSION AND OUTLOOK TO FURTHER [6] M. Burger, P. A. Markowich, and J.-F. Pietschmann, “Continuous APPLICATIONS limit of a crowd motion and herding model: analysis and numerical simulations,” Kinetic and Related Models, vol. 4, no. 4, pp. 1025–1047, In this paper, we have presented a symbolic approach 2011. to derive the mean-ﬁeld PDEs from lattice-based models. [7] M. Burger, M. D. Francesco, P. A. Markowich, and M.-T. Wolfram, We demonstrated the methods in terms of one example in “Mean ﬁeld games with nonlinear mobilities in pedestrian dynamics,” Discrete and Continuous Dynamical Systems - Series B, vol. 19, no. 5, pedestrian dynamics, but the algorithm may also be applied to pp. 1311–1333, 2014. other examples. In [1] different motility mechanisms on regular [8] M. J. Simpson, A. Merriﬁeld, K. A. Landman, and B. D. Hughes, lattices are introduced, which result in nonlinear diffusion “Simulating invasion with cellular automata: connecting cell-scale and equations with different diffusivities. The authors considered population-scale properties,” Physical Review E, vol. 76, no. 021918, various motilities based on attraction or repulsion, i.e. the 2007. transition rate to move away from a neighbouring individual [9] V. J. Blue and J. L. Adler, “Cellular automata microsimulation for increases or decreases respectively. For example, in a minimal modeling bi-directional pedestrian walkways,” Transportation Research Part B: Methodological, vol. 35, no. 3, pp. 293–312, 2001. model the transition rate is given by [10] C. Burstedde, K. Klauck, A. Schadschneider, and J. Zittartz, “Simula- T → − − i i+1 = (1 c )(1 αc − ), tion of pedestrian dynamics using a two-dimensional cellular automa- i+1 i 1 ton,” Physica A: Statistical Mechanics and its Applications, vol. 295, where c again denotes the probability that the lattice site x no. 3, pp. 507–525, 2001. i i is occupied. Hence the transition to move from x to x [11] C. Cercignani, Rareﬁed gas dynamics: from basic concepts to actual i i+1 calculations. Cambridge University Press, 2000. is reduced if the neighbouring site x − is occupied. This i 1 [12] C. Kipnis and C. Landim, Scaling limits of interacting particle systems. phenomenon is known as adhesion and results in a nonlinear Springer Science & Business Media, 2013, vol. 320. diffusion model for the cell density c = c(x, t) of the form [13] A. Stevens, “The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems,” SIAM ∂ c = ∂ (D(c)∂ c), t x x Journal on Applied Mathematics, vol. 61, no. 1, pp. 183–212, 2000. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine##

text: with

text: a

text: diffusivity

text: of

text: the

text: form

text: D(c)

text: see

text: Again

text: in

text: several

text: other

text: transition

text: rates

text: were

text: proposed,

text: which

text: lead

text: to

text: different

text: nonlinear

text: diffusitivies,

text: see

text: Table

text: there.

text: Using

text: our

text: implementation,

text: the

text: entries

text: in

text: that

text: table

text: can

text: be

text: generated

text: automatically.

text: For

text: example,

text: we

text: have

text: tried

text: one

text: of

text: their

text: most

text: complicated

text: models

text: Equation

text: which

text: combines

text: contact-forming

text: or

text: contact-breaking

text: interac-

text: tions

text: with

text: contact-maintaining

text: interactions.

text: Our

text: implementa-

text: tion

text: correctly

text: derives

text: the

text: diffusivity

text: given

text: in

text: Equation

text: where

text: we

text: have

text: chosen

text: the

text: two-dimensional

text: square

text: lattice

text: with

text: Moore

text: interacting

text: neighborhoods.

text: (a)

text: Density

text: of

text: reds

text: at

text: time

text: T

text: ACKNOWLEDGMENT

text: CK

text: was

text: supported

text: by

text: the

text: Austrian

text: Science

text: Fund

text: (FWF):

text: HR

text: and

text: MTW

text: acknowledge

text: support

text: by

text: the

text: Austrian

text: Academy

text: of

text: Sciences

text: O¨

text: AW

text: via

text: the

text: New

text: Frontiers

text: project

text: GR

text: was

text: supported

text: by

text: the

text: Austrian

text: Science

text: Fund

text: (FWF):

text: REFERENCES

text: A.

text: E.

text: Fernando,

text: K.

text: A.

text: Landman,

text: and

text: M.

text: J.

text: Simpson,

text: “Nonlinear

text: diffusion

text: and

text: exclusion

text: processes

text: with

text: contact

text: interactions,”

text: Physical

text: (b)

text: Density

text: of

text: blues

text: at

text: time

text: T

text: Review

text: E,

text: vol.

text: no.

text: M.

text: J.

text: Simpson,

text: R.

text: E.

text: Baker,

text: and

text: S.

text: W.

text: McCue,

text: “Models

text: of

text: collective

text: Fig.

text: Solution

text: to

text: system

text: with

text: cohesion

text: and

text: aversion.

text: cell

text: spreading

text: with

text: variable

text: cell

text: aspect

text: ratio:

text: a

text: motivation

text: for

text: degenerate

text: diffusion

text: models,”

text: Physical

text: Review

text: E,

text: vol.

text: no.

text: M.

text: Burger,

text: M.

text: Di

text: Francesco,

text: J.-F.

text: Pietschmann,

text: and

text: B.

text: Schlake,

text: example

text: we

text: observe

text: lane

text: formation.

text: Since

text: γ

text: γ

text: individuals

text: “Nonlinear

text: cross-diffusion

text: with

text: size

text: exclusion,”

text: SIAM

text: Journal

text: on

text: have

text: a

text: tendency

text: to

text: step

text: to

text: the

text: right.

text: This

text: tendency

text: can

text: also

text: Mathematical

text: Analysis,

text: vol.

text: no.

text: pp.

text: be

text: observed

text: in

text: the

text: formation

text: of

text: the

text: directional

text: lanes.

text: The

text: red

text: M.

text: Burger,

text: B.

text: Schlake,

text: and

text: M.

text: Wolfram,

text: “Nonlinear

text: Poisson–Nernst–

text: Planck

text: equations

text: for

text: ion

text: ﬂux

text: through

text: conﬁned

text: geometries,”

text: Nonlinear-

text: individuals

text: concentrate

text: on

text: the

text: bottom

text: of

text: the

text: corridor,

text: whereas

text: ity,

text: vol.

text: no.

text: the

text: blue

text: individuals

text: move

text: to

text: the

text: top.

text: M.

text: J.

text: Lighthill

text: and

text: G.

text: B.

text: Whitham,

text: “On

text: kinematic

text: waves.

text: II.

text: a

text: theory

text: of

text: trafﬁc

text: ﬂow

text: on

text: long

text: crowded

text: roads,”

text: in

text: Proceedings

text: of

text: the

text: Royal

text: Society

text: of

text: London

text: A:

text: Mathematical,

text: Physical

text: and

text: Engineering

text: Sciences,

text: vol.

text: no.

text: The

text: Royal

text: Society,

text: pp.

text: V.

text: CONCLUSION

text: AND

text: OUTLOOK

text: TO

text: FURTHER

text: M.

text: Burger,

text: P.

text: A.

text: Markowich,

text: and

text: J.-F.

text: Pietschmann,

text: “Continuous

text: APPLICATIONS

text: limit

text: of

text: a

text: crowd

text: motion

text: and

text: herding

text: model:

text: analysis

text: and

text: numerical

text: simulations,”

text: Kinetic

text: and

text: Related

text: Models,

text: vol.

text: no.

text: pp.

text: In

text: this

text: paper,

text: we

text: have

text: presented

text: a

text: symbolic

text: approach

text: to

text: derive

text: the

text: mean-ﬁeld

text: PDEs

text: from

text: lattice-based

text: models.

text: M.

text: Burger,

text: M.

text: D.

text: Francesco,

text: P.

text: A.

text: Markowich,

text: and

text: M.-T.

text: Wolfram,

text: We

text: demonstrated

text: the

text: methods

text: in

text: terms

text: of

text: one

text: example

text: in

text: “Mean

text: ﬁeld

text: games

text: with

text: nonlinear

text: mobilities

text: in

text: pedestrian

text: dynamics,”

text: Discrete

text: and

text: Continuous

text: Dynamical

text: Systems

text: Series

text: B,

text: vol.

text: no.

text: pedestrian

text: dynamics,

text: but

text: the

text: algorithm

text: may

text: also

text: be

text: applied

text: to

text: pp.

text: other

text: examples.

text: In

text: different

text: motility

text: mechanisms

text: on

text: regular

text: M.

text: J.

text: Simpson,

text: A.

text: Merriﬁeld,

text: K.

text: A.

text: Landman,

text: and

text: B.

text: D.

text: Hughes,

text: lattices

text: are

text: introduced,

text: which

text: result

text: in

text: nonlinear

text: diffusion

text: “Simulating

text: invasion

text: with

text: cellular

text: automata:

text: connecting

text: cell-scale

text: and

text: equations

text: with

text: different

text: diffusivities.

text: The

text: authors

text: considered

text: population-scale

text: properties,”

text: Physical

text: Review

text: E,

text: vol.

text: no.

text: various

text: motilities

text: based

text: on

text: attraction

text: or

text: repulsion,

text: i.e.

text: the

text: transition

text: rate

text: to

text: move

text: away

text: from

text: a

text: neighbouring

text: individual

text: V.

text: J.

text: Blue

text: and

text: J.

text: L.

text: Adler,

text: “Cellular

text: automata

text: microsimulation

text: for

text: increases

text: or

text: decreases

text: respectively.

text: For

text: example,

text: in

text: a

text: minimal

text: modeling

text: bi-directional

text: pedestrian

text: walkways,”

text: Transportation

text: Research

text: Part

text: B:

text: Methodological,

text: vol.

text: no.

text: pp.

text: model

text: the

text: transition

text: rate

text: is

text: given

text: by

text: C.

text: Burstedde,

text: K.

text: Klauck,

text: A.

text: Schadschneider,

text: and

text: J.

text: Zittartz,

text: “Simula-

text: T

text: i

text: c

text: αc

text: tion

text: of

text: pedestrian

text: dynamics

text: using

text: a

text: two-dimensional

text: cellular

text: automa-

text: i

text: ton,”

text: Physica

text: A:

text: Statistical

text: Mechanics

text: and

text: its

text: Applications,

text: vol.

text: where

text: c

text: again

text: denotes

text: the

text: probability

text: that

text: the

text: lattice

text: site

text: x

text: no.

text: pp.

text: i

text: i

text: is

text: occupied.

text: Hence

text: the

text: transition

text: to

text: move

text: from

text: x

text: to

text: x

text: C.

text: Cercignani,

text: Rareﬁed

text: gas

text: dynamics:

text: from

text: basic

text: concepts

text: to

text: actual

text: i

text: calculations.

text: Cambridge

text: University

text: Press,

text: is

text: reduced

text: if

text: the

text: neighbouring

text: site

text: x

text: is

text: occupied.

text: This

text: i

text: C.

text: Kipnis

text: and

text: C.

text: Landim,

text: Scaling

text: limits

text: of

text: interacting

text: particle

text: systems.

text: phenomenon

text: is

text: known

text: as

text: adhesion

text: and

text: results

text: in

text: a

text: nonlinear

text: Springer

text: Science

text: Business

text: Media,

text: vol.

text: diffusion

text: model

text: for

text: the

text: cell

text: density

text: c

text: c(x,

text: t)

text: of

text: the

text: form

text: A.

text: Stevens,

text: “The

text: derivation

text: of

text: chemotaxis

text: equations

text: as

text: limit

text: dynamics

text: of

text: moderately

text: interacting

text: stochastic

text: many-particle

text: systems,”

text: SIAM

text: c

text: (D(c)∂

text: c),

text: t

text: x

text: x

text: Journal

text: on

text: Applied

text: Mathematics,

text: vol.

text: no.

text: pp.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##


text: Lry 7201HLΝ0HΤ 2(cid:144)1 11 × − TABLE XV. Left: Calculated half-lives for neutrinoless double-β decay with exchange of heavy neutrinos for η = 2.75 10 7 and | | g = 1.269. Right: Upper limits of η and lower limits of heavy neutrino mass (see text for details) from current experimental A limit from a compilation of Barabash [41]. The value reported by Klapdor-Kleingrothaus et al. [42], IGEX collaboration [36], and the recent limit from KamLAND-Zen [38], EXO [39], and GERDA [40] are also included. | | − (cid:104) (cid:105) Decay τ0νh(1024yr) τ0νh (yr) η (10 6) m (GeV) 1/2 1/2,exp νh → × 48Ca 48Ti 0.72 > 5.8 1022 <0.36 >11.9 → × 76Ge 76Se 1.51 > 1.9 1025 <0.028 >148 × 1.2 1025a 0.035 118 × > 1.6 1025b <0.031 >136 × > 2.1 1025c < 0.027 156 → × 82Se 82Kr 0.55 > 3.6 1023 <0.12 >34 → × 96Zr 96Mo 0.19 > 9.2 1021 <0.46 >9.15 → × 100Mo 100Ru 0.09 > 1.1 1024 <0.028 >146 → 110Pd 110Cd 0.33 → × 116Cd 116Sn 0.19 > 1.7 1023 <0.11 >39.5 → 124Sn 124Te 0.67 → × 128Te 128Xe 6.43 > 1.5 1024 <0.21 >20.2 → × 130Te 130Xe 0.32 > 2.8 1024 <0.034 >123 → 134Xe 134Ba 8.57 → × 136Xe 136Ba 0.50 > 1.9 1025d <0.016 >257 × > 1.6 1025e <0.018 >236 → 148Nd 148Sm 0.36 → × 150Nd 150Sm 0.05 > 1.8 1022 <0.16 >26.3 → 154Sm 154Gd 1.00 → 160Gd 160Dy 0.17 → 198Pt 198Hg 0.48 → 232Th 232U 0.11 → 238U 238Pu 0.03 à 15000..00 æ æ æà EΒC+ΒΒ++ gAIB,eMf−f 2 = 1.269A−0.18, (26) − gISM = 1.269A 0.12 à A,eff à 10.0 à 5.0 and compare them with the unquenched values with à à g = 1.269 also given in Table XVI. Quenching of g A A 1.0 æ æ appears to have a major eﬀect on the calculated half-lives, 0.5 æ by multiplying them by a factor of 6-50.The question of æ æ æ what value of g one needs to use is thus a major concern Ni Zn Kr Ru Cd Xe Ba Ce which needs toAbe addressed. This concern has also been 60 80 100 120 140 discussed recently in Refs. [45, 46]. Mass number (cid:104) (cid:105) FIG. 8. (Color online) Expected half-lives for m = 1 eV, ν g = 1.269 and IBM-2 isospin restored nuclear matrix ele- A ments. The ﬁgure is in semilogarithmic scale. VIII. CONCLUSIONS In this paper, we have introduced a method for restora- 2νββ decay. There is much discussion at the present time tion of the isospin properties of the Fermi transition op- on whether or not 0νββ is equally quenched as 2νββ erator in the calculation of the Fermi matrix elements (and single β-decay). In order to investigate the possi- within of the framework of IBM-2, and done a consis- ble impact of quenching of g we present in Table XVI tent calculation of 0νββ, 0ν ββ, and 2νββ NME in the A h the predicted half-lives under the assumption of maximal closure approximation. With this method, the F matrix quenching elements for 2νββ decay are set to zero, and those for ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Lry

font: default

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text: TABLE

text: XV.

text: Left:

text: Calculated

text: half-lives

text: for

text: neutrinoless

text: double-β

text: decay

text: with

text: exchange

text: of

text: heavy

text: neutrinos

text: for

text: η

text: and

text: g

text: Right:

text: Upper

text: limits

text: of

text: η

text: and

text: lower

text: limits

text: of

text: heavy

text: neutrino

text: mass

text: (see

text: text

text: for

text: details)

text: from

text: current

text: experimental

text: A

text: limit

text: from

text: a

text: compilation

text: of

text: Barabash

text: The

text: value

text: reported

text: by

text: Klapdor-Kleingrothaus

text: et

text: al.

text: IGEX

text: collaboration

text: and

text: the

text: recent

text: limit

text: from

text: KamLAND-Zen

text: EXO

text: and

text: GERDA

text: are

text: also

text: included.

label: table

text: Decay

text: τ0νh

text: (yr)

text: η

text: m

text: (GeV)

text: νh

text: à

text: æ

text: æ

text: æà

text: EΒC+ΒΒ++

text: gAIB,eMf−f

text: gISM

text: à

text: A,eff

text: à

text: à

text: and

text: compare

text: them

text: with

text: the

text: unquenched

text: values

text: with

text: à

text: à

text: g

text: also

text: given

text: in

text: Table

text: XVI.

text: Quenching

text: of

text: g

text: A

text: A

text: æ

text: æ

text: appears

text: to

text: have

text: a

text: major

text: eﬀect

text: on

text: the

text: calculated

text: half-lives,

text: æ

text: by

text: multiplying

text: them

text: by

text: a

text: factor

text: of

text: question

text: of

text: æ

text: æ

text: æ

text: what

text: value

text: of

text: g

text: one

text: needs

text: to

text: use

text: is

text: thus

text: a

text: major

text: concern

text: Ni

text: Zn

text: Kr

text: Ru

text: Cd

text: Xe

text: Ba

text: Ce

text: which

text: needs

text: toAbe

text: addressed.

text: This

text: concern

text: has

text: also

text: been

text: discussed

text: recently

text: in

text: Refs.

text: Mass

text: number

text: FIG.

text: (Color

text: online)

text: Expected

text: half-lives

text: for

text: m

text: eV,

text: ν

text: g

text: and

text: isospin

text: restored

text: nuclear

text: matrix

text: ele-

text: A

text: ments.

text: The

text: ﬁgure

text: is

text: in

text: semilogarithmic

text: scale.

text: VIII.

text: CONCLUSIONS

text: In

text: this

text: paper,

text: we

text: have

text: introduced

text: a

text: method

text: for

text: restora-

text: 2νββ

text: decay.

text: There

text: is

text: much

text: discussion

text: at

text: the

text: present

text: time

text: tion

text: of

text: the

text: isospin

text: properties

text: of

text: the

text: Fermi

text: transition

text: op-

text: on

text: whether

text: or

text: not

text: 0νββ

text: is

text: equally

text: quenched

text: as

text: 2νββ

text: erator

text: in

text: the

text: calculation

text: of

text: the

text: Fermi

text: matrix

text: elements

text: (and

text: single

text: β-decay).

text: In

text: order

text: to

text: investigate

text: the

text: possi-

text: within

text: of

text: the

text: framework

text: of

text: and

text: done

text: a

text: consis-

text: ble

text: impact

text: of

text: quenching

text: of

text: g

text: we

text: present

text: in

text: Table

text: XVI

text: tent

text: calculation

text: of

text: 0νββ,

text: 0ν

text: ββ,

text: and

text: 2νββ

text: NME

text: in

text: the

text: A

text: h

text: the

text: predicted

text: half-lives

text: under

text: the

text: assumption

text: of

text: maximal

text: closure

text: approximation.

text: With

text: this

text: method,

text: the

text: F

text: matrix

text: quenching

text: elements

text: for

text: 2νββ

text: decay

text: are

text: set

text: to

text: zero,

text: and

text: those

text: for

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text: 4 − − TABLE I. IBM-2 nuclear matrix elements M(0ν) (dimensionless) for 0νβ β decay with Argonne SRC, g /g = 1/1.269, and V A isospin restoration. 0+ 0+ 1 2 A M(0ν) M(0ν) M(0ν) M(0ν) M(0ν) M(0ν) M(0ν) M(0ν) GT F T GT F T 48Ca 1.73 -0.30 -0.17 1.75 3.78 -0.27 -0.12 3.82 76Ge 4.49 -0.68 -0.23 4.68 1.95 -0.27 -0.09 2.02 82Se 3.59 -0.60 -0.23 3.73 0.92 -0.13 -0.05 0.95 96Zr 2.51 -0.33 0.11 2.83 0.04 -0.01 0.00 0.05 100Mo 3.73 -0.48 0.19 4.22 0.99 -0.13 0.05 1.12 110Pd 3.59 -0.40 0.21 4.05 0.46 -0.05 0.03 0.52 116Cd 2.76 -0.33 0.14 3.10 0.84 -0.09 0.03 0.93 124Sn 2.96 -0.57 -0.12 3.19 2.21 -0.41 -0.09 2.38 128Te 3.80 -0.72 -0.15 4.10 2.65 -0.47 -0.09 2.85 130Te 3.43 -0.65 -0.13 3.70 2.52 -0.45 -0.08 2.71 134Xe 3.77 -0.68 -0.15 4.05 2.19 -0.36 -0.06 2.35 136Xe 2.83 -0.52 -0.10 3.05 1.49 -0.24 -0.03 1.60 148Nd 2.00 -0.38 0.07 2.31 0.25 -0.05 0.01 0.29 150Nd 2.33 -0.39 0.10 2.67 0.40 -0.06 0.02 0.45 154Sm 2.49 -0.36 0.11 2.82 0.37 -0.04 0.01 0.41 160Gd 3.64 -0.45 0.17 4.08 0.76 -0.11 0.04 0.87 198Pt 1.90 -0.33 0.09 2.19 0.08 -0.02 0.01 0.10 232Th 3.58 -0.44 0.18 4.04 0.12 -0.02 0.01 0.15 238U 4.27 -0.53 0.21 4.81 0.34 -0.05 0.02 0.40 1. Sensitivity to parameter changes, model assumptions TABLE II. Comparison among nuclear matrix elements for − − and operator assumptions 0νβ β decay to ground state, 0+, in IBM-2 with Argonne 1 SRC, g = 1.269, and isospin restoration, QRPA-Tü with Ar- A gonne SRC, g = 1.27, and isospin restoration [11], and ISM The sensitivity of IBM-2 to parameter changes, model A with UCOM SRC and gA = 1.25 [20]. All matrix elements assumptions, and operator assumptions was discussed in are in dimensionless units. great detail in Ref. [5]. We do not repeat this discussion, M(0ν) but only note that because of isospin restoration the sen- sitivity of F matrix elements to isospin purity decreases Decay IBM-2 QRPA-Tü ISM from 10% to 1%, including the case of 48Ca decay, which → 48Ca 48Ti 1.75 0.54 0.85 previously was treated separately from the others. Our → 76Ge 76Se 4.68 5.16 2.81 error estimate for 0νββ is now 16% for all nuclei. In the → 82Se 82Kr 3.73 4.64 2.64 case of 0ν ββ we also estimate a reduced sensitivity of F → h 96Zr 96Mo 2.83 2.72 matrix elements from 10% to 1%, and a reduced sensitiv- → 100Mo 100Ru 4.22 5.40 ity of the matrix elements F + GT to SRC from 50% to → 110Pd 110Cd 4.05 5.76 25%. This sensitivity is estimated by comparing matrix → 116Cd 116Sn 3.10 4.04 elements with Argonne-CD-Bonn and UCOM SRC. Our → 124Sn 124Te 3.19 2.56 2.62 total error estimate for 0ν ββ is now 28% for all nuclei. → h 128Te 128Xe 4.10 4.56 2.88 Our ﬁnal matrix elements, with error estimate are given → 130Te 130Xe 3.70 3.89 2.65 in Table VI. → 134Xe→134Ba 4.05 In addition to IBM-2, QRPA, and ISM, other cal- 136Xe 136Ba 3.05 2.18 2.19 culations have been recently done. In Fig. 4 we com- → 148Nd 148Sm 2.31 pare our results with all available calculations including → 150Nd 150Sm 2.67 density functional theory (DFT) [23] and Hartree-Fock- → 154Sm 154Gd 2.82 Bogoliubov (HFB) theory [24]. → 160Gd 160Dy 4.08 198Pt→198Hg 2.19 IV. RESULTS FOR 0νβ+β+ → 232Th 232U 4.04 → 238U 238Pu 4.81 Matrix elements for double positron (β+β+) decay and the related processes (ECβ+) and (ECEC) can be cal- culated in a similar way. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: TABLE

text: I.

text: nuclear

text: matrix

text: elements

text: (dimensionless)

text: for

text: 0νβ

text: β

text: decay

text: with

text: Argonne

text: SRC,

text: g

text: /g

text: and

text: V

text: A

text: isospin

text: restoration.

text: A

text: GT

text: F

text: T

text: GT

text: F

text: T

text: Sensitivity

text: to

text: parameter

text: changes,

text: model

text: assumptions

text: TABLE

text: II.

text: Comparison

text: among

text: nuclear

text: matrix

text: elements

text: for

text: and

text: operator

text: assumptions

text: 0νβ

text: β

text: decay

text: to

text: ground

text: state,

text: in

text: with

text: Argonne

text: SRC,

text: g

text: and

text: isospin

text: restoration,

text: QRPA-Tü

text: with

text: Ar-

text: A

text: gonne

text: SRC,

text: g

text: and

text: isospin

text: restoration

text: and

text: ISM

text: The

text: sensitivity

text: of

text: to

text: parameter

text: changes,

text: model

text: A

text: with

text: UCOM

text: SRC

text: and

text: gA

text: All

text: matrix

text: elements

text: assumptions,

text: and

text: operator

text: assumptions

text: was

text: discussed

text: in

text: are

text: in

text: dimensionless

text: units.

text: great

text: detail

text: in

text: Ref.

text: We

text: do

text: not

text: repeat

text: this

text: discussion,

text: but

text: only

text: note

text: that

text: because

text: of

text: isospin

text: restoration

text: the

text: sen-

text: sitivity

text: of

text: F

text: matrix

text: elements

text: to

text: isospin

text: purity

text: decreases

text: Decay

text: QRPA-Tü

text: ISM

text: from

text: to

text: including

text: the

text: case

text: of

text: decay,

text: which

text: previously

text: was

text: treated

text: separately

text: from

text: the

text: others.

text: Our

text: error

text: estimate

text: for

text: 0νββ

text: is

text: now

text: for

text: all

text: nuclei.

text: In

text: the

text: case

text: of

text: 0ν

text: ββ

text: we

text: also

text: estimate

text: a

text: reduced

text: sensitivity

text: of

text: F

text: h

text: matrix

text: elements

text: from

text: to

text: and

text: a

text: reduced

text: sensitiv-

text: ity

text: of

text: the

text: matrix

text: elements

text: F

text: GT

text: to

text: SRC

text: from

text: to

text: This

text: sensitivity

text: is

text: estimated

text: by

text: comparing

text: matrix

text: elements

text: with

text: Argonne-CD-Bonn

text: and

text: UCOM

text: SRC.

text: Our

text: total

text: error

text: estimate

text: for

text: 0ν

text: ββ

text: is

text: now

text: for

text: all

text: nuclei.

text: h

text: Our

text: ﬁnal

text: matrix

text: elements,

text: with

text: error

text: estimate

text: are

text: given

text: in

text: Table

text: VI.

text: In

text: addition

text: to

text: QRPA,

text: and

text: ISM,

text: other

text: cal-

text: culations

text: have

text: been

text: recently

text: done.

text: In

text: Fig.

text: we

text: com-

text: pare

text: our

text: results

text: with

text: all

text: available

text: calculations

text: including

text: density

text: functional

text: theory

text: (DFT)

text: and

text: Hartree-Fock-

text: Bogoliubov

text: (HFB)

text: theory

text: IV.

text: RESULTS

text: FOR

text: 0νβ+β+

text: Matrix

text: elements

text: for

text: double

text: positron

text: (β+β+)

text: decay

text: and

text: the

text: related

text: processes

text: (ECβ+)

text: and

text: (ECEC)

text: can

text: be

text: cal-

text: culated

text: in

text: a

text: similar

text: way.

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text: g ffe,A 9 1.4 á from experimental Τ1(cid:144)2 HISML TABLE XIV. Left: Calculated half-lives i(cid:104)n IB(cid:105)M-2 Argonne 1.2 ç gIAS,eMff=1.269A-0.12 SRC for neutrinoless double-β decay for mν = 1 eV and à(cid:144)ò from experimental Τ1(cid:144)2 HIBM-2 CA(cid:144)SSDL gA = 1.269. Right: Upper limit on neutrino mass from 1.0 æ gIAB,eMff-2=1.269A-0.18 current experimental limit from a compilation of Barabash á á [41]. The value reported by Klapdor-Kleingrothaus et al. 0.8 ç [42], IGEX collaboration [36], and the recent limits from ç çá Ca à àò çá ç KamLAND-Zen [38], EXO [39], and GERDA [40] are also 00..46 àæ Gæe Sæe ZòærMòæ o Còæd Tàòæàeæ Xàáòæ e Nòæd iDncelcuadyed. τ10/ν2(1024yr) τ10/ν2,exp(yr) (cid:104)mν(cid:105) (eV) → × 0.2 48Ca 48Ti 1.33 > 5.8 1022 < 4.8 → × 76Ge 76Se 1.95 > 1.9 1025 < 0.32 0.0 × 40 60 80 100 120 140 160 1.2 1025a 0.40 × Mass number > 1.6 1025b < 0.35 × > 2.1 1025c < 0.30 → × 82Se 82Kr 0.71 > 3.6 1023 < 1.4 FIG. 5. (Color online) Value of gA,eff extracted from exper- 96Zr→96Mo 0.61 > 9.2 × 1021 < 8.1 iment for IBM-2 and ISM. → × 100Mo 100Ru 0.36 > 1.1 1024 < 0.57 → 110Pd 110Cd 1.27 → × 116Cd 116Sn 0.63 > 1.7 1023 < 1.9 TABLE XIII. 2νβ+β+, 2νECβ+, and 2νECEC nuclear ma- → 124Sn 124Te 1.09 trix elements (dimensionless) to the ground state (columns 2 → × and 3) and to the ﬁrst excited state (columns 4 and 5) using 128Te 128Xe 10.19 > 1.5 1024 < 2.6 → × IBM-2 with isospin restoration and and Argonne SRC in the 130Te 130Xe 0.52 > 2.8 1024 < 0.43 → closure approximation. 134Xe 124Ba 10.23 → × 136Xe 136Ba 0.74 > 1.9 1025d < 0.20 0+1 0+2 > 1.6 × 1025e < 0.22 → Nucleus M(2ν) M(2ν) M(2ν) M(2ν) 148Nd 148Sm 1.87 GT F GT F → × 150Nd 150Sm 0.22 > 1.8 1022 < 3.5 58Ni 2.11 -0.00 2.34 -0.00 → 154Sm 154Gd 4.19 64Zn 5.20 -0.01 0.71 -0.00 → 160Gd 160Dy 0.63 78Kr 3.67 -0.01 0.83 -0.00 → 198Pt 198Hg 2.77 96Ru 2.17 -0.00 0.05 -0.00 → 232Th 232U 0.44 106Cd 2.57 -0.00 1.47 -0.00 → 238U 238Pu 0.13 124Xe 4.24 -0.01 0.71 -0.00 130Ba 4.22 -0.01 0.30 -0.00 136Ce 4.17 -0.01 0.34 -0.00 156Dy 2.20 -0.00 1.15 -0.00 164Er 2.58 -0.00 0.90 -0.00 Using the best ﬁt values [34] 180W 3.09 -0.01 0.12 -0.00 (cid:12) (cid:12) (cid:12) (cid:12) sin2 ϑ12 = 0.213, sin2 ϑ13 = 0.016, (cid:12) (cid:12) × − | | (cid:12)(cid:104)m (cid:105)(cid:12)2 sin2 ϑ23 = 0.466×, δm−2 = 7.67 10 5 eV2, (21) f(m , U ) 2 = ν . (19) ∆m2 = 2.39 10 3 eV2 i ei m e The average light neutrino mass is constrained by at- we obtain the values given in Fig. 7. In this ﬁgure mospheric, solar, reactor and accelerator neutrino oscil- we have added the current limits, for g = 1.269, com- A lation ex(cid:12)periments to be [34] (cid:12) ing from CUORICINO [35], IGEX [36], NEMO-3 [37], (cid:12) (cid:12) KamLAND-Zen [38], EXO [39] and GERDA [40] exper- iments. Also, henceforth we use c=1 to conform with (cid:104) (cid:105) m(cid:0)ν = c213c212(cid:1)m1 +c213s212m2e(cid:18)iϕ2 + s213m3eiϕ3 , (cid:19) standard notation. c = cos ϑ , s = sin ϑ , ϕ = [0, 2π], ij ij ij ij 2,3 B. Heavy neutrino exchange m2 + m2 −δm2 δm2 ± m2, m2, m2 = 1 2 + , + , ∆m2 . 1 2 3 2 2 2 (20) The half-lives for this case are calculated using the for- mula ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 3 cept what has been until recently the intuitive answer TABLE I: Quantum trajectory time evolution according to to the previous question, namely that nothing (besides Kominis’ approach. Hamiltonian evolution) happens to surviving RPs. As Name Event Probability of event mentioned in Section IIA, the rates k and k are pa- (cid:104) | | (cid:105) S T K1 projection to the singlet dp = dtkS+kT ψ Q ψ rameters entering into the master equation, which obvi- √ | (cid:105) S 2 S state (cid:104)QS| ψ | (cid:105) ously must be valid for any choice of those parameters. ψ QS ψ (cid:104) | | (cid:105) The case kS = kT is rather simple dynamically, since in K2 projection√to th| e(cid:105)triplet dpT = dtkS+2kT ψ QT ψ th≡is case RP population decays exponentially at a rate state (cid:104)QψT|QψT|ψ(cid:105) (cid:104) | | (cid:105) ksurvikvSin=g kRTP,sw. itThhoiust mtheeandsectahyeafﬀoellcotwininggt.heCsotnasteidoerf tfhoer K3 singlet recombination dr = k dt ψ Q ψ S S (cid:104) | S | (cid:105) example a 50/50 mixture of singlet and triplet RPs, hav- K4 triplet recombination dr = k dt ψ Q ψ H − T − T − T− ing no magnetic interactions ( = 0). If kS = kT, the K5 hamiltonian evolution 1 dpS dpT drS drT same number of singlet and triplet RPs will recombine in the interval dt, hence at time t + dt the mixture will still be 50/50, albeit having a smaller total population. (cid:54) the results of the master equation (1)-(4). We will check On the other hand, if k = k , the spin character of this S T whether this is the case in the context of CIDNP observ- mixture would change, becoming more (less) singlet if ables in Section IV. k < k (k > k ). S T S T III. HABERKORN APPROACH TO RADICAL-PAIR QUANTUM DYNAMICS To summarize, (i) assuming that having a diﬀerent fun- damental theory for diﬀerent radical pairs (i.e. diﬀerent combinations of k and k ) is not an acceptable option, A. Master equation approach S T and (ii) being unable to propose what happens in the gen- eral case to non-recombining RPs from Haberkorn’s point The traditional, or Haberkorn master equation reads of view, except in the special case k = k , where consis- S T dρ − H − k − k tency forces one to accept that nothing else happens be- dt = i[ , ρ] 2S (QSρ + ρQS) 2T (QTρ + ρQT) (5) sides unitary evolution (to be proved in the following), we take this to be the general answer, and hence Haberkorn’s Interestingly, Haberkorn’s master equation follows from quantum trajectories are formed by the three events pre- our master equation (1)-(4) by forcing p to be zero at sented in Table II. coh all times. In any case, as we have done for our approach in the previous section, Haberkorn’s approach must be able to provide the equivalent picture of single-molecule TABLE II: Quantum trajectory time evolution according to quantum trajectories. However, the concept of quantum Haberkorn’ approach. trajectories has not been utilized in spin chemistry so far. Name Event Probability of event Furthermore, although a recent experiment [46] provided (cid:104) | | (cid:105) evidence for the physical reality of the S-T decoherence H1 singlet recombination drS = kSdt ψ QS ψ (cid:104) | | (cid:105) process we introduced, a general consensus on what ex- H2 triplet recombination dr = k dt ψ Q ψ T T T − − actly is the quantum state evolution of surviving RPs is H3 hamiltonian evolution 1 dr dr S T still missing. From our perspective, we do not see how Haberkorn’s approach, being phenomenological, can in- corporate quantum trajectories without going into the There are two comments to be made. First, the recom- (cid:82) derivations of [39, 43], which lead to events K1 and K2, bination probabilities dr and dr are the same in both S T but we leave it as an open question to be addressed by Haberkorn’s approach and ours, since both theories agree the proponents of the conventional theory. Nevertheless, in how the singlet and triplet reaction yields, Y = dr , j j we will here outline some rather strong guidelines as to with j = S, T, are calculated. Second, we can now easily how the consistency check of Haberkorn’s approach can prove our previous statement about H1-H3 ensuring con- in principle unfold. sistency in the special case k = k . Indeed, taking into S T B. Quantum trajectory approach account the complet≡eness relation QS+QT = 1, it follows that when k = k k, Haberkorn’s master equation (5) S T − H − − From Haberkorn’s theory point of view, it is clear that becomes dρ/dt = i[ , ρ] kρ. Deﬁning ρ = e ktR, − H one cannot agree with possibilities K1 and K2, since these it follows that dR/dt = i[ , R]. It is thus evident { } lead to our approach. Hence one has to suggest what is that apart from an exponential decay of Tr ρ , the only the speciﬁc state evolution of surviving RPs, i.e. what is, radical-pair state change is due to the Hamiltonian evo- if any, the state change of radical-pairs until the instant lution. In terms of the quantum trajectories H1-H3, we − H − of their recombination into a neutral product. can retrieve the master equation dρ/dt = i[ , ρ] kρ as { } We will now show that there is limited freedom in do- follows. Since QS + QT = 1, it is drS +{ d}rT = kdtTr ρ . ing so. This is because in order to secure consistency The single-RP density matrix is ρ/Tr ρ , hence averag- in the dynamically simple case k = k , one has to ac- ing H1-H3 leads to S T ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: in

text: the

text: special

text: case

text: k

text: k

text: where

text: consis-

text: S

text: T

text: dρ

text: H

text: k

text: k

text: tency

text: forces

text: one

text: to

text: accept

text: that

text: nothing

text: else

text: happens

text: be-

text: dt

text: i[

text: ρ]

text: (QSρ

text: ρQS)

text: (QTρ

text: ρQT)

text: sides

text: unitary

text: evolution

text: (to

text: be

text: proved

text: in

text: the

text: following),

text: we

text: take

text: this

text: to

text: be

text: the

text: general

text: answer,

text: and

text: hence

text: Haberkorn’s

text: Interestingly,

text: Haberkorn’s

text: master

text: equation

text: follows

text: from

text: quantum

text: trajectories

text: are

text: formed

text: by

text: the

text: three

text: events

text: pre-

text: our

text: master

text: equation

text: by

text: forcing

text: p

text: to

text: be

text: zero

text: at

text: sented

text: in

text: Table

text: II.

text: coh

text: all

text: times.

text: In

text: any

text: case,

text: as

text: we

text: have

text: done

text: for

text: our

text: approach

text: in

text: the

text: previous

text: section,

text: Haberkorn’s

text: approach

text: must

text: be

text: able

text: to

text: provide

text: the

text: equivalent

text: picture

text: of

text: single-molecule

text: TABLE

text: II:

text: Quantum

text: trajectory

text: time

text: evolution

text: according

text: to

text: quantum

text: trajectories.

text: However,

text: the

text: concept

text: of

text: quantum

text: Haberkorn’

text: approach.

text: trajectories

text: has

text: not

text: been

text: utilized

text: in

text: spin

text: chemistry

text: so

text: far.

text: Name

text: Event

text: Probability

text: of

text: event

text: Furthermore,

text: although

text: a

text: recent

text: experiment

text: provided

text: evidence

text: for

text: the

text: physical

text: reality

text: of

text: the

text: S-T

text: decoherence

text: singlet

text: recombination

text: drS

text: kSdt

text: ψ

text: QS

text: ψ

text: process

text: we

text: introduced,

text: a

text: general

text: consensus

text: on

text: what

text: ex-

text: triplet

text: recombination

text: dr

text: k

text: dt

text: ψ

text: Q

text: ψ

text: T

text: T

text: T

text: actly

text: is

text: the

text: quantum

text: state

text: evolution

text: of

text: surviving

text: RPs

text: is

text: hamiltonian

text: evolution

text: dr

text: dr

text: S

text: T

text: still

text: missing.

text: From

text: our

text: perspective,

text: we

text: do

text: not

text: see

text: how

text: Haberkorn’s

text: approach,

text: being

text: phenomenological,

text: can

text: in-

text: corporate

text: quantum

text: trajectories

text: without

text: going

text: into

text: the

text: There

text: are

text: two

text: comments

text: to

text: be

text: made.

text: First,

text: the

text: recom-

text: derivations

text: of

text: which

text: lead

text: to

text: events

text: and

text: bination

text: probabilities

text: dr

text: and

text: dr

text: are

text: the

text: same

text: in

text: both

text: S

text: T

text: but

text: we

text: leave

text: it

text: as

text: an

text: open

text: question

text: to

text: be

text: addressed

text: by

text: Haberkorn’s

text: approach

text: and

text: ours,

text: since

text: both

text: theories

text: agree

text: the

text: proponents

text: of

text: the

text: conventional

text: theory.

text: Nevertheless,

text: in

text: how

text: the

text: singlet

text: and

text: triplet

text: reaction

text: yields,

text: Y

text: dr

text: j

text: j

text: we

text: will

text: here

text: outline

text: some

text: rather

text: strong

text: guidelines

text: as

text: to

text: with

text: j

text: S,

text: T,

text: are

text: calculated.

text: Second,

text: we

text: can

text: now

text: easily

text: how

text: the

text: consistency

text: check

text: of

text: Haberkorn’s

text: approach

text: can

text: prove

text: our

text: previous

text: statement

text: about

text: ensuring

text: con-

text: in

text: principle

text: unfold.

text: sistency

text: in

text: the

text: special

text: case

text: k

text: k

text: Indeed,

text: taking

text: into

text: S

text: T

text: B.

text: Quantum

text: trajectory

text: approach

text: account

text: the

text: complet≡eness

text: relation

text: QS+QT

text: it

text: follows

text: that

text: when

text: k

text: k

text: k,

text: Haberkorn’s

text: master

text: equation

text: S

text: T

text: H

text: From

text: Haberkorn’s

text: theory

text: point

text: of

text: view,

text: it

text: is

text: clear

text: that

text: becomes

text: dρ/dt

text: i[

text: ρ]

text: kρ.

text: Deﬁning

text: ρ

text: e

text: ktR,

text: H

text: one

text: cannot

text: agree

text: with

text: possibilities

text: and

text: since

text: these

text: it

text: follows

text: that

text: dR/dt

text: i[

text: R].

text: It

text: is

text: thus

text: evident

text: lead

text: to

text: our

text: approach.

text: Hence

text: one

text: has

text: to

text: suggest

text: what

text: is

text: that

text: apart

text: from

text: an

text: exponential

text: decay

text: of

text: Tr

text: ρ

text: the

text: only

text: the

text: speciﬁc

text: state

text: evolution

text: of

text: surviving

text: RPs,

text: i.e.

text: what

text: is,

text: radical-pair

text: state

text: change

text: is

text: due

text: to

text: the

text: Hamiltonian

text: evo-

text: if

text: any,

text: the

text: state

text: change

text: of

text: radical-pairs

text: until

text: the

text: instant

text: lution.

text: In

text: terms

text: of

text: the

text: quantum

text: trajectories

text: we

text: H

text: of

text: their

text: recombination

text: into

text: a

text: neutral

text: product.

text: can

text: retrieve

text: the

text: master

text: equation

text: dρ/dt

text: i[

text: ρ]

text: kρ

text: as

text: We

text: will

text: now

text: show

text: that

text: there

text: is

text: limited

text: freedom

text: in

text: do-

text: follows.

text: Since

text: QS

text: QT

text: it

text: is

text: drS

text: d}rT

text: kdtTr

text: ρ

text: ing

text: so.

text: This

text: is

text: because

text: in

text: order

text: to

text: secure

text: consistency

text: The

text: single-RP

text: density

text: matrix

text: is

text: ρ/Tr

text: ρ

text: hence

text: averag-

text: in

text: the

text: dynamically

text: simple

text: case

text: k

text: k

text: one

text: has

text: to

text: ac-

text: ing

text: leads

text: to

text: S

text: T

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text: The data sets consists of the yearly count of the population of whole Bulgar- ian cities from 2004 till 2011, as recorded by the National Statistical Institute of the Republic of Bulgaria (http : www.nsi.bg). Thus, the hypotheses to be discussed are : • Hypothesis 1: The growth rate of the ”rescaled city population” is inde- pendent on the size of the population. • Hypothesis 2: The settlement formation follows a Yule process, in which the initial populations of the settlements are distributed according to the log-normal function. The evolution of the city populations of the formed settlements follows a stochastic process, like the geometric Brownian mo- tion. Hypothesis 1 is a formulation of the Gibrat’s law which leads to power law distributions of the rescaled sizes of a system of cities. Hypothesis 2 gives the necessary conditions for describing the distribution of the (non-rescaled) city population sizes of a system of cities along the double Pareto log-normal distribution. 2. Test of Hypothesis 1 The i-th city rescaled size is deﬁned as S (t) = N (t)/N(t), (2) i i at time t, where N (t) is the population of the i-th city and N(t) is the i population of all cities in year t. The (rescaled city) sizes are ranked such that ≥ r = 1, 2, ... with N (t)/N(t) N (t)/N(t). In so doing, the (rescaled) rank r r+1 (r)-size relationship reads − ln(r) = α β ln(S) (3) where α and β are parameters. After visual inspection and much statistical analysis of the data, i.e., adapting the limits of the relevant intervals by trial and error proce- dures, it is found that the cities are best grouped into four classes: large cities (class 1); medium size cities (class 2); small cities (class 3) and very small cities (class 4). A comment on the various sizes has to be found below, after the data speciﬁc analysis. Fig. 1 outlines the rank-size relationship for the rescaled sizes S of the system of Bulgarian cities, in 2004. The ﬁgures for other analyzed years, 2005-2011, are not shown, for conciseness, but are very similar to the 2004 case. It is found that the rank-size relationship for the classes 1, 2, and 4 can be approximated by a straight line (Zipf’s law), Eq.(3). The parameters α and β for these classes are given in Table 1. Note that for the case of large cities 3

text: The

text: data

text: sets

text: consists

text: of

text: the

text: yearly

text: count

text: of

text: the

text: population

text: of

text: whole

text: Bulgar-

text: ian

text: cities

text: from

text: till

text: as

text: recorded

text: by

text: the

text: National

text: Statistical

text: Institute

text: of

text: the

text: Republic

text: of

text: Bulgaria

text: (http

text: Thus,

text: the

text: hypotheses

text: to

text: be

text: discussed

text: are

label: list

text: Hypothesis

text: The

text: growth

text: rate

text: of

text: the

text: ”rescaled

text: city

text: population”

text: is

text: inde-

text: pendent

text: on

text: the

text: size

text: of

text: the

text: population.

text: Hypothesis

text: The

text: settlement

text: formation

text: follows

text: a

text: Yule

text: process,

text: in

text: which

text: the

text: initial

text: populations

text: of

text: the

text: settlements

text: are

text: distributed

text: according

text: to

text: the

text: log-normal

text: function.

text: The

text: evolution

text: of

text: the

text: city

text: populations

text: of

text: the

text: formed

text: settlements

text: follows

text: a

text: stochastic

text: process,

text: like

text: the

text: geometric

text: Brownian

text: mo-

text: tion.

text: Hypothesis

text: is

text: a

text: formulation

text: of

text: the

text: Gibrat’s

text: law

text: which

text: leads

text: to

text: power

text: law

text: distributions

text: of

text: the

text: rescaled

text: sizes

text: of

text: a

text: system

text: of

text: cities.

text: Hypothesis

text: gives

text: the

text: necessary

text: conditions

text: for

text: describing

text: the

text: distribution

text: of

text: the

text: (non-rescaled)

text: city

text: population

text: sizes

text: of

text: a

text: system

text: of

text: cities

text: along

text: the

text: double

text: Pareto

text: log-normal

text: distribution.

text: Test

text: of

text: Hypothesis

text: The

text: i-th

text: city

text: rescaled

text: size

text: is

text: deﬁned

text: as

text: S

text: (t)

text: N

text: (t)/N(t),

text: i

text: i

text: at

text: time

text: t,

text: where

text: N

text: (t)

text: is

text: the

text: population

text: of

text: the

text: i-th

text: city

text: and

text: N(t)

text: is

text: the

text: i

text: population

text: of

text: all

text: cities

text: in

text: year

text: t.

text: The

text: (rescaled

text: city)

text: sizes

text: are

text: ranked

text: such

text: that

text: r

text: with

text: N

text: (t)/N(t)

text: N

text: (t)/N(t).

text: In

text: so

text: doing,

text: the

text: (rescaled)

text: rank

text: r

text: (r)-size

text: relationship

text: reads

text: ln(r)

text: α

text: β

text: ln(S)

text: where

text: α

text: and

text: β

text: are

text: parameters.

text: After

text: visual

text: inspection

text: and

text: much

text: statistical

text: analysis

text: of

text: the

text: data,

text: i.e.,

text: adapting

text: the

text: limits

text: of

text: the

text: relevant

text: intervals

text: by

text: trial

text: and

text: error

text: proce-

text: dures,

text: it

text: is

text: found

text: that

text: the

text: cities

text: are

text: best

text: grouped

text: into

text: four

text: classes:

text: large

text: cities

text: (class

text: medium

text: size

text: cities

text: (class

text: small

text: cities

text: (class

text: and

text: very

text: small

text: cities

text: (class

text: A

text: comment

text: on

text: the

text: various

text: sizes

text: has

text: to

text: be

text: found

text: below,

text: after

text: the

text: data

text: speciﬁc

text: analysis.

text: Fig.

text: outlines

text: the

text: rank-size

text: relationship

text: for

text: the

text: rescaled

text: sizes

text: S

text: of

text: the

text: system

text: of

text: Bulgarian

text: cities,

text: in

text: The

text: ﬁgures

text: for

text: other

text: analyzed

text: years,

text: are

text: not

text: shown,

text: for

text: conciseness,

text: but

text: are

text: very

text: similar

text: to

text: the

text: case.

text: It

text: is

text: found

text: that

text: the

text: rank-size

text: relationship

text: for

text: the

text: classes

text: and

text: can

text: be

text: approximated

text: by

text: a

text: straight

text: line

text: (Zipf’s

text: law),

text: The

text: parameters

text: α

text: and

text: β

text: for

text: these

text: classes

text: are

text: given

text: in

text: Table

text: Note

text: that

text: for

text: the

text: case

text: of

text: large

text: cities