Exact Solution and Multifractal Analysis of a Multivariable Fragmentation Model

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Exact Solution and Multifractal Analysis of a Multivariable Fragmentation Model

D. Boyer, G. Tarjus, P. Viot

To cite this version:

D. Boyer, G. Tarjus, P. Viot. Exact Solution and Multifractal Analysis of a Multivariable Fragmen- tation Model. Journal de Physique I, EDP Sciences, 1997, 7 (1), pp.13-38. �10.1051/jp1:1997125�.

�jpa-00247321�

Exact Solution and Multifractal Analysis of

_a

Multivariable

Fragmentation Model

D.

Boyer, G. Tarjus

and P. Viot

(*)

Laboratoire de

Physique

Thdorique des

Liquides~

Universitd Pierre et Marie Curie.

4

place Jussieu,

75252 Paris Cedex 05, France

(Received

20

May

1996, received in final form 19

September

1996, accepted 30

September 1996)

PACS.05.20.-y

Statistical mechanics

PACS.02.50.-r

Probability theory.

stochastic _processes, and statistics

Abstract. Most theoretical kinetic

approaches proposed

_so far to describe fragmentation

processes

rely

_on the

assumption

that the

fragments

_can be characterized

by

_a unique relevant

variable,

for

example

their size

or mass h. We investigate ^the consequences of

introducing

additional variables

by considering

_a

fragmentation

model in which the

fragments

_are described by two internal variables, h, _w, ^and _a

breakup

rate

proportional

to

h"w~.

_{The exact solution.}

already partly presented

in reference

ill,

is derived and discussed in details with _an

emphasis

on the

fragment

distribution function. It is shown that~ because of the random

multip)icative

nature of the _process,

no reduction to _an effective one-variable

problem

is

usually possible.

The

fragment

_mass distribution has _no

simple scaling properties;

it rather exhibits

multiscaling,

_a

feature that is

interpreted

in the framework of _a multifractal formalism.

R4sumk. La

plupart

des

approches thdoriques

cin6tiques

propos4es jusqu'ici

_pour ddcrire les processus de

fragmentation

reposent _sur

l'hypothAse

_que les

fragments

peuvent Atre caract6risds

par une seule variable

pertinente,

_par

exemple

leur taille _ou leur _masse h. Nous examinons

les

cons6quences

de l'introduction de variables

supp16mentaires

_en considdrant _un modAle de

fragmentation,

dans

lequel

les

fragments

sont d6crits _par deux variables internes,

h,

_w, et _un taux de brisure

proportionnel

h

h"w~.

_{La solution exacte,}

d6jl partiellement pr6sent6e

dans la rdf6rence

ill,

est d6rivde et discutde _en ddtails _en mettant l'accent _sur la fonction de distribution des

fragments.

On montre h _cause de la nature a16atoire multiplicative du _processus,

qu'aucune

r6duction h _un

problAme

elfectif h _une variable n'est

possible.

La distribution de _masse des

fragments

n'a _pas de

propridtds

de lois d'dchelles

simples;

it

apparait plut6t

des 6chelles

multiples,

une

caractdristique

qui est interprdtde dans le cadre du formalisme des 6chelles

multiples.

1. Introduction

Fragmentation

is _a

usually

irreversible _process which _occurs in _a

variety

of

physical

situations like

polymer degradation [2-5j, vaporization

of

droplets

_[6],

breaking

of

compact macroscopic objects ii, _8j

_or of hot nuclei

[9, loj.

These

phenomena

have

recently

stimulated

increasing

ex-

perimental

work and various theoretical

approaches. Although they

_may involve _very different

physical systems,

some

generic

features

usually

_come out of the

experiments,

such _as

power-law

(*)

^{Author for}

correspondence (e-mail: viotfilptl.jussieu.fr)

Q

Les Editions de Physique 1997

behaviors of the

fragment

mass distribution function in the small-mass

region ill _j.

Some stud- ies have drawn _a

parallel

with critical

phenomena

in

equilibrium

_systems in order to understand

the final distributions observed in _some nuclear collisions

[12j.

For similar

problems

the _use of

percolation theory happened

to be

adequate

_ai well

[9,13-16j;

it may also allow to discuss the

intermittency

in the

fragment

distributions

[15,16j.

Other

approaches

used

probabilistic

and combinatorial

techniques iii, 18j.

Fragmentation

_can also be

investigated

via _a kinetic

description.

We will

adopt

here this

point

^of view and will devote

particular

attention _to the

long

time behavior of the distribution

functions. The

approach

consists in

modeling

the

phenomena

_as

sequential

_processes in which

fragments

break _up in cascades. This may represent an

oversimplication

for

fragmentation

processes that

produce

_a

large

number of

fragments

_very

rapidly

and in _a very

complex

_way,

but it _can be taken _{as a} first

step

^towards more

microscopic description (see

_e-g- Ref.

[19jl.

Moreover, fragmentation occurring

over

relatively large

time scales

(of

the order of _one

minute)

have also been

experimentally

observed in the _case of silica colloidal

aggregates [20j.

For such systems, each

fragment

breaks _up

independently

from the others and the _{process can} indeed be taken _as

sequential.

A kinetic

formalism,

in which the time evolution is

governed by

_a linear rate

equation accounting

for creation and destruction of the

fragments,

_was first introduced

by Filippov _[21j

and later

developed by

Ziff and

McGrady _{[22, 23]}

and

Cheng

and Redner

[26].

Additional mechanisms like inactivation

II

_7] or mass loss [25] were also considered.

A

particle

_or

fragment

issued from _an initial

system

^is

generally

characterized

by

variables like its _mass

(size), geometry (elongation _{), charge,} excitation,

_or kinetic _energy. We will restrict ourselves to

systems

for which the break _up is driven

by

external _causes. The kinetics then

depend

_on the

coupling

between the external conditions and the

physical _properties

of the

fragments.

Collisions and recombinations between

fragments

_are

neglected, although they

_may

be

important

in _{some cases}

[26]. Introducing

a set of internal variables

(h~)

to describe of _a

given fragment,

the

fragment

distribution function

g((hi),t)

at time t is assumed to follow _a

linear rate

equation:

~~~()~'~~

=

-allhil)gllh~l.t)

₊

/ _{lallhll) gllhll,t)} fllhllllhil) IIdhl. Ii)

The

quantity a((h~)) represents

the overall

breakup

rate of the

fragments

characterized

by (h~)

and

f((h[))(h~))

is the conditional

probability

that a

fragment (h~)

is

produced

from _a

fragment (h[).

The first term in

equation (I)

takes into account the loss of the

fragments

due to their scission and the second _term describes their creation from

larger

_ones.

Each internal variable _can

play

a role _or not in the

breakup

of _a

particle. Nearly

all

previous

studies assumed that _a

single

internal

parameter

is sufficient to describe _a

fragment, usually

its _{mass or} size.

Equations

similar to

equation (I),

but reduced to _a one-variable

description,

were used _to model atomic collision cascades [27]

problems

of

degradation

of

polymers _[2, 28],

or

aggregate fragmentation [20].

However~ such _a restriction may ^not be

legitimate

for _many

physical

systems. ^For

instance,

the

fragment

_mass distribution obtained from the crush of

macroscopic objects

made of _{gypsum or}

glass [7,8) depends

on the initial

geometry

^of the

objects (spherical, cylindric~ ellipsoidal ).

In this _case, the

breakup

_{rate can} be function of both _mass and

shape.

Other

examples

include

degradation

of _a

polyelectrolyte

that _can

a

priori

be controlled

by

its

length

and its

charge

and

disintegration

of

heavy

ions and atomic

aggregates

that may

depend

_upon both their _mass and their excitation energy, or their

velocity

and their kinetic _energy.

A

description

ⁱⁿ terms of _a

unique variable,

_e.g. the _mass h of the

fragment usually

contains the

implicit

or

explicit

_[10]

assumption

that all other variables _can be

replaced by

their _mean value. It _means that

a(h,h~)

_ci

_a(h~ (h~)h)

and

f(h',h[)h,h~)

_ci

f(h'~ (h~)h,)h, (h~)h),

where

(al lbl

Fig.

I. Illustration of _a two-variable uniform

fragmentation

of

an initial

rectangle,

after two

breakup

events:

a) binary fragmentation (the

hatched

regions correspond

to the

rectangles

that _are considered

for further

breakups); bj four-piece fragmentation (the

total _area is conserved

during

the

process).

the averages are taken _over all

fi.agments

of _{same mass.} In the

long,time limit,

the number of

fragments

increases and their _mean size goes ^to zero; the

asymptotic

evolution of _{a mass-}

dependent

_process is thus controlled

by

the form of the

breakup

rate a in the small-mass

region.

If

a(h)

_goes to a nonzero constant, the scission is random. The other _cases contained in the

more

general homogeneous

form

a(h)

m~ h°

(2)

have been studied

by

Ziff and

McGrady [22, 23), Cheng

and Redner

[26],

^{and Ernst and Szamel}

[24].

The purpose of this article is to

study

how the main characteristics of linear

fragmentation

processes are modified

by

the introduction of additional variables. A

preliminary

account of

our work _was

published

in reference

ill.

We consider for

simplicity

_a two-variable

description

and _assume that both

variables,

denoted h and _w and

arbitrary

called ~'mass" and

~~energy",

are conserved at each

breakup

event. If

a(h, _w)

becomes

independent

of _w when _{w goes} to

zero, the

fi.agments

_can be

asymptotically

described

by

_a

unique

variable h. More

generally,

we consider the form

a(h, w)

m~

h°w~, ₍₃₎

which represents a natural extension of

equation (2).

If

fl # 0,

two

fragments

^of _same mass

may have

quite

different

probabilities

of

breaking

_up. When _a and

fl

have

opposite signs,

one

expects competing

effects which _are not contained in the one-variable _case.

We consider the

binary

scission of _a

fragment (h', w')

into

(h' h,

w' _-w

)

and

(h,

_w

),

where h and _{w are}

randomly

and

uniformly

chosen in the intervals

[0, h']

and

[0, w'), respectively.

This

process is thus conservative for the _mass h and the energy w and is ruled

by

the

following

kinetic

equation:

~~~jl'~~

~

~h~'~~glh''~'t)

_{+ 2}

/~ /~ h'~~~~°~~~~glh~'~°" t) ^dh~d~°~' ₁₄₎

W

The choice of _a uniform conditional

probability f simplifies

the

description

without

imposing

too much restriction. It is worth

noting

that

equation (4)

^has a

simple geometrical interpreta-

tion. It describes the uniform

binary fragmentation

of _an initial

rectangle

in smaller _ones

(see Fig. la).

The

breakup

rate of _a

given rectangle (of length

h and width

w)

hence

depends

_on

its _area hw and its aspect ratio,

h/w,

_as

(hw)@(h/w)~

_The

process is

anisotropic

since

h and _w do not

generally play equivalent

roles. If the factor _{2 in}

equation (4)

is

replaced by

_a factor

4,

the _process is not

qualitatively

modified: for each

breakup

event, four

rectangles

_are

created and the total _area is conserved

(Fig. lb). Krapivsky

et al. [29) studied the

particular

case a =

fl

= 1

corresponding

to a rate

just proportional

to the

rectangle

_area.

We first

briefly

recall the main features of the one-variable model with _a

breakup

rate

given by equation (2).

We then consider the two-variable model introduced above in

equation (4)

and

~v.e

give

a detailed derivation of the exact solution for the

fragment

distribution function and its moments, some of the results

having

been

briefly presented

in reference

ill.

More technical

aspects are collected in _an

appendix.

The model does not

generally

reduce to an effective one-variable

problem.

In

particular,

_we show that _no

scaling

solution _can be

found, thereby indicating

the presence of _an infinite number of _mass scales in the

system. Also,

_as discussed in reference

iii,

_{a new} kind of

shattering

'~transition" in which _mass is lost with _{a power} law

decay

in time to _a dust

phase

is

obtained,

when the parameters a and

fl

have

opposite signs.

To

provide

more

insight

into the

fragmentation

_{process, we}

study

the behavior of the

fragment

mass distribution function in the small _mass

region

and its variation with _a and

fl. Finally,

_we

investigate

the

multiscaling property

of the

system by

_means of _a multifractal

analysis

which

amounts to

study

how the

fragment

distribution

depends

_on initial conditions.

2. One-Variable

Fragmentation.

A

Summary

We

briefly

summarize the main features of the standard one-variable _process

proposed by

Ziff and

McGrady _[22].

In this _case,

equation (I)

with

a(h)

₌ h~ becomes

~~~'~~

⁼

-h"g(h, t)

+

/~ h'° f(h)h')g(h'~ t) dh'. (5)

t h

An exact solution _was derived for _a

binary

uniform

breakup I.e., f(h)h')

=

2/h'. Cheng

^and

Redner [26] further considered _cases where the

breakup

conditional

probability f

has the _more

general

form

f(h)h')

=

b(h/h')/h'.

In the

large-time limit,

the _mass distribution function shows

interesting scaling

law behavior. For _a >

0,

the function

g(h~ t)

_can indeed be written in the form

glh,t)

~f

31t)~~~lh/31t)),

₁₆₎

where

s(t)

is _a

typical particle

_mass, and the exponent -2 is _a consequence of the _mass conservation. The

scaling

ansatz,

equation (6),

reduces the

description

of the _process with variables h and t to _a

single

variable

h/s(t).

It also

requires

that

s(t)

be

proportional

_to

the inverse number of

fragments, ff

dh

g(h,t). Therefore, s(t)

is the

only

characteristic size of the

system

^{and it behaves}

asymptotically

like

t~~/°

_For

large

values of _x

=

h/s(t),

~(x)

m~

x~(~~~~e~~"/° _[26].

For _a <

o,

the

scaling description

breaks down and _a

striking

behavior _appears: the total

mass

Mj

₌

ff

dh

hg(h, t)

decreases

exponentially

with time. It is the

signature

of _a "shat-

tering"

transition that takes

place

in

parameter

space at _a

= 0

[22j.

A finite fraction of the initial _mass is lost in _an infinite number of zero-size

particles,

called the dust

phase.

The _mass

apparently

^decreases because

equation (5) only

describes the evolution of

fragments

of _nonzero mass, that do not

belong

to the dust

phase.

The total _mass

Ml

+

Md,ist

is

actually

conserved.

In the

long-time

and small-mass

regime,

the distribution function behaves _as

[26j

g(h,t)

m~

e~~ h'°'~~ (7)

3. Two-Variable Model

The model is defined

by equation (4),

to which _we add

monodisperse

initial

conditions, g(h,w,t

=

o)

₌

d(h ho )d(w wo). (Because

of the

linearity

of the rate

equation,

_any

general

initial condition is reducible to this

monodisperse condition).

We further

simplify

the

problem by rescaling

the

variables, h/ho

-

h, w/wo

_{- w,}

th$w(

- t, so that _{one can} set

ho

_{= wo =} I in what follows. It is convenient to introduce the moments of the

fragment distribution,

co co

fi£(~,

_~t,

t)

= dh dw

h'iu"g(h,

_w,

_{t). (8)}

M(I,

_~t,

t)

also

represents

^{the double Mellin transform of the function}

g(h,

_w,

t).

In

physical terms, M(1, 0, t)

is the total _mass of the

fragments, M(0,1, t)

the total energy, and

M(0, 0, t)

the number of

fragments

created at time t. With the

monodisperse

initial condition~ the

moments must

satisfy:

M(~,

_{p, t}

=

0)

= 1.

(9)

Combining equation (8)

with

equation (4),

_one obtains the

following equation

for the moments:

~~li

^{" ~~}

=

Ii>

₊

^~)~

+

i~ i) ^fi~i>

⁺ a ~1 +

fl t). (lo)

The time derivative of _a

given

moment is thus

proportional

to the moment obtained

by

trans- lation

by

_a vector with

components (a,fl)

in the

(I,~t)-plane.

We stress that

equation (10)

is

only

valid if the moments involved _are defined _or finite. Different _cases will have to be

considered, depending

_on the

signs

of the parameters a and

fl.

We will

particularly

discuss the

long-time limit,

since it is

physically

the most

interesting regime.

The

knowledge

of all the moments leads the to the distribution function itself _ma two suc-

cessive inverse Mellin transforms:

91h

_W

t)

-

~il~~ [

_d>

I _{dP Ml>}

i l~ i

t) h'

W».

iii)

Since it is not

always

_easy to calculate _nor

study

such

integrals,

we also introduce various _mass distribution functions,

g~(h, t),

_as

g~jh,i)

=

/"

dw

w»gjh, w,1). j12)

The function _go

(h, t) simply represents

^the

density

of

fragments having

_{a mass} between h and h +

dh,

and it is the

only

distribution that is encountered in the one-variable model. In the

case where

fl

₌

0,

it is easy ^to check that

equation (4)

^reduces to the one-variable

equation

for go

(h, t).

The behavior of the functions

g~(h, t)

in the small-mass

region

_can be inferred from _a

study

of the domain of definition of the moments. At fixed

parameters

a and

fl,

for _any value

~t, there is _a value

lc(~1)

such that

l~~li~i~~) ~~j~~~r~~~~~~i.

~~~~

As

M(I,

_~t,

t)

=

f~" h'g~(h, t) dh,

the

leading

behavior of gp for small ^{h is} then

given by 9p(h,t)

m~

h~l'~(~~+~~ (14)

This

technique

has also been used in the one-variable model

[26].

^More details will be

given

below.

4. Solution for the Case _{a >}

0, fl

> 0

4,I. SOLUTION FOR THE MOMENTS. We first consider this _case because it does not _ex-

hibit any

shattering phenomenon. Equation (10)

_can be solved

by following

the method that

Charlesby

used in the framework of _a one-variable model for

polymer degradation _[28].

It consists in

expanding

the _moments in power of t:

co

~kmjj,p,o)t~

(15) M(I,

_~t,

t)

" 1+

~

fitk k!

When _a and

fl

_are

positive, h'+k°w~+kP

_<

h'w",

so that if

M(1,

~1,

t)

is

finite,

then

lzI(I

₊

a, ~1+

fl, t)

is also finite. Each

partial

derivative at t ₌ 0 _can be

computed recursively

from

equation (10)

and the initial condition

equation (9).

It

yields

2

)

akmjjj ^/1°)

=

j~

₊

ii~ (16)

+

i~ II Iii

+

i~

_{i~~y +}

i)i~t

₊

ik i)o

₊

i)

With

_thesual

_(a)k

=

_a(a

+

I)..

la _k

Mi>, J1, t) = ~ iii ₊

1))())))~) ^)/o)~

here the

oefficients -4

and B

are

iven by

A

=

~~~

+

~)

+

fill

and

This

result, already given

in reference

iii,

is _a

generalization

of the solution obtained

by

Ziff and

McGrady

for the one-variable

problem.

This latter involves _a

generalized hypergeometric

function of lower order:

M(I, t)

=

ifi III I) la. (I

+

I) la, -t). Taking

the limit

fl

_- 0 in

equation (20) gives

this result back.

It is easy ^to see from

equation (lo)

that all defined _moments

M(1,

_~1,

t)

which _are such that

(I +1)(~1+1)

= 2 _are

independent

of time.

Contrary

to the one-variable model there is _an infinite number of conservation laws for _{a >} 0 and

fl

> 0: not

only

the total _mass and the total energy are

conserved,

but also _an

infinity

of other nontrivial moments of the distribution.

The

long-time

behavior of the _moments is obtained from the

study

of the

2F2

functions

[30j.

Their

asymptotic

^behavior in time is

algebraic,

which leads to

iii

4

3

-

t-i~i

~

~

+l)(~t+1)=2

tlBi

0

0 2 3 4

x+1

Fig.

^2.

Hyperbola

of conserved moments in the

(~

+ l~_{p +}

I)-space

for _{a >} 0 and

fl

_> 0. The space is

separated

into two

regions:

the moments increase

(respectively decrease)

with time in the lower left

(respectively

_upper

rightl

part.

where r denotes the gamma

function;

A and B _are defined

by equations (18,19).

In the

(1,

_~1)-

plane,

the moments _are defined when the

arguments

in the _gamma functions _are

positive, I.e.,

in the

region (I

+1 >

0, ~1+1

>

0). Figure

2

displays

the

hyperbola

of the conserved

moments that separates the domain of definition of the moments in two

parts:

in the lower left

one the _moments

(like

the number of

fragments)

increase with time and

they

^decrease in the

upper

right

_one. A remarkable characteristic of this model is the irrational

dependence

of the

exponent

B _{on a} and

fl.

This results from the correlations between the two variables h and

w. As _an

illustration,

the total number of

fragments

increases like flJ2+8mp-a-p

N(t)

=

M(0, 0, t)

m~ t 2nP

,

(22)

whereas in the one-variable model it goes as

N(t)

m~

t~'° (23)

4.2. ABSENCE _OF SCALING SOLUTION _{IN THE} TWO-VARIABLE MODEL. We _now show

that the function

g(h,

_w,

t)

does not

obey

a

scaling form, contrary

to the

Ziif-McGrady

model.

Consider _a two-variable

scaling

ansatz,

91h,

_W,

t)

_~~f

kit) ~ (j. fij

,

j~~~

where

s(t)

and

a(t)

would

represent

^the

typical

_mass and the

typical

_energy of the

fragments

at time

t, respectively.

Whereas the _mass is the

only

conserved variable when _{a >} 0 and

fl

₌

0,

_an ^infinite number of _moments

M(1, _~1,t)

_are conserved for _a > 0 and

fl

> 0. It is

straightforward

to derive that the

scaling form, equation (24),

combined with the conservation

of all moments such that

(I +1)(~1+1)

₌ 2 would

imply

that the functions

k(t), s(t),

and

a(t)

satisfy

_an infinite number of conditions:

kji)sji)>+iaji)i

=

c,

,

j25)

for any value of I > o. As

Ci

is

independent

of

time,

the above condition is

only

fulfilled when

k(t), s(t)

and

a(t)

_{are constant:} from

equation (lo),

_{one can} check that this

corresponds

to the

unacceptable

solution

g(h,

_iu,

t)

= 0.

Consequently,

the

system

cannot be described

by

_a

single pair

^of mass or energy scales. It rather

requires

_an

infinity

of scales. We will further

investigate

the absence of

scaling

in Section 7

by studying

the multifractal

properties

of the

fragment

distribution.

As done

by Krapivsky

et al.

[29]

^{in their model of scission of} a

rectangle,

_{one can} also calculate the correlations between the variables h and _iu. We thus consider the _averages

(h~iu~)

which

are taken _over the

fragment

distribution and which _can be derived from the

knowledge

of the

moments

according

to

~

M(n,

_m,

t)

i~~~

"

Mj0, 0, t) i~~)

Comparing

the _average

(h~w~)

with the

product

of averages

(h~)(w~)

illustrates how the

fragment

distribution deviates from the

scaling

form

given by equation (24). By using equations jig, 21, 26),

_one obtains for instance for the _{case m}

= n and _{a =}

fl:

)~)lW~)

'~

~~

" ~~~~

The above ratio goes

asymptotically

to _zero for

n >

o,

whereas it would reach _a fixed _nonzero

vilue

_{in the}

case of _a distribution

obeying

a

scaling

form. The _same feature holds for the _mass

distribution,

_go

(h, ii

a

f/

_du,

_g(h.

w,

t),

which has _no

scaling

form either. The ratio

(h~) / (hi

^"

increases

unboundedly

with time for _n >

i, lh~)

~_

~>~/~ap _j~~~

jhjn

^'

with

An =

in Ii ~/jo _oj2

₊

₈₀₀

₊

~/ijn

_{+ I}

in oj2

₊

8ap nj20

₊

o). j29)

which is the

signature

of _a broad distribution of _masses. The one-variable

fragmentation

model does not

produce large fluctuations,

since the ratio

(h") / (hi"

reaches _a constant _nonzero value at

large

^{times. A}

unique

_mass scale is _not sufficient to describe the distribution. As time passes, more and _{more mass} scales _appear in the system: it is the

signal

of the

'~multiscaling"

property.

4.3. FRAGMENT DISTRIBUTION FUNCTION. To

explain

these

large fluctuations, deeper insight

into the

fragment

distribution is

required.

We _now derive the exact

expression

^of the distribution function

g(h,w,t) by taking

the inverse Mellin transform of the solution that

we have

already

obtained for the moments. The factor associated with each power of _t in

equation (17)

contains _a finite number of

poles.

With the

help

of the residue

theorem,

_{one can}

express

g(h,

_w,

ii

_{as a}

uniformly convergent

series in powers of

h°, ^wP

and t:

g(h,

_w,

t)

=

6(w 1)6(h I) ^e~~

+ 2t

~~

~

l( _f~

~afl(i~-j)2

^{~~~~~~ ~~'°~~~}

~

~~~~

~

=~~

J

+f If ⁽

^~~

^jj

^~

-i)h°(~~l)~p(j-I) t~_

k=2 j=I i=I, i#j

~~~

_~~~

t=I,1#~,l#j

~~~~

_~~~~ ~~ ~~

The function

g(h,

_w,

t)

reaches _a finite value when h _{or w goes} to _zero. We thus check that the moments _are finite for I+1 > 0 and

~1+1

> 0. We recall that

equations (17,

20 for the moments are

only

valid for

positive

values of the

parameters

_a and

fl. Nevertheless, equation (30)

is

always

the proper solution of the _rate

equation, equation (4),

whose structure is

independent

of the

signs

of _a and

fl.

It is valid whatever the

signs

of _a and

fl, but, unfortunately,

it is

only

tractable

numerically

for

positive

a and

fl.

In the _case where at least _one of the

parameters

is

negative,

the series converge

slowly,

and this situation

requires

_a

special study

which is done in the

following

sections.

For _{a >} 0 and

fl

>

0,

_we

paid particular

attention to the energy distribution _among

fragments

of fixed _mass h.

Figure

3a

displays

the energy

probability

distribution function for three different values of the

fragment

_mass at _a

given time,

while

Figure

3b shows the _same function at two different _times for _a

given

_mass. The rather flat

shape

of these _curves indicates

large

fluctuations in energy among

fragments

of _{same mass.} Two

fragments

of _{same mass may} have

quite

different

energies

with

comparable probabilities

and _may thus evolve _very

differently.

Without _a

typical

_energy, the _process cannot be reduced to a one-variable

model,

since it would be incorrect to

replace

the energy ^w of each

fragment

^of mass h

by

the _mean value

(w)h, assuming

that fluctuations _around this _mean value _are

negligible.

The

multiscaling

property

revealed

by equations (28,29)

_can thus be viewed _as

resulting

from the fluctuations of the second variable _iu;

The _mass distribution function itself _can be obtained

directly by

_a

simple

Mellin transform

of the moment

M(I _{1, 0,} ii.

We

get

go(h, t)

=

6(h I) ^e~~

₊ 2t

(31)

~

f (

2

jj

2

ij ^{ha(J-i) t~}

~_~ ~~~

l ₊

(j I)fl

~~ _~~~

°(i j)(I

+

Ii

l

)fl)

kl

The

asymptotic

behavior of the above

expression

_can ^{also be derived}

by

_an inverse transform of the

asymptotic expression

of the moments:

~i~'~~

_~

( _{(~ FjAn)I)} ~/~~)Fj(I-

En) ~~~~~~~' _i~~~

with

("

= n +

Ill _/fl

₊

_n)2

₊

/(all)j

(33)

n

2

d

z 5

(al 16)

4

5 t=io

h=0

jj

3

h=0

(

~j

~

_$

Z

t ₌

05

0 0

0 02 04 06 o-B 0 0.2040608

w w

Fig.

3.

Fragment

distribution function

g(h,

_w,

t)

_{versus energy w} for _o > 0 and _> 0:

(a)

at _a

given

time t

= 1for three different _masses h

(h

= 0.1, 0.5,

0.9),

and

(b)

at _a

given

mass h

= 0.5 for

two different times t

it

₌

1,10).

As the

exponent -Bn

increases

indefinitely

with _n, the _mass distribution function _go has _no

simple asymptotic

form when t _{- oo.} Note that

equation (31)

is still valid for _{a <}

0, provided fl

> 0. When _a

= 0 and _>

0,

the scission is random _as far _as the _mass is concerned and go

(h, t)

does not

depends

_on h. When _{a >} 0, the distribution is characterized

by

_a finite value

at h ₌

0,

_as it is shown in

Figure

4.

5. Solution for the Cases

ad

< 0

When at least _one of the parameters ^is

negative, according

to the

equation (8),

the

inequality fit(~

_{+ a, p + @,}

t)

_<

M(~,

_p,

t)

is _not

always

valid since

h~+~ diverges

_more

rapidly

than

h~

~vhen h goes ^to zero.

Charlesby's

method _{can no}

longer

be used. We rewrite the monlent

equation (10)

in the form

~~~~~

^~~

~ ₊

)

p +

1j

~~~

_{~ °'~ ~}

~'~~'

_~~~~

°

d

where the coefficients A and B

depend

_on ~. /1, a, as

given by equations (18,19). Equa-

tion

(34)

is

only~valid

for well-defined moments and may ^not be

applicable everywhere

in the

region ((~ +1)

>

0, (p +1)

>

0).

The

divergence

of _some moments is linked to _a

shattering

phenonlenon.

^If one assunles that there exists values of _a and such that the "total" _nlass

5

4

t ₌ i o

_

/

3

o

~° 2

t=I

*

0

0 02 04 06 08

h

Fig.

4. Mass distribution function

go(h, t)

_versus h for _two given times t

it

₌

1,10)

for _o > 0 and

> 0.

M(1, 0, t)

of finite size

particles decreases,

it _means that

fit(1+

_a,

fl, t)

is infinite. More _gener-

ally,

if _some of the nloments

fit(~,

_p,

t)

such that

(~ +1)(p +1)

₌ 2 decrease. the

corresponding

moments

M(~

+ a, y +

fl, t)

nlust be infinite.

We

emphasize that,

when

shattering

_occurs,

equation (4)

cannot describe the time evolution of the dust

phase composed by particles

of _{zero mass or zero} energy. As the nloments defined

by equation (8)

are derived from the

equation (4), they

do not include contributions from the

singular

part of the distribution function. I,e. front the dust

phase.

5.I. SOLUTION _{FOR THE} MOMENTS. To

get

an idea of the

long

^{t1nle behavior of the}

nloments, ^{it is} useful to consider their

Laplace transform, 3i(~,

_p,

_s)

=

ff

_dt

e~~~M(~, y,t),

for the value _s

= 0. The

expression

is

readily

obtained and reads:

l~

⁺

^j

^{/1+ 1}

^j

~~~~'~'~~

^~

(A 1)(B 1)

~~~~

When

iI(~, _y,0)

_is

infinite, &1(~,

_/1,

t)

is infinite _or else decreases _nlore

slowly

than

1It.

The solution in

equation (20)

does

satisfy equations (10)

_or

(34)

and the proper initial condition

&1(~,

_/1,

0)

= 1, but it has _an

unacceptable asynlptotic

behavior _{as soon as a or}

fl

is

negative:

it behaves _as

t~~

_when

a < 0 and

fl

> 0 and _as

t~~

when _{a >} 0 and

fl

<

0,

which is not

conlpatible

with

equation (35). Actually, provided

that the moments still possess a

power-law asynlptotic

behavior, _one must have

Ml~,11,tl~t~~ if°<°andfl>°

+~t~~ ifa>0andfl<0, j36~

with A and B defined in

equations (18,19).

The method of resolution relies then _on

developing

the _{moments on a set} of functions than include the 2F2

given

in

equation (20)

and have similar

asynlptotic

behaviors

(t~~, t~~,...).

Such _a set of functions is

given by

all the solutions of the

hypergeometric

differential

equation

^satisfied

by 2F2(-4, B, (~ +1) la, (p +1)/fl; -t).

An

equivalent

formulation is obtained

by expressing

the monlents with

Meijer G-functions,

which

are extensions of the

generalized hypergeonletric

functions

[30j.

^{Details of the resolution} are

given

ⁱⁿ the

appendix.

^The solution for _{a <} 0 and

fl

> 0 _can be written _as follows:

~j~

_~

t)

=

~~~ ~~~ l~1

Gi t Ill

+ i

137)

riB)r ~

~

~)

°'~

°

'~ ~

'

or,

equivalently,

~~~'~'~~ ~'~~'~'"'~'

_~~

~~~~~~~~~~)/~~~iP~ ^~

^~~~~

t~~" 2F2(1

+ A u,1 + B _u, 2

u,1

+ v ~L;

-t),

with u =

(~ +1) la

and _v

=

(/1+1) Id.

In the l1nlit _-

0,

_{one recovers} the fornlula

given by

Ernst et al. [24] for the nloments of the one-variable

problenl.

When _{o >} 0 and <

0,

^the solution is deduced from

equation (37) by permuting

A with B and

(~ +1) la

with

(p +1) Id:

P(1-B)P ~~~) 1-A,1-B

Mj~,

_~t,

t)

=

~

) Gll

t

~,

^{P +}

I,1_

^{~ + l}

¹³⁹⁾

~~~~~ ~

d ^~

^~

or

Mj~.v,t)

=

~F~jA,B.u,v;-t)- jij)jiU)[ijjB)rii,j(-v) j40j

ti-v ~f~ji+A-v,i~j ~,2-~v,~~jj~ ;fit).~

with

again

_u

=

(~ +1)la

and _v

=

(/1+1)/fl.

The above

equations (37)

and

(39)

have been

previously presented

in reference

[1].

5.2. SHATTERING TRANSITION. With the exact solutions in

hand,

_{one can}

study

the main features of the

shattering

transition.

Using

the

properties

^of the

Meijer

G-functions

[30],

^the

asynlptotic

behavior of the solution for _{o <} 0 and

fl

> 0 is shown to have the fornl [1]

~~~'~'~~

)~ ~j _{j[1~~ ~)/~} )

~'

~~~~

which

corresponds

to the

expected

result

(Eq. (36)).

When _a and have

opposite signs, only part

^{of the}

hyperbola (~ +1)(/1+1)

= 2 in the

(~, /1)-plane corresponds

to conserved monlents.

For

_{a <} 0 and

fl

>

0, equation (19)

shows that the

exponent

^B in the above

asymptotic

fornlula is

equal

_{to zero on} the

hyperbola only

if

fi

~ + l 2

f ⁽⁴²⁾

_

+

~

(~+l)(/L+1)=2

I I

I I

' '

~°~ ~~ '

', (2, 1)

~1~+

°

~ x~+ⁱ -jai x~+1

~ + i

Fig.

5. Domain of definition of the moments in the

(~

+1,

~1+1)-space

for _a < 0 and > 0. The

conserved moments

belong

to that part of the

hyperbola

for which ~ > ~o

(full curve).

The _moments decrease in the upper

right region, including

the left part of the

hyperbola (dashed curve); they

_are

not defined in the hatched

region

and increase with time in the remaining.

The moments associated with the upper part of the _curve decrease; _see

Figure

5. An apparent

loss of _mass is observed if the

point (~

+1 ₌

2, /1+1

₌

1) belongs

to this _upper

part;

^{it is the}

case when

p

_<

'j' j43j

Similarly,

when _a > 0 and

fl

_<

0,

the moments of the

hyperbola (~ +1)(/1+1)

= 2 decrease if ~ +1 >

20/(fl(.

The total _nlass is then

decreasing

with t1nle when

(p(

>

a/2. So,

to

sunlmarize,

the total _nlass decreases

algebraically

in the

region fl

<

-o/2, according

to

Mji, o, t)

r~

t-11++) _j44)

A _new

type

^{of behavior is} thus observed:

contrary

to what _occurs in the one-variable model

(and

to the

present

model when _a < 0 and

fl

<

0),

the nloments that _are associated with the

shattering

transition

(like M(1,

0,

t)

when

fl

<

a/2)

do not

decay exponentially

with time but rather

algebraically.

This results from the

competing

effects that are

generated by

the

opposite

fi~

M(0,1)~tiz/P-ila0

Mji 0)~ t~ii/p-z/rail

M(I,o)~ e~t MID-1)~ e~t

~

M(1 0)~ L~izla-</iPiJ

M(o,1)~'

~

/

i~

aj bj

Fig.

6.

a)

Phase

diagram

of the

shattering

^{transition in} the

(o, fl)

parameter _space. The total _mass

M(1, 0)

is conserved in the hatched

region,

decreases

algebraically

in the lower

right

sector

(bounded by

the two

straight

lines

id

₌ 0, fl _{= -a}

/2)

and in the upper left sector

(bounded

by the two

straight

lines

(o

= 0,

fl

=

-a/2)),

and decreases

exponentially

for _a < 0, < 0.

b)

Similar to

(a),

_except that the

shattering

transition is _now defined

by

_a decay of the total energy

M(0,11.

signs

of _a and in the overall

breakup

rate and lead to a slow

(or

"critical" evolution. If _a < 0 and

fl

>

0,

the

disintegration

^cascade

primarily

affects the variable h. A

singular

distribution of

fragments

_appears, but the

fragmentation involving

_w makes the whole _process much less

effective than when _a and

fl

are both

negative.

There is thus _some

anlbiguity

in

defining

the

shattering

transition. One could choose _as its definite

signature

either the appearance

of

a

sing~iar

distribution of

fragments

_or the

decay of

the moment

M(1,0,t)

associated with the total _nlass

(or

else the

decay

of another

normally

conserved moment, like that associated with the energy,

&1(0,1,t) ).

The latter choice is _more restrictive and is not

equivalent

to the

former since _a

singular

distribution of

fragments

in h

= 0 appears for all

pairs (a

_<

0, fl

>

0).

Nevertheless,

this

singularity

_may not be strong

enough

to affect the conservation of the total

nlass.

Retaining

now the

decay

of the _nlass to define

shattering,

_{we can} draw the

phase diagram

of the

shattering'transition

in the

(o, fl)-plane.

It is shown in

Figure

6a. Note

that, although

the model is

symnletric

in h and _w, the

phase diagram

^is not

symmetric

because of the choice of

fit(1,0, t)

to characterize

shattering.

If _one chooses instead the total _energy

fiI(o,1, t),

_one

obtains _a different

phase diagram displayed

in

Figure

6b.

It is

quite

remarkable that the total _mass is conserved when _a <

0, provided fl

>

-a/2.

Such _a feature is

totally

absent from one-variable models.

Synlmetrically,

a

decay

of the _mass is observed for _a >

0, provided fl

<

-a/2.

The

shattering

transition is then

triggered by

the

disintegration

cascade

primarily involving

the _{energy w} via the correlation between h and _w.

The

part

^{of the}

diagranl corresponding

_to the _{case a <} 0 and

fl

< 0 will be discuss later _on.

5.3. MAss DisTRiBuTioN FuNcTioN. When _o < 0 and

fl

>

0,

the moments _are defined

(I.e.,

finite and

positive)

in the domain of the

(~, /1)-plane

where the arguments of the gamnla functions in

equation (41)

_are all

positive

^and the exponent B is real.

M(~,

_/1,

t)

exists in the

following

conditions:

l~

+ i +

(a(

)lJ~ + i

fl)

> 2 if ~ _{+ i} <

(a(

^-1 +

~

~ ~ ~~

~

~ ~

))j

^{~ + ~ >}

2/j

if ~ _{+ i} >

(a(

+

~~

_~~~~

a >

o, fl

< o

~~ ~

~~~~

~ ~ ~~~~ ~ ~ ~~ ~ ~ ~ _~

~

~

~~

j46)

~

~

(~(

~

~~

~~ ~ ~ ~ ~ ~

~

~

~~

This is illustrated in

Figure

5, where the hatched _zone denotes the

region

where the moments are not defined. This

region

is delimited

by

_a

portion

of

hyperbola

and _a

portion

of

straight

line. With the

property

discussed in and around

equations (13,14),

_we deduce the

leading

behavior of the _mass distribution function in the small-mass

region.

For

all

<

0, gain, t)

+~

h'°"'@~

if

fl

> i oi a >

-11

[~~ 147)

and

~

2fl

~~~~~_~~ ~

~ ^~ _{~ ~} ~~ ~ ~ ~~~ _{~ ~}

Ii _fl)~

^~~~~

The value of the exponent of the

algebraic

^law _can be either

positive

_or

negative.

For

strong disintegrations (o sufficiently negative,

for

example)

the initial

fragment disappears

almost

instantaneously

to

produce

dust

particles:

the _mass distribution vanishes _as h

- 0 because small

(but finite) fragments

_occurs with very low

probability. Conversely,

in

disintegrations

^of

intermediate

strength,

^the initial

fragment

^has the

opportunity

to break _up in _numerous small

(but finite)

_ones, while dust is

produced

at the _same time. It is

important

to notice that the

mass

exponent

^is

always larger

than

-2, insuring

that the

corresponding

total _mass is

always finite,

which is not the _case of the number of

fragnlents

for instance.

Figure

7 _{sums up} the different behaviors encountered in the

(a, fl)-plane.

On the

fl

₌ 0 axis, _{we recover} the results of the one-variable model

[22, 26]

^{for which} a transition between two kinds of _nlass distribution

occurs at ac = -2. The introduction of _a second variable nlodifies the scission of the snlall

fragments

and

logically

shifts the value _ac to smaller values when

fl

> 0 and to

larger

valiies when

fl

< 0.

It is also worth

noting

that contrary ^{to the} one-variable

model,

the

present

_{one nlay} lead to a

power-law divergence

of the _mass distribution in the small-nlass

reg1nle,

_even when

shattering

does not occur, as is indeed observed

experimentally.

This

requires

^the presence of _a second

variable

which,

when _{a >}

0,

boots the

production

^{of the} small but finite

fragments (case fl

_<

-a/8).

6. Solution for the Case _{a <} 0 and

fl

< 0

6.I. SHATTERING TRANSITION. We first show

by

heuristic

arguments

that the

system

necessarily produces

_a dust

phase

for all

pairs (o

_<

0, fl

<

0),

and that the

(apparent)

_mass

loss is

exponential

in time. Consider the moment

equation, equation (10),

in which _{one sets}

~ ₌ 1 +

(a(

and _/1

=

(S(:

0Mil

₊

(a(, ifli,t)

~

~

i) ^{Mji,o,t). 149)}

0t 12 +

jai)11

₊

_ifli

-~ I

~

fi

o

no ~

~

~d h h

~

~~

d

noi

j~

~

~

~

h

i

no h