Monte Carlo simulations of aggregation phenomena

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Submitted on 1 Jan 1993

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N. Menci, S. Colafrancesco, L. Biferale

To cite this version:

N. Menci, S. Colafrancesco, L. Biferale. Monte Carlo simulations of aggregation phenomena. Journal

de Physique I, EDP Sciences, 1993, 3 (5), pp.1105-1118. �10.1051/jp1:1993259�. �jpa-00246783�

Classification Physics Abstracts

64.60 82.20 02.50

Monte Carlo simulations of aggregation phenomena

N. Menci

(~),

^S. Colafrancesco

(~)

^and L. Biferale (~)

(~ Dipartimento di Fisica II Universitl di Roma, via delta Ricerca Scientifica I, 1-00133 Roma, Italy

(~) Osservatorio Astronomico di Roma, via dell'osservatorio,1-00040 Monteporzio, Italy

(Received

16 July 1992, revised 23 December1992, accepted 22

January1993)

Abstract. We discuss the evolution of many-body _systems under

merging

interactions between their components; the realizations that _are considered here _can be related _to various physical systems from the evolution ofsystems of galaxies to the growth ofstructures in colloids.

We _run Monte Carlo simulations of aggregation _processes of clusters with _mass M, for various merging probabilities K

+J

M',

_{and study the evolution of the resulting}

mass distribution

(MD).

We find out evidence for the known phase transition around the critical value I ₌ I. We extend the analysis of the aggregation to the _case of spatial fluctuations of the cluster distribution by

means of _a multifractal analysis. The distribution of clusters _{turns out to} be fractal in the

mass space and multifractal in the real _space. The phase transition produces _an enhancement of the spatial dishomogeneity for large _masses, that is quantified in terms of different values of the generalized fractal dimension

Dr(q).

A correspondence between the effects of the phase

transition in the _mass space and in the coordinates is also found.

1 Intro duction.

The kinetics of

aggregation

_processes have been studied in _a

variety

of

fields,

from statistical

physics

to

chemistry

_{[1, 2]} to

planetary

formation [3] ^{and also in}

astrophysics

and

cosmology

[4, 5]:

specifically,

there has been _a renewed interest for the

study

of

merging phenomena

occurring

between

galaxies

in groups or in the

field,

_or between substructures in

forming

clusters of

galaxies

[6,

ii.

The classical

approach

to

merging phenomena

is based _on the Smoluchowski [8]

aggregation equation

N(M,t)

₌

~j K(M', M",t) N(M',t) N(M",t)

M>+M,'=M

~x>

-N(M,t) £ K(M,M',t)N(M',t) (I.I)

M,=1

Send o1Tprint requests to: S. Colafrancesco.

describing

the evolution with time t of the _mass distribution

N(M,t)

of clusters with _mass M under

binary

collisions. The

quantity K(M, M', t)

is the

aggregation

rate and is

given by K(M', M", t)

₌ fl

(L(M', M",t) V),

where fl ^{is the} average

density

of

aggregating clusters,

L is their _cross section for

aggregation averaged

_over their relative

velocity

V.

Analytical

studies of

aggregation phenomena

described

by equation (I.I)

have been

performed

for

separable

homogeneous

interaction kernels

verifying

K(aM, aM',t)

=

a'K(M, M')F(t) (1.2)

where

F(t)

-J

if.

The behaviour of the solutions

depends critically

_on ^the

homogeneity degree

I [1, 9] and _on the exponent

f

[6,

ii.

Specifically,

for I _< I _a self-similar

scaling

form

N(M,t)

-~

M.(t)~~ #(M/M.) (13)

is

obtained,

^where the _mean cluster size

M.(t)

remains finite at every time t. In terms of the adimensional variable

m %

M/M.,

one can

verify [ii

that

#(m)

-J

m~f

(with (

>

0)

^for m <

I,

while for _m »

I, #(m)

-J

m~'ezp(-m)

holds. This leads to the conservation of the total _mass

fi4,

_as

fir

=

£ _MN(M,t)

= 0.

(1,4)

A different situation arises when I _> I and

f

> -I; ^{in this} _case, a substitution of the

simple scaling (1.3)

in

equation (I.I)

leads to _a

divergence

of

M.(t)

after _a critical finite time tc -J

1/fi4 (see [10], [iii).

This

implies

_a

scaling

form

#(m)

-J

m~l'+~)/~

_{at all}

masses

which,

in turn,

yields

fir

=

£ Mk(M, _t)

_{< 0}

_(1.5)

for _{t >} tc. ^This

phenomenon

_may ^be

regarded

_{as a}

phase

transition

(known

_as

gelation) occurring

at the critical time tc, with order parameter 1-

fi4(t)/fi4(0).

Numerical solutions of the _mean field

equation(I -Ii

lead _to the _same results

iii.

The usual

interpretation

of the result in

equation (1.5)

is that _a part ^{of the} system ^is no

longer

described

by equation (I.I):

this part ^consists of _a

large

_merger which

gains

mass from the rest of the system with rate

ji.

In

chemistry,

this result has been

interpreted

_as

responsible

for the formation of _an 'infinite' cluster in

suspensions

of

particles (aereosol),

_{see 11, 12] and} references therein. It has been also

considered _{as one}

possible

mechanism for the formation of massive

(cD-like) galaxies

present in the central

regions

of dense _groups of

galaxies

_as well _as for the _erasure of substructures

developing

in the late

phases

of the overall

collapse

of

gravitationally

bound structures [6,

ii.

In this context, ^the

colliding

bodies _are

galaxies

in _{groups or} in clusters. For such systems, the kernel is characterized

by

_a mixed _cross section [13] ^{which is} a sum of two terms, whose

scalings

are

parametrized by1

₌

2/3

and 1

=

4/3, respectively.

The relative

importance

of

the two terms

depends

_on the detailed characteristics of the environment [6,

ii.

The

previous analytical approach

based _on the solutions of

equation (I.I)

reveals several limitations in the

description

of the system at all times.

Firstly,

it does not describe the evolution of the whole system for t > tc in the _cases with I _>

I, f

> -I

(gelling systems).

The absence of detailed

dynamical

information about the total _mass distribution

(including

the

merger)

is

replaced by integrated information, namely,

the _appearence of _a finite _mass flux

fir

< 0.

Secondly,

the _mass distribution obtained _{as a} solution of

equation (I.I)

is

averaged

_over the ensemble of the

single

realizations of the

aggregation

_process. The

peculiar

aspects of each

realization _can be revealed

only through

_{a more} detailed

analysis.

J

~

M=M+M

~~~~ ^~

i~

I

~4

M=M+M

M

M t=1

~~ /~

M~

M~

M~

M M~ ^t=o

~'

I ~~'~~')

ij~''~'~j ^dN(M)

~ ^/ ~~

M

£

Fig. 1. The inverse-cascading ^scheme shown in this figure reproduces the aggregation mechanism and the resulting kinetics described

(only

_on

average)

by equation

(1.1).

The _{tree structure} reproduces schematically the evolution of the system if _we consider discrete time steps. ^{The links} are drawn _every- time that two cluster coalesce, with _a

probability

proportional to

KIM, M',t),

the kernel

describing

the

aggregation

rate

(see ii).

Thirdly,

the

spatial

fluctuations _{are not} accounted for in

equation (I.I);

it is

important

to know whether

they

affect _or not the

evolutionary

^behaviour of the _mass distributions.

In this _{paper we} deal with the

previous points by studying

the evolution of

aggregating

systems

by

_means of Monte Carlo simulations of

merging

_processes. The simulations _{are con-} structed

by

_an

inverse-cascading

scheme

(Fig. I)

that

reproduces

the

aggregation

mechanism and the

resulting

kinetics described

(only

_on

average) by equation (I.I).

We

perform

the anal-

ysis

for various interaction kernels with different values of I and

f.

We follow the evolution of the system with I _> I

beyond

the critical time tc and extract detailed information about the

growing dishomogeneity

in the _mass distribution

by

_means of _a fractal

analysis.

The role of

spatial

fluctuation is also

investigated by assigning

_a

spatial

structure to the system ^of

interacting particles.

2. Monte Carlo simulation of

cascading

systems.

The structure of

equation (I.I)

_suggests the

possibility

of

describing

_an

aggregation

_process

by

_means of the _more

general

^formalism of random

cascading

models. The first _{term on} the

right-hand

side of

equation I-I)

describes the increase in

N(M,t)

due to the coalescence of clusters of _masses M' and M" with clusters of _mass M'+ M"

= M, while the second term describes the decrease _on

N(M,t)

due to the coalescence of clusters of _mass M with other clusters of different _masses.

Thus, considering

discrete time steps, it is

possible

to look at the evolution of the system as a

possible configuration

^of a tree structure

(see Fig. Ii,

where links _are drawn

everytime

that two clusters

coalesce,

with _a

probability proportional

to the

aggregation

rate

K(M, M', ii.

The

picture

of the

aggregation

_{processes we are}

discussing

is similar to that of random _cas-

cading models,

and the relation between these two formalisms _can also be studied

analytically;

this leads to _a

correspondence

of the solutions of

equation (I.I)

with the

generating

functionals of the

partition

function of _a

Cayley

tree

averaged

_over all the

possible configurations

of the

tree

[14].

This

approach, although

_very attractive,

provides again only

an average

description

of the system

evolution,

and thus contains the

problems previously

discussed

Alternatively,

_{one can use} the

cascading approach

and

study

the

configurations

of the tree

by

_a Monte Carlo simulation. In this paper we follow this second

approach.

The technical

procedure

is _as follows:

a)

We distribute _a total number fit of clusters in the _{mass space,}

assigning

_{a mass} M to each cluster

according

to the initial distribution function

N(M,t~).

b)

We select _a

couple

of clusters with _masses M' and M" with _a

probability proportional

to

N(M', _t~) N(M",t~).

The selected

couple

then aggregates to form _a

larger cluster,

with _mass M = M'+

M",

with

a

probability p(M', M")

_c~

I((M', M",t~). Thus,

at the next time step,

we will find _one cluster with _mass M

=

M'+

M" with _a

probability

_p, and the two selected clusters M'and M" with

probability

I _p. The step

b)

is

repeated

until _every cluster present

at t ₌ t~ ^{has been} selected.

c)

We re-define the mass-distribution

simply by counting

the _masses of the clusters that _we have

generated.

We then iterate the

procedure

of steps

b)

and

c)

in the whole time interval

considered.

Doing this,

we obtain

configurations

of _a tree structure that _are

reproduced,

_{on average,}

by equation (I.I).

In this

approach,

we can

study

in

principle aggregation

mechanisms characterized

by

the _more

general

kind of kernels and _{we can}

investigate

the

properties

of the

single

realizations of the

aggregation

_process.

3.

Analysis

in the _{mass space.}

We have

performed

simulations with _an initial number of

particles

^fit ₌ 5 X 10~ and with different initial distributions of _masses in the range M~ _< M _< 50M~, ^with M~ the unit _mass

taken to be ₌ I.

We have checked that

already

for values of fit

-J

O(I

X 10~) ^{the main} statistical features of

our results _are

independent

_on the size of the system for all values of I and

f

chosen hereafter.

The first feature _we want to

investigate

is the mechanism

leading

to the

phase

transition at the critical time tc, characteristic of kernels with I > I and

f

_> I. The

plots

in

figure

2 show the

quantity N(M,t) M,

normalized to the total _mass

fi4,

_{as a} function of the _mass formed at different times t. The left column of

figure

² illustrates the behaviour of the system for _a

A=2/3 A=4/3

4 4

3 3

) )

~ ~ z t=io

~ ~

~ %

i i

O O

20 40 60 SO O 50 LOO ISO ZOO

M ~

4 4

3 3

f j

~ 2 ~ t=35

> ~

% ii

i i

O

O 20 40 60 80 O 50 loo 150 ZOO

M M

4

3 3

~

~

t=loo ,

~

~ z

j

~

£

i i

O

O 20 40 60 80 O 50 LOO 150 ZOO

M M

Fig. 2. The _mass distribution of the system ^at different times. The plots show the fraction of total

mass

N(M,t) M/fi4 (y-axis)

found at the _mass M

(z-axis)

at three different times t ₌ 10, t ₌ 35 and t ₌ 100, in units of the time step used in the numerical calculations.

a)

The three plots in the

left column show the behaviour of the system ^for _a ^{kernel with 1} ₌

2/3, b)

The three plots in the

right column show the evolution of the system for _a kernel with 1

=

4/3.

The _mass axis has been

chosen different in the two _cases in order to show _more details of the _mass distribution in the

case

1

=

2/3

that evolves at slower rates. In both _cases

(represented

in the two

columns)

_we kept f

= 0

(the

aggregation rate does not depend explicitly _on

time).

kernel with 1

=

2/3,

while the

right

column refers to 1

=

4/3.

In both _{cases we}

kept f

₌ 0

(the aggregation

rate does not

depend explicitly

_on

time).

The

analytical

^{result of} the

phase

transition _appears here _{as a}

stronger dishomogeneity

of the

gelling

system

(I

>

I) compared

with the

relatively homogeneous

distribution

produced by non-gelling

systems

(I

<

I).

In

particular, gelling

systems _are characterized

by

_a

branching

in the _mass distribution after _a short time t _cS tc

(the

critical

time),

with the

high-mass

branch

gel-phase) rapidly concentrating

the main part of the total _mass of the system. The low-mass branch

(sol-phase)

evolves

smoothly

while the

gel phase

evolves

through

_a

nearly

self-similar series of

branchings.

The time evolution of the MD is characterized

by

_an initial transient for t

$

T -J

(fit(LV))~~ (the typical

^{timescale for} the evolution of the

system)

where the relative increase of the

gel

_mass is

small,

because the flux of clusters from the sol _to the

gel phase

consists

predominantly

of small _masses; the further

branchings

take

longer

and

longer

to take

place.

To maki the

previous

considerations

more

quantitative,

_we

analyse

the _moments

(M~)

₌

£ N(M)

M~ of the _mass distribution. In

figure

3 _we

plot

the ratio

(M~) /(M~)~

_{as a} function of time for

gelling

systems with 1

=

4/3, f

₌ 0

(solid line),

for

non-gelling

systems with

1 =

2/3, f

= 0

(dotted line).

If the _mass distribution had _a self similar form of the kind

(1.3),

the ratio

(M~) /(M~)~

would stay constant in time. From the behaviour of

(M~) /(M~)~

plotted

in

figure 3,

_{one can} conclude that this is

actually

the _case for the

non-gelling

systems.

A different situation holds for

gelling

systems ^after the time tc, because of the formation of

a

large

_merger. We also

investigated

the role of the

explicit t-dependence

of the interaction kernel. In

general,

^the

larger

is

f,

the faster is the evolution. The value

f

₌ -I is

critical,

the evolution

being appreciable only

for

substantially larger

values of

f.

In this respect, we

verified that the

gelling phase

transition _{occurs even} for _a time

decreasing

kernel

provided

the

limiting

condition

f

_> -I holds.

zsoo

zero

«

j

^X=4/3

~

1500

j

~ fl

1000

j

li

500

0 loo 200 300 400

t

Fig. 3. The ratio

((M M)~)/(k)~

is shown _as

a function of time for gelling

(solid line)

and

non-gelling (dotted line)

systems. A value f

= 0 is used throughout.

To

analyse quantitatively

the actual _mass distribution of _a

single

realization

(without making

any kind of

average),

_we

study

the

degree

of

dishomogeneity by

_means of _a fractal

analysis

in the _{mass space}

(see

[15] for _a

review).

To this

aim,

_we divide the whole _{mass range} in boxes of size _£, and _we define the _measure of the I-th box _as

Pi(£)

+

Ni/fit, (3.1)

where Ni is the number of clusters in the I-th box and fit is the total number of clusters. The set of

general12ed

dimension is then defined

by

(q i)D(q)

_%

lim<-o

^~~

()l'~~, ^(3.2)

with

Z(q,

_{£) =}

LP>(£)~. (3.3)

For the fractal

analysis

of the _mass distribution of

clusters,

the number of boxes with N;

#

0 needed to cover the M-axis scales _as

£~m(°),

where

Dm(0)

is the fractal dimension in the _mass space. Such

a dimension is _{a measure} of the

homogenity

of the distribution of

particles

in the considered _{mass range.}

From the

analysis

of the distribution

produced by

the

aggregation

at each fixed

time,

_we find

that,

for _{every q,} the function

Z(q)

scales with _{£ as a power} law

-J £(~~~)~m(~)

(in

the limit

£ <

I)

with

Dm(0)

< 1.

Thus,

the realizations of

aggregation

_{processes seem} to distribute the

mass

according

to _a fractal distribution in the

mass space.

To understand wheather the distribution follows _{a pure} fractal _{or a} multifractal

scaling,

_we

study

the

q-behaviour

of the function

T(q)

+

(q I) Dm(q). According

to _a

general

theorem

[15],

a multifractal distribution

produces

_{a convex} function of_q while _a linear

dependence

is the

signature

of _a

purely

fractal behaviour. We find that the

previous

function shows _a linear

trend, providing

evidence for _a

simple

fractal

scaling,

_so

that,

^for _{every q,}

Dm(q)

₌

Dm(0)

e Dm.

The dimension Dm

changes

with time

following

the

developing dishomogeneity

of the system.

Its time behaviour will

depend

_on the

strength

of the

interaction,

I-e-, on I and

f.

We illustrate in

figure

4 the evolution of

Dm(t)

_{as a} function of time for 1

=

2/3

and 1=

4/3 (with f

=

0).

For

increasing time,

_{a gap}

develops

between the characteristic dimensions Dm for I < I and I > I. This indicates _a different

growth

of the

dishomogeneity

for the two _cases that reflects two different _ways of

producing

_a

large

mass distribution

starting

from _a uniform initial condition.

In the

non-gelling

case

high

masses are

produced by

_a diffusion-like

homogeneous

_process,

while the

gelling phase

tends to create

large

attractive clusters at the expense of many smaller

interacting

bodies.

After the initial transient, the systems evolves toward _a stable state, characterized

by

the

plateau

of

Dm(0),

that defines the

asymptotic

mass distribution.

A final remark _concerns the

reliability

of the statistic: with the time

going

_on, the number of

particles decreases,

but

always

stays

large enough

to allow _a feasible estimate of the smaller

moments of _mass

distribution,

in both

gelling

and

non-gelling phases (see

_[1,

6, ii

for _a direct

calculation of the total number of clusters _{as a} function of

time).

5

$ ^X=2 /3

1 ^'"

5

X=4/3

0

0 20 40 60 fi0

t

Fig. 4. The dimension

Dm(0)

_{as a} function of time is plotted for different values of I

(and

with

f = 0). ^The dotted line

(upper curve)

refers to the non-gelling _case 1

=

2/3

while the solid line

corresponds to the gelling-case 1=

4/3.

4. The role of

spatial

fluctuations.

We _now turn to

considering

the effect of

spatial

fluctuations of the cluster

density

_{p =}

p(r).

In order _to obtain _a

spatially

resolved

representation

of the

merging

_{process, we} distribute the

fit _masses in _a 2-dimensional box of size R. We also

assign

coordinates to each

particle.

The

Monte Carlo

procedure

described in §2 is then modified _as follows:

To select _a

couple

of

interacting clusters,

_we extract first _one

particle,

and _we

explore

_a

region

inside _a

sphere

with radius V

At,

centered _on such _a

particle.

The cluster

velocity

V is here assumed constant and At is the time step.

A second

particle

is selected

only

if it is inside such _a

region.

In _case of

aggregation,

the _new cluster will be

placed

in the center of the _mass of the two

interacting particles.

The

remaining

criteria for the

particle

selection and

aggregation

_are those described in §2; the cluster

positions

_are

changed

at every interaction

by

_an amount V At in _a

randomly

chosen

direction.

The described

procedure

will allow to interact

only

clusters which _are able to meet in the

sample

time interval

At;

the interaction

probability

in the time At will be

proportional

_to the interaction volume L VAt

(as

for the

homogeneous case),

thus

reproducing

the

aggregation

rate

entering

the Smoluchowski

equation.

The

point

is

that,

for

given

L

VAt,

the

aggregation

rate will be

proportional

to the number of

particles

which is inside such _a

volume,

I-e- to the

density

_p. However, the

density

here is not constant, _so that the

merging

rate for the

homogeneous

_case c~ L V At turns out _now to be modulated

by

the local

density

P(r)

=

All

+

f(r)1 13.4)

The initial form of the

two-point

correlation function

((r)

is

provided by

the

physical

back-

ground

of the

aggregation

_process. We

investigated

different scenarios characterized

by

dif- ferent initial correlation functions:

homogeneous ((~(r)=

const.

),

or

power-law correlations, f~(r)

_c~ r~~ with ₇

= 0.8 and _{7 =} 1.5. The value of ₇

depends

_on the

physics

of the

particular

system considered. For

instance,

in the _case of the distribution of cosmic structures

(from galaxies

to clusters and

larger structures),

_a value _{7 cS} 1.8

(with

_a

flattening

^towards ₇ cS I _on

large scales)

is indicated

by cosmogonical

theories and relevant observations

[16, Iii.

The correlation function will evolve

according

to the

aggregation

mechanism and it is stud- ied

by

_a multifractal

analysis.

We have verified

that,

_as for the simulations with _no

spatial

structure, the initial total number of clusters fit

= 5 _X 10~ is

large enough

to prevent detectable finite size effects. A discussion about the

dependence

of _our results _on size is

given,

in quan-

titative terms, at the end of this section. The detailed

shape

of the initial _mass

distribution,

N(M, t~),

turns out to be

unimportant

for the

asymptotic

time evolution.

RESULTS. A first result is

that,

the initial

spatial

correlations do not

produce appreciable

variation _as for the appearence of the

phase

transition in the I _> I _cases,

indipendently

of the

particular

initial

density

distribution. On the other

hand,

the correlations tend to shorten the critical time tc

by

_a few percent.

Moreover,

the

spatial

fluctuations affect the

asymptotic spatial

distribution of clusters. To

perform

_a fractal

analysis

of the _space distribution of the

clusters,

_we divide the total volume in boxes of size _£, and _we define _a local

spatial

measure as the _mass

density

in the I-th box

normalized to the total mass, I-e-, _{pi =}

Mi1fi4tot.

The

generalized

dimensions,

Dr(q),

in the coordinate _space

depend

_on the initial

configuration.

As _a first

aim,

_we

study

the time evolution of

Dr(q).

In the

homogeneous

case

(no

initial

correlations) Dr(q)

stays constant _{as a} function of both _q and time.

So,

in absence of correla-

tions,

the initial random fluctuations

are not sufficient _to

produce

an

increasing disomogeneity during

the time evolution for _every value of I.

A different situation arises for the

initially

correlated

configuration, provided

the condition I _> I holds. The behavior of

Dr(q)

for _q

=

0,

2, 4 is

plotted

ⁱⁿ

figure

5 _{as a} function of time for _an initial distribution defined

by

_a correlation function

fo(r)

_c~ r~ with ₇

=

0.8,

in the _case

1 =

4/3.

A transition from the initial to the

asymptotic

values of

Dr(q)

_appears after _a short

time, similarly

to what

previously

resulted in the _{mass space}

analysis (cf. Fig. 4).

The values of

Dr(q)

in the

plateau

thus define the

asymptotic spatial

distribution of the clusters. Similar

results _are obtained for different initial values of ₇ > 0 and I > I.

The

complete

multifractal spectrum in the

asymptotic region

has been also evaluated _{as a} function of the initial

spatial configuration.

The results for 1=

4/3

_are illustrated in

figure

6 for different initial

distributions,

and

compared

with the spectrum of the initial condition. This is not flat because of the _non-zero initial correlation

(the

exponent ₇ is

actually

related to the fractal dimension

D(2)

of the initial

distribution, computed

from the _measure

pi(£)

₌

Ni IN,

by

the standard

relationship

_{7 =} D~

D(2),

where D~ is the dimension of the

embedding

space, 2 in _our

case).

An evolution from the initial condition is present, ^{but it is}

appreciable only

for _q

l$

2. This is

because,

while

Dr(0)

represents ^the

simple counting

of boxes with _{a non-zero measure,} Dr

(q)

with _q

#

0

weights preferentially

the

peaks

of the _mass distribution

describing

the intermittent behaviour of the system. Thus the time evolution

changes mainly

the distribution of massive

clusters, leaving

^the distribution of the smaller

background particles unchanged.

This is _a

further consequence of the strong

dishomogeneity

in the _mass

distribution, resulting

after the

phase

transition.

In the _cases with I _< I, _no

significant

evolution of the multifractal spectrum ^is observed for

any initial

condition,

which

implies

the _same self-similar

clustering

for small and

large

^clusters.

z

q=o

1.5

O

0 20 40 60 fi0

t

Fig.

5. The time evolution for three generalized fractal dimensions,

Dr(q),

evaluated for 1

=

4/3

and for _q

= 0

(solid line),

_{q =} 2

(dashed line),

_q

= 4

(dotted line).

^{The critical} time tc corresponds to the beginning of the asymptotic region.

In order to test _our

statistics,

_we have _run the _same simulations for _a total number of

particles

from fit

= 5 _X 10~ _up to fit

= 5 _X 10~. The

scalings

with

log(£)

of the

logarithm

of the

partition

function

Z(q,

_£) defined in

equation (3.3),

are

plotted

in

figure

7 for _q

= 0 and _q

=

4,

for different values of fit. The

scalings

refer to the _case 1 ₌

4/3

with _an initial ₇

= 0.8. As

appears from the

figure (and

from the

reported

variance of the data from the

least-square fit),

the

scalings

for fit

=

O(10~)

_are

already quite good. They

show the _same

slopes (I.e.

fractal

dimensions)

for the various values of

fit,

with _{an error on} the dimension estimate

decreasing

for

larger

values of fit. We have verified

that,

at least for moments up to order _q

= 5 6,

the

scalings

_are

again

well stable and reach _a

good reliability,

_so that the

previously presented

results _are robust relative to variations of fit.

The results shown in

figure

7 _are

actually

_{very common} in multifractal systems where

only

high

moment distributions _are affected from the convergence

problem

due to the

possible

presence of

strong-fluctuations.

5. Discussion.

In this _{paper we} show that

a statistical numerical

approach

to the

study

of

aggregation

_prc-

cesses allows _us to describe in detail the statistics of the _mass distribution that is

represented

only

_{on average}

by

the _mean field Smoluchowski

equation.

With such _an

approach,

_we also extended the

study

to the role of

spatial

correlations.

To

analyze

the different behaviour for

non-gelling

and

gelling

_{cases, we}

performed

_a fractal

analysis,

both in the _mass and in the coordinate spaces.

From the

analysis

in the _{mass space, we} find that the mechanism

driving

the evolution of the

3

y=08

z

o

O 1 2 3 4 5 6 7

3

y=i.s

2

1

"

",

"&m.-,,-_

~~~~~"~~-.--

~^~4

2 3 4 5 6 7

Fig.

6. The comparision between initial and asymptotic values of _some generalized dimensions for different initial conditions, and for

=

4/3.

(a)

The solid line represents

Dr(q(t

₌

0),

while the dotted line is the _same quantity in the stationary regime t > tc, ^{with initial} condition _{7 "} 0.8.

(b)

The _{same as} in

(a),

but for the _case with initial condition ₇

" 1.5.

system

depends

_on the

strength

of the interaction

(parametrized by

the

homogeneity degree

of the

Kernel)

in _a critical _way. For I _< I

(non-gelling systems),

the system ^tends to create still

larger objects by

_a

diffusion-like

_{process among} clusters of

comparable

size. For I _> I

(gelling systems) instead,

the evolution is driven

by

the accretion of

large

clusters

(formed

at _a finite time t ₌

tc)

at the expense of smaller

interacting

bodies.

Thus,

in the latter _{case, a mass} gap is created between two

phases, resulting

in _a

highly dishomogeneous

mass distribution.

By

_means of _our numerical

approach,

_we follow the behaviour of the system

beyond

^the

phase

n(oj=1 9a, a=s io'~

~

n(4)=o _{7, am} to'~

~

~

m

f l'

~

M a~° _.-"

lG /

£

^{X=S io~}

2

~ ~

~ O

fl

)

2 ~

"

,~d

~ ."'

d ^%

z

~

i ,,'

»'~

." ^»"

0

0 1 2 3 4 5 0 l 2 3 4 5

tog(I/£) lag(t/£)

o(o)=19a, a=lo'~

~

^{n(4)=069, a*z lo'~}

m '

Q ^{' 3}

W ~°° d X=5 to'

2 u' ~

4

,

z _~

~

fl''

,'

z ." fl'

_o ^,

'

"' x

0 0

0 1 2 3 4 5 0 1 2 3 4 5

log(I/£) log(I/£)

to s

8 D(0)=2.02,

=5.10"/D(4)=0

69, a=5.t0~~

6 q

~

~o

3

~

_~d

£

^{X=5 Id}

.O £

,

°'

fl

~'~

,' ,,'

»"

/ fl

'~

,°' ^.

_/ ^,

0 1 2 3 4 5 0 t 2 3 4 5

tog(I/£) log(I/£)

Fig.

7. The

logarithm

of

Z(q, e)/(q i)

_versus

log(e).

The slopes of the resulting _curves in the scaling regions give the generalized fractal dimensions. The three plots _on the left column correspond

to q = 0 for fit

= 5 _x 10~, 5 x 10~, 5 x 10~, from top to bottom. On the right column the scalings for q = 4 _are plotted for the _same values of fit. _The least-squares fit

(dotted lines)

_are evaluated in the

linear

region.

The resulting slope (I.e.

Dr(q))

and the variance _on their estimate _are also given.

transition,

where the MD evolves

through

_a

repeating

series of

branchings (Figs.

2 and

3).

The fractal

analysis quantifies

such results. A _pure fractal

scaling

for the dimension

Dm(q),

related _to the _mass

distribution,

results. The fractal dimension reaches _a

stationary

^value

(depending

_on

I)

after _a transient

time,

thus

describing

the

asymptotic

mass distribution. A

gap between the

asymptotic

values of Dm for

gelling

and

non-gelling

systems is

found,

due to different

dishomogeneities

in the _mass distributions.

We have

investigated

the role of

spatial

fluctuations for different initial

spatial configurations

and at different

times,

and found _some

interesting

results.

In the _case of

gelling

systems, the multifractal spectrum

(I.e.,

the

generalized

dimension

Dr(q)

_{as a} function of

q)

^{reflects the}

different

behaviour and

spatial

distribution of

large

massive clusters and small

background particles.

In

fact,

_an evolution with respect ^{to the} initial conditions is present

only

for _q

l$ 2,

that

is, only

when

large

_masses dominate the

statistics.

The

dependence

from the initial condition is _as

expected; increasing

the initial

correlation, produces

_a stronger

dishomogeneity

of the system,

expressed by

the lower values of

Dr(q)

at

large

_q.

As for the time

bebaviour,

the system ^reaches an

asymptotic regime

characterized

by

the

presence of _a

plateau

for

Dr(q)

with _q

l$

2. Such evolution is

analogous

to that of the fractal

dimension

Dm

in the _mass space on the _same time scale.

Thus,

the

phase

transition characteristic of

gelling

systems is marked

by analogous

be- haviours in the _mass and in the coordinate _space. Such behaviours affect each other: the concentration of

particles

in

strongly

correlated

regions

enhances the

probability

of

produc- ing large clusters;

_on the other

hand,

the _presence of

large

clusters enhances the

merging probability (which

is

proportional

to

M')

with other

clusters,

thus

changing

the local

spatial

distribution.

The

non-gelling

systems show _a smoother behaviour both in the _mass and in the coordinate

space. In view of the

previous considerations,

this is due to the fact that the lower M-

dependence

of the

aggregation probability

not

only produce

less massive

objects,

but also affects the

spatial

distribution in _a minor _way.

Thus,

in this _case, the _mass distribution is almost

homogeneous,

while _no time evolution of

the _space distribution is present. ^Such a result is the

signature

of similar

aggregation

behaviour

and

clustering

of

large

and small

clusters,

which

have, however, always comparable

_masses.

A few comments about the relation with

physical

systems are in order: the

physical

time scale is

given

by ^I

/nLV

₌ t,

/(fit/R~)LR

with t, ₌

R/V,

where R and V _are now the

physical

size of the system ^{and the}

physical particle velocity, respectively.

The

assignment

of _some

computing

units to V and

R,

for _a

given

number of

particles fit, provides

_a

relationship

between the

computing

and the

physical

time

scale,

relative to the

particular

system ⁱⁿ consideration.

As _an

example,

the critical time t~

resulting

from the simulations for _a of group of

galaxies

with R _ci 250

kpc

and

galaxy velocity dispersion

V _cf 500

km/s, containing

_a number fit

= 10~

galaxies,

is of the order of 5 6 _X 10~ _yrs.

Our results _compare

interestingly

to some

astrophysical problems.

For

example, they

_are

analogous

to those

usually

found in

cosmology

for the different matter and

light

distributions

[18,

19,

20].

^{In this} _case, the luminous visible

baryonic

material in

fully

evolved cosmic structures, is located

preferentially

at the

peaks

of their

density

distribution and is

segregated

from the

underlying

distribution of dark collisionless material.

Actually,

_a

multifractal distribution of the luminous matter _seems to fit the observations

[21,

22] and the result of

N-body cosmological

simulations [23] are in

qualitative

agreement with _our

picture

that predicts _a

larger clustering

for _more massive clusters. A

closer, quantitative

relation of the described

picture

of

aggregation

with the referred

astrophysical problems

_may be _proven

JOURNAL DE PHY~IQLE I I N' ~ MAY lwl 19

by extending

_our study to three dimensional distributions.

We believe that the present numerical

approach

could be

easily

extended also to

study

the statistics of _very

general aggregation

mechanisms based _on

cascading (or inverse-cascading)

processes.

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Ernst M-H- 1986, ^{in 'Fractals} in Physics', L. Pietronero and E. Tosatti

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