of aggregation phenomena ~j - International Journal of Limnology

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May 18, 1993 - di Roma, via dell'osservatorio,1-00040. Monteporzio, Italy. (Received 16 July 1992, revised. 23. December1992, accepted 22. January1993).
J. Pllys.

I

France

(1993)

3

MAY1993,

l105-ll18

l105

PAGE

Classification

Physics

Abstracts

64.60

82.20

02.50

simulations

Carlo

Monte

(~), S.

N.

Menci

(~

Dipartimento Italy

(~)

Osservatorio

(Received

We their

systems

We

Monte

mass

the

of

the

of

of

for

the

known

the

to

analysis.

Scientifica

accepted

are

January1993)

22

galaxies

of

processes

here

the

to

case

of

spatial

can

distribution

The

of

clusters

ofstructures with

distribution

value

I

=

to

I.

We

various colloids. various

(MD). extend

by

distribution

cluster out

to

in

M, for

mass

mass

of the turns

interactions

related

be

growth

clusters

fluctuations

merging

under

systems

considered

Roma,

I, 1-00133

Monteporzio, Italy

study the evolution of the resulting phase transition around the critical

and

aggregation

of

aggregation

of

M', +J

that

ofsystems

evolution

Ricerca

many-body

of

realizations

simulations K

multifractal

a

be

fractal

in

the

multifractal in the real space. The phase produces an transition enhancement dishomogeneity for large masses, that is quantified in of different values terms Dr(q). A correspondence between the effects of the phase generalized fractal dimension and

space

spatial

transition

1

of the

evolution

the

(~)

dell'osservatorio,1-00040

via

December1992,

23

the

the

Carlo

evidence

out

analysis

means

from

probabilities

We find the

discuss

Biferale

Roma, via delta

di

Roma,

di

revised

components;

physical

(~) and L. Universitl

II

Astronomico

Abstract.

run

Fisica

July 1992,

16

between

merging

Colafrancesco

di

aggregation phenomena

of

in

the

space

mass

and

in

coordinates

the

is

also

found.

duction.

Intro

of aggregation have been studied in a variety of fields, from statistical processes chemistry [1, 2] to planetary formation and also in astrophysics and cosmology [3] interest for the study of merging phenomena [4, 5]: specifically, there has been a renewed 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 The

kinetics

physics

to

N(M,t)

~j

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

=

M>+M,'=M

£ ~x>

-N(M,t)

M,=1

Send

o1Tprint

requests

to:

S.

Colafrancesco.

K(M,M',t)N(M',t)

(I.I)

JOURNAL

1106

describing

the

with

evolution

time

of

t

PHYSIQUE

DE

the

I

N°5

N(M,t)

distribution

mass

quantity K(M, M', t)

of

clusters

with

mass

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 their relative velocity V. Analytical studies is their section for aggregation averaged over cross described by equation (I.I) have been performed for separable of aggregation phenomena kernels verifying interaction homogeneous binary

under

M

The

collisions.

is the

=

K(aM, aM',t) if.

F(t)

where

behaviour

The

for I < I

where

fi4,

for

m

the

variable

adimensional

%

m

M/M.,

one

M.(t) holds.

£

fir different

situation

when

arises

I

This

and

I

>

finite

f

leads

at

every

time

-J

conservation

-I;

this

case,

-J

yields

£ Mk(M, t)

fir =

for t

occurring of the

the

critical

field

equation(I -Ii

time

with

tc,

lead

regarded

be

may

order

to

as

a

parameter

the

results

same

of the




0)

for

(1,4) in

-J

turn,

>

of the

terms

0.

simple scaling (1.3) in equation (I.I) leads to a divergence of M.(t) after m~l'+~)/~ 1/fi4 (see [10], [iii). This implies a scaling form #(m) tc in

In

t.

m~f (with (

the

to

=

>

(13)

#(m)

that

MN(M,t)

=

A

remains

verify [ii

can

m~'ezp(-m)

-J

as

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

-~

size

cluster

mean

I, #(m)

»

homogeneity degree

the

scaling form

self-similar

a

N(M,t) obtained,

on

exponent f [6, ii.

the

on

Specifically,

while

depends critically

solutions

of the

(1.2)

-J

I [1, 9] and

is

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

=

is

no

from

ji.

rate

interpreted as responsible for the formation of an 'infinite' suspensions of particles (aereosol), see 11, 12] and references therein. It has been also (cD-like) considered possible mechanism for the formation of massive galaxies present as one in the central regions of dense of galaxies well the of for substructures groups as as erasure developing in the late phases of the overall collapse of gravitationally bound structures [6, ii. For such In this context, the colliding bodies are galaxies in groups or in clusters. systems, characterized by a mixed cross the kernel is section [13] which is a sum of two terms, whose 2/3 and 1 4/3, respectively. The relative importance of scalings are parametrized by1 characteristics environment depends on the detailed of the the two terms [6, ii. solutions of equation (I.I) reveals several The previous analytical approach based on the 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 absence of detailed dynamical information about the I > I, f > -I (gelling systems). The (including merger) is replaced by integrated total distribution the information, namely, mass In

cluster

has

been

in

=

=

the

of

appearence

Secondly, ensemble realization

the

of the can

finite

a

mass

single be

mass

flux

distribution

realizations

revealed

fir

0.




t=35

~

%

ii

i

i

O

20

O

40

60

80

50

O

loo M

M

150

ZOO

4

3

3

~

~ ,

~

~

z

j

t=loo

~

£

i

i

O

20

O

40

60

80

O

50

Fig.

left

t

=

100, in

column

right =

2/3

the

in

the

evolves

(the aggregation

of

the

show

different that

units

show

column

chosen 1

distribution

mass

rate

of the found

the

time

behaviour evolution two at

does

ZOO

M

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

mass

and

The

2.

150

LOO

M

cases

slower not

system

at

step of

mass

used

in

the

in

order

to

In

both

depend

the

show

explicitly

cases on

kernel

a

more

different

with

details

(represented time).

1

=

1

=

of the in

the

the

mass

two

of

fraction

times

a) The 2/3, b) The 4/3. The

calculations. with

kernel

a

three

at

numerical

plots show

The

times.

(z-axis)

M

system for system for

the

of

rates.

different

at

the

t

=

three

axis

distribution

columns)

t

total 35

=

plots in the plots in the

three

mass

10,

in we

has

the

kept f

been case =

0

JOURNAL

1110

with

kernel

1

=

(the aggregation

2/3,

while

rate

does

the not

right

PHYSIQUE

DE

column

refers

depend explicitly

to 1

I

N°5

4/3.

=

In

both

cases

we

kept f

=

0

time).

on

here as a stronger dishomogeneity of analytical result of the phase transition appears distribution produced the gelling system (I > I) compared with the relatively homogeneous characterized by a branching by non-gelling systems (I < I). In particular, gelling systems are after a short time t cS tc (the critical time), with the high-mass branch in the mass distribution gel-phase) rapidly concentrating the main part of the total mass of the system. The low-mass self-similar branch (sol-phase) evolves smoothly while the gel phase evolves through a nearly characterized by an initial transient for evolution of the MD is series of branchings. The time (fit(LV))~~ (the typical timescale for the evolution of the system) where the relative t $ T because the flux of clusters from the sol to the gel phase increase of the gel mass is small, predominantly of small masses; the further branchings take longer and longer to take consists place. The

-J

maki

To

£ N(M) time

of 1

M~

of the

(1.3), plotted

the in

considerations

mass

1

with

the

ratio

figure 3,

one

can

=

distribution

mass

stay

conclude

quantitative, we analyse the (M~) moments /(M~)~ plot the ratio function 3 we (M~) as a (solid line), non-gelling for 0 systems with

=

In figure 4/3, f

=

(dotted line). If (M~) /(M~)~ would 0

=

more

distribution.

gelling systems

for

2/3, f

=

previous

the

in

constant

that

had

time.

a

actually the

this is

self the

From case

similar

form

for the

of

the

kind

(M~) /(M~)~ non-gelling systems.

behaviour

of

gelling systems after the time tc, because of the formation of large also investigated the role of the explicit t-dependence of the interaction We merger. a kernel. The value f In general, the larger is f, the faster is the evolution. -I is critical, the evolution being appreciable only for substantially larger values of f. In this respect, we that the gelling phase transition verified for a time decreasing kernel provided the occurs even limiting condition f > -I holds.

A

situation

different

holds

for

=

zsoo

zero

«

j

X=4/3 1500

~

j

~ fl

1000

j li 500

Fig.

3.

The

ratio

0

loo

((M

M)~)/(k)~

non-gelling (dotted line) systems.

A

value

200 t

is

shown

f

=

0 is

as

300

a

used

function

of

throughout.

400

time

for

gelling (solid line)

and

MONTE

N°5

To

CARLO

SIMULATIONS

OF

PHENOMENA

AGGREGATION

iii1

analyse quantitatively the actual mass distribution of a single realization (without making of average), we study the degree of dishomogeneity by of a fractal analysis in means divide the whole space (see [15] for a review). To this aim, we mass range in boxes mass

kind

any

the

of size £,

and

define

we

the

of the

measure

Pi(£) Ni is the

where

of

set

number

general12ed

clusters

of

is

dimension

(3.1)

and fit is the

box

total

number

clusters.

of

The

by

defined

i)D(q)

(q

as

Ni/fit,

+

I-th

in the

then

box

I-th

lim