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
