We have studied (in and
) the arrangement of the grains
inside the box and the correlations between their positions. The main
feature observed in this study is the appearance of spatial
clustering for inelastic gases (
) in the colliding regime (
). We firstly give detailed results
for
and, after, similar measurements for
.
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In Fig. fig_density_1d we show a profile of the density (not normalized) of the gas obtained by means of a coarse graining. The pictures are taken for two different choices of the parameters: one in the collisionless regime (and near elasticity), the other in the colliding regime. A guideline indicates the average (uniform) density. The colliding regime is characterized by fluctuations of the density very much stronger than in the collisionless regime. We call these fluctuations ``clusters''.
In Fig. fig_density_dist_1d and
Fig. fig_density_dist_1d_DSMC we show the probability distribution of
the ``M-cluster mass'' defined as the number of particles found
in a box of volume
. The two figures are similar in the choices
of the parameters, but the first is taken from a MD simulation, while
the second from a DSMC run. These figures are important because show
an impressive qualitative agreement between the two algorithms used in
this work: in the collisionless regime (
) the
number of particle in a box is Poisson distributed, i.e.:
where
is the expected value of
. This
distribution is the signature of the fact that the positions of the
particles are independently distributed in the space, i.e. no
correlations emerge from the dynamics in this regime. On the other
side, in the colliding regime (
), the distribution
is very different. In both the figures it can be fitted by a curve
. This curve is the product of a
negative power law with a negative exponential cut-off. In most
observed situations we have measured
and
slightly greater than
. The power law is the signature of
self-similarity in the distribution of clusters: in the colliding
regime structures emerge with no characteristic mass. The exponential
cut-off is due to finite size effects (
is finite and is a
normalizing constraint). We stress that the agreement between MD and
DSMC results is striking: these observations suggest strong
correlations between grains, a fact that cannot be considered obvious
in a DSMC simulation (where some correlations are artificially
removed).
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We have characterized the departure from the Poisson distribution of
the cluster masses by means of an entropy functional which is
defined as
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(3.15) |
where the system has been divided in boxes of volume
and the
quantity
is the number of particles in the
-th box. This
quantity attains its maximum value
when the number of particle
is the same,
, in every box and decreases as the density becomes
more and more clusterized. As we have seen, the homogeneous regime is
characterized by fluctuations of the quantity
, following the
Poisson distribution of Eq. (3.15); in this case the value
can be numerically calculated. We call this ``homogeneous''
value
. The value
has the same properties of stationarity
found for the kinetic energy or the dissipated energy. In Figure
Fig. fig_entropy_1d_MD it is shown the quantity
, where
and
. We observe that the departure
from the collisionless regime is accompanied by a sensible decrease of
the entropy of the distribution of the masses, which is a signature of
the organization in dense clusters and almost empty regions. The
clusters are rapidly evolving, they break and form again, so that the
entropy
is statistically stationary on large observation times
(time averages).
Furtherly, to obtain more information on the kind of spatial
correlations among grains, we have studied the correlation dimension
(Grassberger and Procaccia [100]) that is defined
in the following way. The cumulated particle-particle correlation
function is:
After having checked that the system has reached a stationary regime,
we have computed its time-average . In the
Fig. fig_d2_1d_MD and Fig. fig_d2_1d_DSMC we show the
vs.
for different regimes and with different algorithm. We observe a power
law behavior in both colliding and collisionless regimes, and with
both algorithm, MD and DSMC (this is again a surprising agreement):
This relation defines the correlation dimension .
In the case of homogeneous density is expected to be the
euclidean dimension
. The case
is a definition of fractal density. We find a homogeneous density in the collisionless
regime, and fractal density in the inelastic colliding regime (
and
). This is consistent with the power law distribution
of cluster masses observed before in the same regime.
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In Fig. fig_d2_1d we show the stationary values of
obtained for different choices of the parameters
and
(keeping almost fixed
). We stress the fact that the DSMC
algorithm imposes the absence of correlations among colliding
particles: this is observed in the plot of
where, in the
colliding regime, the slope
is observed only for
, with
(see Fig. fig_d2_1d_MD and
Fig. fig_d2_1d_DSMC) . For a discussion on the Bird radius
we
refer to Appendix A.
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We have repeated all the analysis of the density for the system in
, using only DSMC simulations, and finding the same phenomena:
homogeneous density in the case
and clusterized
fractal density when
and
. In
Fig. fig_density_2d we propose a ``photography'' of the system with
the positions of the particles, in an instant of the simulation in the
clusterized regime (
and
). In
Fig. fig_density_dist_2d we show the probability distribution of
cluster masses, as defined above, for the two different regimes: we
observe again the Poisson distribution in the homogeneous situation
and the power law (plus the finite size exponential cut-off) in the
clusterized situation.
Finally, we have studied the correlation function defined in
Eq. (3.17) and its power-law behavior
characterized by the exponent . The function is shown in
Fig. fig_d2_2d.
In Fig. fig_summary we have plotted a summary of the
measurements of several quantities in
: the granular temperature,
the correlation dimension
, the collision rate
and the
diffusion coefficient
. The diffusion coefficient is discussed in a
paragraph 5.66.
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