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4 Explaining DCA Dynamics
DCA behavior exhibits a strong analogy with the behavior of dissipative sys-
tems [11], e.g., Benard's cells. A fluid between two plates is in thermodynamic
equilibrium if no thermal energy flows from the external to perturb the equi-
librium. In presence of small differences between the temperature of the two
plates, the thermal energy is still not enough to perturb the fluid, and energy
flows between the two plates in the form of thermal diffusion. However, as soon
as the temperature gradient reaches a critical point, thermal flow in the fluid
starts occurring via convection. This motion does not occur in a disordered way:
regular spatial patterns of movement emerge, with wide-range and symmetry
breaking correlation among cell movements. This behavior is maintained until
the temperature gradient between the two plates become too high, in which case
the regular patterns disappear and the fluid motion becomes turbulent.
By analogy, we conjecture that the behavior of DCA might be subject to the
same phenomenon, where the temperature gradient between the two plates is
substituted by the ratio
λ
e
/λ
a
. When this ratio is 0, the system is in equilibrium,
and no perturbation from the external occur. For very small perturbation, the
dynamic behavior of the DCA does not substantially change. As soon as the ratio
becomes high enough, the DCA dynamics changes and regular spatial patterns
appears. For very high ratio, spatial patterns disappear and the DCA dynamics
becomes highly disordered.
A rough measure of the emergence of macro-level structures can be pro-
vided by the compression percentage achieved by compression algorithms. The
higher the compression factor, the lower the randomness of the CA configura-
tion. Although this measure does not directly evidence long-range correlations, it
nevertheless provides meaningful information about the amount of structure of a
CA state. Fig. 7 shows typical results obtained with DCA with different number
of neighbors and transition function. The states of DCA have been measured
once the equilibrium have been reached
1
. As we can observe, when the ratio
λ
e
/λ
a
approaches a critical value
θ
1
the compression ratio
cr
abruptly increases.
This corresponds to the onset of structure in the system.
cr
reaches a maximum
approximately located at
λ
e
/λ
a
≈
0
.
05. Then it decreases till reaching again
the initial values, indicating the disappearance of macro-level structures in DCA
states. We observe that, for the first two DCA in Fig. 7,
cr
quickly decreases, on
the opposite of the last one, for which it seems that structures are still present
for higher values of the ration
λ
e
/λ
a
. In general, for all the experiments per-
formed, we observed that the critical value for the onset of structured patterns
is approximately
λ
e
/λ
a
≈
0
.
05.
The above similarity suggests that the same causes that determine the be-
havior of Benard cells also determine the behavior of DCA.
Without any perturbation, or in the presence of small one, each autonomous
component (a molecule or a DCA cell), acting asynchronously accordingly to
1
The compression algorithm used is that provided by usual compression utilities like
gzip
, at maximum compression level.
