Figure 1.
Schematic representation of the construction of the accessible states.
In this example we have a array (left panel) and we choose the embedding dimensions
and
. In the right panel we illustrate the construction of the states. We first obtain the sub-matrix corresponding to
and
that have as elements
and, after sorting, this sub-matrix leads to the state “0132”. We thus move to next sub-matrix
and
which have the elements
and that, after sorting, leads to the state “1023”. The last two remaining matrices lead to the states “1230” and “0132”. Finally, we estimate the probabilities
, that are,
,
and
which are then used in the equations (1) and (2), leading to
and
.
Figure 2.
Examples of fractal surfaces obtained through the random midpoint displacement method.
These are surfaces (
) for different values of the Hurst exponent
. For easier visualization, we have scaled the height of the surfaces in order to stay between
and
. We note that for small values of
the surfaces display an alternation of peaks and valleys (anti-persistent behavior) much more frequent than those one obtained for larger values of
. For larger values of
, the surfaces are smoother reflecting the persistent behavior induced by the value of
.
Figure 3.
Dependence of the complexity-entropy causality plane on Hurst exponent h.
We have employed fractal surfaces of size (
). In (a) we plot
and
versus
for the embedding dimensions
and
(circles) and also for
and
(squares). We note the invariance of the index against the rotation
and
. In (b) we plot the diagram for
. We observe changes in the scale of
and
caused by the increasing number of states. In both cases, as
increases the complexity
also increases while the permutation entropy
decreases. This behavior reflects the differences in the roughness shown in Fig. 2. For values of
the surface is anti-persistent which generates a flatter distribution for the values of
leading to values of
and
closer to the aleatory limit (
and
). For values of
there is a persistent behavior in the surfaces heights which generates a more intricate distribution of
and, consequently, values of
and
that are closer to the middle of the causality plane (region of higher complexity).
Figure 4.
Characteristic textures of a lyotropic liquid crystal at different temperatures and phases.
The lyotropic system used here is a mixture of potassium laurate , decanol
and deuterium oxide
– suitable concentrations in order to get a isotropic
nematic
isotropic phase sequence [38]. These images were constructed by observing the optical microscopy of a flat capillary which contains the mixture at different temperatures. Here, we have used the average value of the pixels of the three layers (RGB) of the original image and a rescaled temperature.
Figure 5.
Dependence of the entropic indexes on the temperature of a lyotropic liquid crystal.
We plot versus the temperature in (a) and
versus the temperature in (b), where we employ
. Figures (c) and (d) present the results for
and
, and also for
and
. The different shaded areas represent the different liquid crystal phases. Note that the phase transitions are properly identified in all cases. Due to the asymmetry of the elongated capillary tube where the liquid crystal sample is placed,
and
present slight differences under the rotation
and
.
Figure 6.
Complexity-entropy causality plane evaluated for several liquid crystal textures [[39]].
Here, we have used the averaged pixel values of the three layers (RGB) of the original image and and
. The image sizes are about
pixels. We note that each texture has a unique position in the causality plane which indicates that the permutation quantifiers are capable of differentiate not only transitions involving the isotropic phase, but also smoother phase transitions. We further observe that some high ordered phase such as the blue phase are located at the central part of the causality plane (region of higher complexity), while other phases which present a large number of defects such as the Smectic B and C are closer to the aleatory limit (
and
).
Figure 7.
Examples of Ising surfaces for three different temperatures.
These surfaces were obtained after Monte Carlo steps for three different temperatures: below
, at
and above
. In these plots, the height values were scaled to stay between
and
. We note that for temperatures higher or lower than
, the surfaces exhibit an almost random pattern. For values of the temperature closer to
the surfaces exhibit a more complex pattern, reflecting the long-range correlations that appear among the spin sites during the phase transition.
Figure 8.
Dependence of the entropic indexes on the reduced temperature for Ising surfaces.
(a) The permutation entropy and (b) the complexity measure
versus the reduced temperature for
and
, and also for
and
. We note invariance of indexes under the rotation
and
. (c) A 3d visualization of the Ising model phase transition when considering
. The gray shadows represent the dependences of
on
and of
on
.
Figure 9.
Dependence of the entropic indexes on the number of Monte Carlo steps.
Here, denotes the number of Monte Carlo steps and the reduced temperatures are indicated in the plots. In (a) we show
versus
and in (b)
versus
for
. We note the stability of both indexes after
Monte Carlo steps.