![All time series meet the conditions to be described as chaotic, so they form two chaotic systems [35,36,38,39,40,41,42,43,44,45,46,47]: the urban meteorology system (T, WS, and RH) and the pollutant system (PM10, PM2.5, and CO). The results of the calculations are in Table A1, Table A2 and Table A3 of Appendix A.](https://www.researchgate.net/profile/E-Mera/publication/389918642/figure/fig4/AS:11431281325951827@1743015112878/All-time-series-meet-the-conditions-to-be-described-as-chaotic-so-they-form-two-chaotic_Q320.jpg)
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Academic Editor: David F. Plusquellic
Received: 20 January 2025
Revised: 5 March 2025
Accepted: 8 March 2025
Published: 17 March 2025
Citation: Pacheco, P.; Mera, E.;
Navarro, G.; Polo, S. Interaction
Between Maximum Entropies of
Urban Meteorology and Pollutants:
Effects on Relative Humidity and
Temperature in the Boundary Layer of
a Basin Geomorphology. Atmosphere
2025,16, 337. https://doi.org/
10.3390/atmos16030337
Copyright: © 2025 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license
(https://creativecommons.org/
licenses/by/4.0/).
Article
Interaction Between Maximum Entropies of Urban Meteorology
and Pollutants: Effects on Relative Humidity and Temperature in
the Boundary Layer of a Basin Geomorphology
Patricio Pacheco 1, *, Eduardo Mera 1, Gustavo Navarro 2and Steicy Polo 1
1Departamento de Física, Facultad de Ciencias Naturales, Matemáticas y Medio Ambiente,
Universidad Tecnológica Metropolitana, Las Palmeras 3360, Ñuñoa, Santiago 7750000, Chile;
emera@utem.cl (E.M.); spolo@utem.cl (S.P.)
2Departamento de Ciencias Exactas, Facultad de Ingeniería, Arquitectura y Diseño,
Universidad de San Sebastián, Bellavista 7, Recoleta, Santiago 8420000, Chile; gustavo.navarro@uss.cl
*Correspondence: patricio.pachecol@utem.cl
Abstract: Using chaos theory, maximum entropies are calculated for 108 time series, each
consisting of 28,463 hourly data of urban meteorology and pollutants. The series were mea-
sured with standardized and certified instruments (EPA) in six locations at different heights
and in three periods (2010/2013, 2017/2020, and 2019/2022) in a basin geomorphology.
Each urban meteorology series corresponds to relative humidity (RH), temperature (T), and
wind speed magnitude (WS), and each pollutant series corresponds to 10
µ
m particulate
matter (PM
10
), 2.5
µ
m particulate matter (PM
2.5
), and carbon monoxide (CO). These pollu-
tants are in the top three places of presence in the studied geomorphology and in incidence
in population diseases. From the calculated entropies, a quotient is constructed between
the entropies of each of the first two urban meteorology variables (RH and T) and the sum
of maximum entropies of the time series of anthropogenic pollutants, demonstrating the
gradual decay in time of the quotient that is dominated by the maximum entropies of the
pollutants. The latter leads to a more excited and warm boundary layer, due to thermal
transfers, which makes it more unpredictable, increasing its capacity to contain water. It is
verified that the diffusion is anomalous with alpha < 1 and that the contamination has a
high probability, using a heavy-tailed probability function, of causing extreme events by
influencing urban meteorology.
Keywords: relative humidity; chaos; entropy; enthalpy; heavy-tail probability
1. Introduction
Various investigations address meteorology and atmospheric pollution from the per-
spective of chaos theory. From the point of view of meteorology, in [
1
], it is indicated that
atmospheric flows, which are a case of turbulent fluid flows, show fractal fluctuations on
all space–time scales, from the turbulence scale of mm/sec to climatic scales of thousands
of kilometers/years [
1
]. Fractal fluctuations show a form of inverse power law, meaning
long-range correlations described as dynamical systems having critical points as an at-
tractor in their time evolution (self-organized criticality), and are present in dynamical
systems in nature. These systems are revealed with sensitive dependence on the initial
condition (deterministic chaos) in restricted precision computer elaborations of nonlinear
mathematical models of real-world dynamical systems [
1
]. While the self-similar nature
of atmospheric flows has been analyzed for some time, the correct physical mechanism
has not yet been identified. Basic physical concepts of nonlinear dynamics and chaos need
Atmosphere 2025,16, 337 https://doi.org/10.3390/atmos16030337
Atmosphere 2025,16, 337 2 of 30
to be developed and incorporated into classical meteorology for more realistic predictive
models of weather and climate [
1
]. On the other hand, outdoor air pollution has a high
impact on human health. This drives the search for a better understanding of the dynamics
and statistics of measured air pollution concentrations, including temporal variations in
observed concentrations and spatial heterogeneities. Analysis of extensive measured data
from Europe [
2
] resulted in the construction of probability density functions (PDFs) of
air pollution concentrations which were shown to be highly dependent on spatial loca-
tion and pollutant type [
2
]. Thus, time-series data from 3544 different monitoring sites
showed that the PDFs of nitric oxide (NO), nitrogen dioxide (NO
2
), and particulate matter
(PM
10
and PM
2.5
) concentrations exhibit generally heavy tails. These are approximated
asymptotically by q-exponential distributions for a given entropic index q and width pa-
rameter (
λ
). The power law parameter (q) and width parameter (
λ
) vary widely for different
spatial locations. Data analysis allows in [
2
] to construct a map showing the relevance of q
and
λ
in a given region. The resulting spatial patterns can be correlated with properties
of the regional geography. Results are also obtained for typical time scales associated
with dynamic behavior. Another study [
3
] analyzed the definition, consequences, and
applications of an entropy, called Sq, that generalizes the usual Boltzmann–Gibbs entropy
SBG
(S1 = SBG),
the basis of usual statistical mechanics, which is known to be applicable
whenever ergodicity is satisfied at the microscopic dynamical level. Such an entropy Sq is
based on the notion of q-exponential and exhibits properties that are not shared by other
available alternative generalizations of SBG. The thermodynamics arising in this way is
generically non-extensive [
3
]. This approximation describes very well a vast class of natural
and artificial systems that are not ergodic or close to it [4].
The study presented here includes concepts and ideas indicated above, but it is carried
out on a basin geomorphology and is based on measurements to which the chaos theory
is applied [
1
]. This gives rise to the basic question that guides the authors (Pacheco et al)
in this article: is it possible, based exclusively on measurements, with data records (using
calibrated, certified, and standardized instruments according to EPA), in the form of time
series, with variables to be measured without boundary condition restrictions, to reveal
that the nature of the processes that have been incubated is chaotic, thus favoring entropic,
dynamic, anomalous subdiffusion and heavy-tailed statistics?
For this purpose, the measuring instruments were located at a height between 4 and
10 m (from the ground) depending on the communes. Each commune was treated, in a
first approximation, as an island up to a radius of 5 km. Each of the three measurement
periods covered 3.25 years, and the measurement periods were 2010/2013, 2017/2020,
and 2019/2022. In each of the six commons studied, three urban meteorological variables
(temperature (T), wind speed magnitude (WS), and relative humidity (RH)) and three
atmospheric pollutant variables (10
µ
m particulate matter (PM
10
), 2.5
µ
m particulate matter
(PM
2.5
), and carbon monoxide (CO)) were measured as hourly time series. This gives a
total of 36 time series containing 1,024,668 data for each period. In total, 3,074,004 data were
involved in the three periods. The measurement area corresponds to 641 km
2
and is subject
to increasing processes of intensive high-rise construction, population growth that currently
borders on 7,500,000 (data recording area), polluting vehicles, built-up area, reduction of
agricultural areas and green areas, drought, low relative humidity, piping and channeling of
water courses, etc. Through chaos theory, the chaotic parameters are calculated for each time
series:
λ
(Lyapunov coefficient) > 0,
0<DC(correlation dimension) < 5,
S
K
(Kolmogorov
entropy) > 0, 0.5 < H (Hurst coefficient) < 1, LZ (Lempel–Ziv complexity) > 0, D (fractal
dimension) > 0, <
∆
I> (information loss) < 0, demonstrating that the 108 time series are
chaotic. From the Kolmogorov entropy, calculated for the time series, the quotient is
constructed between the sum of the Kolmogorov entropies of the relative humidity (of each
Atmosphere 2025,16, 337 3 of 30
commune and for each period) and the sum of the Kolmogorov entropies of the pollutants
(of each commune and for each period), S
K,RH
/S
K,P
(same procedure for temperature),
which allows us to show the following in a relatively simple way:
1.
The interactive decay, as the periods advance toward the present, of the quotient.
This decay is compatible with what has been observed (of the relative humidity with
respect to the pollutants and of the temperature with respect to the thermal flow of
the pollutants).
2. The presence of entropic dynamic
3.
The manifestation and permanence of a subdiffusive phenomenon in the
studied basin.
4. The presence of statistics of extreme events through the heavy-tail analysis.
1.1. Basic Concepts
1.1.1. Water in the Atmosphere
Although the amount of water contained in the atmosphere (in the form of vapor,
clouds and small ice crystals) constitutes a minimal part of the amount existing on the planet,
of the order of 0.001%, its importance is vital: it participates in temperature regulation
processes, in the water cycle, in climatic phenomena, and even in natural disasters [
5
].
This very small proportion of water in the atmosphere is at the center of the processes that
determine climate, the hydrological cycle, atmospheric chemistry, and the development of
life. The main form of atmospheric water is water vapor, and the quantity of this content in
the air is called humidity. Although not as visible as liquid or solid forms (clouds, fog, rain,
snow, and hail), water vapor is always present in the atmosphere, even in deserts.
The behavior of water in the atmosphere; what its quantity depends on; when and
where it will fall; whether it will manifest itself as rain, snow, or frost; and whether there
will be too little or too much, are issues that have always concerned humans. Over time,
progress has been made in its study, although it has also introduced greater complexities,
such as air pollution, changes in land use, and water catchment areas, to name a few.
The troposphere is the lower part of the atmosphere (from 0 to approximately 12 km),
where climate and life develop. It contains three-quarters of the mass of the atmosphere and
around 99% of atmospheric water; its depth varies with latitude and the season of the year.
Solar energy does not heat this layer appreciably, so its heating is due to radiation coming
from the Earth’s surface. Its temperature therefore decreases with height, approximately
6.5
◦
C per kilometer [
6
], corresponding to the thermal gradient of the standard atmosphere
in a static state. Deviations from this value cause thermal convection.
1.1.2. Atmospheric Composition
Although the concentration of water vapor in the atmosphere is highly variable
spatially and temporally, its average proportion in a mixed volume of air is of the order of
1 percent, so it can be considered the third most abundant gas in the atmosphere. Dry air [
7
]
is composed of nitrogen (78.08%), oxygen (20.88%), and argon (0.93%). About 0.1% of the
total volume corresponds to other greenhouse gases, such as carbon dioxide, methane,
nitrous oxide, and ozone. The atmosphere also contains aerosols (small solid and liquid
particles in suspension) that interact with solar and terrestrial radiation, depending on their
spatial location, chemical composition, and concentration. In addition, half of atmospheric
water, present in its liquid and solid phases, is concentrated in the first two kilometers of
altitude. This article, when analyzing water in the atmosphere, will refer to that which is
present in the troposphere and in the boundary layer in particular (from 1 to 2 km).
When the vapor entering the atmosphere reaches the saturation point of the air (that
is, when it contains all the water vapor that it can contain at that temperature), the excess
molecules condense, forming droplets (dew, rain, and fog) or ice crystals. Evaporation and
Atmosphere 2025,16, 337 4 of 30
condensation are continuous processes that generate and redistribute heat, transporting it
and transforming it all the time between vapor, liquid, and ice crystals, depending on the
temperature and pressure of the air. The higher the temperature of the air, the greater its
capacity to contain water vapor. Hence, the concept of relative humidity serves to indicate,
in percentage, the amount of water vapor that a portion of air contains, in relation to what
it can contain given its temperature. A relative humidity of 100 percent indicates that that
portion of air cannot contain more vapor. If the temperature of the air drops, the amount of
vapor that it can contain decreases, and all the excess will condense. If the temperature of
the saturated air increases, its capacity to contain water vapor will increase and the relative
humidity will decrease.
Since the atmosphere is a small reservoir of water compared to oceans and continents,
transfer to other reservoirs must be rapid. The turnover time of all atmospheric water
is estimated to be about ten days [
8
,
9
]. Compared to the 10-year atmospheric lifetime of
methane and the more than 50 years of carbon dioxide, the short average life of water does
not allow for a more homogeneous mixture, resulting in an uneven global distribution.
1.1.3. Pollution
Some gases penetrate the drops, changing the characteristics of rainwater. This is the
case of acid rain, which negatively impacts the sites where it falls, contaminating lakes and
other ecosystems and damaging buildings [
10
–
13
]. The gases involved in these reactions
are the product of human activity, mainly sulfuric acid and nitric acid.
Pollution also has climatic effects, since the excess of anthropogenic aerosols (produced
by man), which act as condensation nuclei, causes an increase in the number of drops
formed. This results in an increase in the albedo of the cloud, as well as the formation
of smaller drops (less influenced by gravity), thus decreasing the possibility of rain and
increasing the lifetime of the cloud.
1.1.4. Climate Change and Atmospheric Water
Water in the atmosphere plays a very active role in climate: it participates in the entry
and exit of energy, the redistribution of heat in the atmosphere, and the movement of water
between distant regions. Any spatial or temporal modification of the water content in the
atmosphere results in a change in the conditions of the Earth’s surface. Droughts, floods,
hurricanes, and tornadoes are related to the amount of water in the atmosphere and its
behavior, hence the interest in any signal that may indicate long-term changes [
13
–
16
].
Global data from the last 30 years show an increase in precipitable water throughout the
world except in a portion of Canada, where it decreased. It is not unusual for temperature
increases to result in increases in evaporation; but all the consequences that this can have
when combining situations such as greater cloud formation with smaller droplets that
do not precipitate are not clear. The change in rainfall patterns in different regions has
consequences for the economy and the lives of many people.
1.1.5. Relative Humidity (RH)
Intensive variables are those whose value does not depend on the size or quantity
of matter in the system (they have the same value for a system as for each of its parts
considered as subsystems of the same), as is the case of temperature and pressure. In
thermodynamics, a system is said to be in thermodynamic equilibrium when the inten-
sive variables that describe its state do not vary over time. If the system is not isolated,
thermodynamic equilibrium is defined in relation to its surroundings.
Relative humidity (RH or
Φ
) is the ratio of the partial pressure of water vapor (p
H20
) in
the mixture to the equilibrium vapor pressure of water (
p*
H2O
) at a given temperature. Rela-
tive humidity depends on the temperature and pressure of the system of
interest [17–21].
Atmosphere 2025,16, 337 5 of 30
The same amount of water vapor produces a higher relative humidity in cold air than in
warm air. One parameter of the relationship is the dew point. The correspondence between
the pressures on a flat surface of pure water at a given temperature is written as follows:
RH =Φ=pH2O/p*
H2O(∼according to ideal gas law ∼nv
nvs ⌊T
), (1)
RH is usually expressed as a percentage; a higher percentage means that the air–water
mixture is more humid. At 100% relative humidity, the air is saturated, cannot hold more
vapor, and, as noted, is at its dew point. Pollutants transfer thermal energy to the water
vapor system of the atmosphere, adjacent to the ground, and which has a certain RH. H
2
O
molecules have the energy to disperse toward volumes of lower thermal energy, decreasing
their diffusion possibilities and favoring their concentration. The polluting system can
cause a decrease in relative humidity, contributing to aridity in a basin morphology. In
this geomorphology, pollutants experience a subdiffusive process, since their concentration
grows steadily over time due to the permanent contribution of heat from human activity
(anthropogenic causes), which stabilizes a drying dynamic. Due to conservation of mass,
water vapor cannot disappear; the concentration and rehydration of geographies and
fragments of atmospheric layers become chaotic and unpredictable; and its emergence can
generate high-risk events.
If the temperature of the saturated air increases, due to the effect of the thermal transfer
of the same pollutants, its capacity to contain water vapor will increase and the relative
humidity will decrease. From a dynamic perspective, if the entropic forces due to pollutants
are greater than those of urban meteorology and RH in particular, they can favor an increase
in environmental dryness. When considering the forces of an entropic nature (2) originated
by the entropy surface, S (x, y, z; Figure 1) [22–25], we have the following:
→
F→
X0=T∇S→
X|→
X=→
X0, (2)
Atmosphere 2025, 16, x FOR PEER REVIEW 6 of 33
Figure 1. Representation of an entropy surface, S (x, y, z), which has maxima of entropy of time
series that can be measurements of urban meteorological variables or of pollutants in different loca-
tions. From S(x,y,z) the entropic forces, we can calculate the F
.
The pressure is p = F/A, so then it is possible to write using (2), following the notation
of (1), and we obtain the following expression:
RH =
∗=
∗=∇
(∇)
∗, (3)
Expression that gives a relationship between relative humidity and the gradient of
the entropic surface. Assuming that the process occurs in a spherical volume, with the
operator ∇ only having radial dependence (r), we obtain the following:
RH =
∗=
∗=
∗~∆
(∆)
∗~∆
∆
(∆)
∗
∆ ~,
(),
∗, (4)
which indicates for RH a dependence, considering the assumptions indicated, on the Kol-
mogorov-type entropies [26–30].
As a note, if we perform the calculation, we obtain the following:
∇S,
S, =S,∇S,−S,∇S,
S,=T
∇S,−S,
S, T
∇S,
T
S, =F
−S,
S, F
T
S,
T
>0,S,>0→∆
=F
−S,
S, F
=<0
,F
is greater
=0
,not observed
> 0
,F
is greater
where F
is the entropic force associated with relative humidity, and F
is the entropic
force associated with pollutants
If the volume fraction of the vapor is saturated, at dew point, its entropy is minimal.
It will experience a phase transition releasing heat. Under standard atmospheric condi-
tions (ideal). As the contaminant system increasingly invades the boundary layer of the
atmosphere with sufficient energy to affect it, the atmospheric air is more likely to expand,
there is more space to contain water vapor. This phenomenon is not static in time, it is
extremely dynamic, the entropic layers are very variable, very rough. The entropic force
Figure 1. Representation of an entropy surface, S (x, y, z), which has maxima of entropy of time series
that can be measurements of urban meteorological variables or of pollutants in different locations.
From S(x,y,z) the entropic forces, we can calculate the →
FENTROPIC.
Atmosphere 2025,16, 337 6 of 30
The pressure is p = F/A, so then it is possible to write using (2), following the notation
of (1), and we obtain the following expression:
RH =pH2O
P*
H2O
=F
AH2O
F
A*
H2O
=
∇SH2O
(∇S)*
H2O
, (3)
Expression that gives a relationship between relative humidity and the gradient of the
entropic surface. Assuming that the process occurs in a spherical volume, with the operator
∇only having radial dependence (r), we obtain the following:
RH =pH2O
P*
H2O
=F
AH2O
F
A*
H2O
=
∂SH2O
∂r
∂S
∂r*
H2O
∼∆SH2O
(∆S)*
H2O
∼
∆SH2O
∆t
(∆S)*
H2O
∆t
∼SK,H2O
(S)*
K,H2O
(4)
which indicates for RH a dependence, considering the assumptions indicated, on the
Kolmogorov-type entropies [26–30].
As a note, if we perform the calculation, we obtain the following:
∇SK,RH
SK,P =SK,P∇SK,RH −SK,RH ∇SK,P
SK,P2=T∇SK,RH −SK,RH
SK,P T∇SK,P
TSK,P
=
→
FRH −SK,RH
SK,P
→
FP
TSK,P
T>0, SK,P >0→→
∆=→
FRH −SK,RH
SK,P
→
FP=
<
→
0 , →
FPis greater
=→
0 , not observed
>
→
0 , →
FRH is greater
where
→
FRH
is the entropic force associated with relative humidity, and
→
FP
is the entropic
force associated with pollutants
If the volume fraction of the vapor is saturated, at dew point, its entropy is minimal.
It will experience a phase transition releasing heat. Under standard atmospheric condi-
tions (ideal). As the contaminant system increasingly invades the boundary layer of the
atmosphere with sufficient energy to affect it, the atmospheric air is more likely to expand,
there is more space to contain water vapor. This phenomenon is not static in time, it is
extremely dynamic, the entropic layers are very variable, very rough. The entropic force is
very variable. This increases the low predictability of the events. What formula
→
∆
states is
that an entropic force of urban meteorology is related to the entropic force of pollutants
(and vice versa) and to C
K
. Equilibria have been lost and the systems, which have entered
into irreversible processes, are complex, connected and chaotic. Equation (4) is a very crude
approximation of the current processes in the atmosphere. But it is possible to show from
the entropy ratio the turbulence and disorder of the interactive systems.
1.1.6. Thermodynamic Aspects Associated with Relative Humidity
Thermodynamic Principles
RH is a thermodynamically controlled property of air. Thermodynamics provides
substance to psychometry [
31
], the study of the transformations of air necessary for its use
in a multitude of fields: food preservation in chambers, air conditioning of premises, drying
processes and drug manufacturing, metrology, explosive atmospheres, environments in
computer rooms, textile industry, clean rooms, etc. [32,33].
Atmosphere 2025,16, 337 7 of 30
There are four laws of thermodynamics, which govern the behavior of energy and
matter and are basic to understanding the behavior of air when it reaches saturation and a
relative humidity of 100% [34].
The first law of thermodynamics states the conservation of energy in the context of
thermodynamics and indicates that if work (W) is performed on a system or it exchanges
heat (Q) with another, the internal energy (U) of the system will change. This law is
important to understand the exchange of energy that occurs in the atmosphere when the
air reaches saturation.
The second law of thermodynamics states that in every irreversible process, the entropy
of the universe increases: Isolated systems, when evolving, tend to become disordered,
never ordered. Entropy is a measure of the disorder or randomness of a system. When the
air is saturated with humidity, any addition of humidity will result in the condensation of
water vapor into liquid water, which will release latent heat and increase the entropy of
the system.
The Gibbs free energy equation [G] describes the relationship between changes in
enthalpy, H, entropy, S, and temperature, T of a system, as dG = dH −TdS.
If the variation is: less than 0, the reaction is spontaneous.
greater than 0, the reaction is not spontaneous (that is, work needs to be performed for
it to happen).
is equal to 0, the reaction is in equilibrium.
As the relative humidity of the air reaches 100%, which occurs at the dew point
temperature, the moist air reaches its maximum because the specific volume of the air
increases with its moisture content. This causes the Gibbs free energy of the system to reach
its minimum value, leading to a state of equilibrium.
Latent Heat
This is the amount of energy required for a substance to change phase, from solid to
liquid (heat of fusion) and from liquid to gas (heat of vaporization) [
6
]. It should be noted
that this energy in the form of heat is invested in the phase change and not in an increase in
temperature. Air is saturated in a given volume when it can no longer support water vapor.
Any excess beyond this point results in phenomena such as dew, rain, fog.
When a phase change occurs (i.e., when a substance passes from one region of its
phase diagram to another), Figure 2:
Atmosphere 2025, 16, x FOR PEER REVIEW 8 of 33
Figure 2. Phase diagram of water. A: triple point; B: melting point; C: boiling point; D: critical pres-
sure; E: critical point.
The entropy of that substance changes, even if its temperature remains the same. A
substance in solid phase has a low entropy; in liquid phase, a medium entropy; in gas
phase, a high entropy. The entropy of fusion increases when a substance melts. This is
almost always positive, since the degree of disorder increases in the transition from an
organized crystalline solid to the disorganized structure of a liquid; the only known ex-
ception is Helium. It is denoted as ΔS fusion and usually expressed in J mol−1 K−1.
2. Materials and Methods
2.1. Study Area
The city of Santiago is located at 33.5° S and 70.8° W. It contains a population of
8,420,000 inhabitants, representing 40% of the total population of the country, on a surface
of approximately 641 km2. It is located in the middle of the country, at a height of about
520 m.a.s.l. The altitude above sea level increases from west to east. It is surrounded by
two mountain chains: the Andes and the Coastal Mountain range. Its climate is Mediter-
ranean (Figure 3). The driest and warmest months are from December to February, his-
torically reaching maximum temperatures of around 35 °C in the shade (air temperature
in the sun); however, these temperatures are currently exceeded. Given its topography
and the dominant meteorological conditions, there is, in general, a strong horizontal and
vertical dispersion of pollutants generated by an important number of sources in the city
(heating, vehicles, industries, etc.), especially during fall (20 March–21 June) and winter
(21 June–23 September). The emissions have a tendency to increase given the also increas-
ing population density, which implies an increase in fixed and mobile sources. In addition,
the number of vehicles has increased rapidly in recent years.
Figure 2. Phase diagram of water. A: triple point; B: melting point; C: boiling point; D: critical
pressure; E: critical point.
Atmosphere 2025,16, 337 8 of 30
The entropy of that substance changes, even if its temperature remains the same. A
substance in solid phase has a low entropy; in liquid phase, a medium entropy; in gas phase,
a high entropy. The entropy of fusion increases when a substance melts. This is almost
always positive, since the degree of disorder increases in the transition from an organized
crystalline solid to the disorganized structure of a liquid; the only known exception is
Helium. It is denoted as ∆S fusion and usually expressed in J mol−1K−1.
2. Materials and Methods
2.1. Study Area
The city of Santiago is located at 33.5
◦
S and 70.8
◦
W. It contains a population of
8,420,000 inhabitants, representing 40% of the total population of the country, on a sur-
face of approximately 641 km
2
. It is located in the middle of the country, at a height
of about
520 m.a.s.l.
The altitude above sea level increases from west to east. It is sur-
rounded by two mountain chains: the Andes and the Coastal Mountain range. Its climate is
Mediterranean (Figure 3). The driest and warmest months are from December to February,
historically reaching maximum temperatures of around 35
◦
C in the shade (air temperature
in the sun); however, these temperatures are currently exceeded. Given its topography
and the dominant meteorological conditions, there is, in general, a strong horizontal and
vertical dispersion of pollutants generated by an important number of sources in the city
(heating, vehicles, industries, etc.), especially during fall (20 March–21 June) and winter
(21 June–23 September).
The emissions have a tendency to increase given the also increas-
ing population density, which implies an increase in fixed and mobile sources. In addition,
the number of vehicles has increased rapidly in recent years.
Atmosphere 2025, 16, x FOR PEER REVIEW 9 of 33
Figure 3. Santiago de Chile is located in a basin geomorphology.
2.2. Kolmogorov Entropy.
Using the Chaos Data Analyzer (CDA) program [35,36], 108 time series (each with
28,463 data) were processed, representing a total of 3,074,004 data points, corresponding
to variables of temperature (T), wind speed magnitude (WS), relative humidity (RH), con-
centration of particulate maer of 10 µm (PM10) and 2.5 µm (PM2.5), and carbon monoxide
(CO). For the series to be chaotic, the fundamental parameters to be calculated must satisfy
the following:
λ (Lyapunov coefficient) > 0, 0 < DC (correlation dimension) < 5, SK (Kolmogorov
entropy) > 0, 0.5 < H (Hurst coefficient) < 1, LZ (Lempel–Ziv complexity) > 0, D (fractal
dimension)> 0, <ΔI> (information loss) < 0.
It is common that in the ordered pairs that make up the data sequence of the time
series, there are missing measurement discrepancies. This happens, for example, due to
an electrical fault, a defect in the measuring equipment, etc. The missing data in a series
were completed using the Kriging technique [37] (this is indicated in the flowchart in Fig-
ure 4). Figure 4 is a flowchart showing the process followed to apply chaotic analysis soft-
ware (CDA) [36] to time series.
Figure 3. Santiago de Chile is located in a basin geomorphology.
2.2. Kolmogorov Entropy
Using the Chaos Data Analyzer (CDA) program [
35
,
36
], 108 time series (each with
28,463 data) were processed, representing a total of 3,074,004 data points, corresponding
to variables of temperature (T), wind speed magnitude (WS), relative humidity (RH),
concentration of particulate matter of 10
µ
m (PM
10
) and 2.5
µ
m (PM
2.5
), and carbon
monoxide (CO). For the series to be chaotic, the fundamental parameters to be calculated
must satisfy the following:
Atmosphere 2025,16, 337 9 of 30
λ
(Lyapunov coefficient) > 0, 0 < D
C
(correlation dimension) < 5, SK (Kolmogorov
entropy) > 0, 0.5 < H (Hurst coefficient) < 1, LZ (Lempel–Ziv complexity) > 0, D (fractal
dimension) > 0, <∆I> (information loss) < 0.
It is common that in the ordered pairs that make up the data sequence of the time
series, there are missing measurement discrepancies. This happens, for example, due to an
electrical fault, a defect in the measuring equipment, etc. The missing data in a series were
completed using the Kriging technique [
37
] (this is indicated in the flowchart in Figure 4).
Figure 4is a flowchart showing the process followed to apply chaotic analysis software
(CDA) [36] to time series.
Atmosphere 2025, 16, x FOR PEER REVIEW 10 of 33
Figure 4. All time series meet the conditions to be described as chaotic, so they form two chaotic
systems [35,36,38–47]: the urban meteorology system (T, WS, and RH) and the pollutant system
(PM10, PM2.5, and CO). The results of the calculations are in Tables A1–A3 of Appendix A.
3. Results
3.1. Temperatures
The ability of air to hold water vapor depends entirely on its temperature. As air
temperature changes, the ability to hold moisture increases or decreases, thus affecting
relative humidity. Table 1 is a comparative chart, by measurement periods, of temperature
and relative humidity.
Figure 4. All time series meet the conditions to be described as chaotic, so they form two chaotic
systems [
35
,
36
,
38
–
47
]: the urban meteorology system (T, WS, and RH) and the pollutant system
(PM10, PM2.5, and CO). The results of the calculations are in Tables A1–A3 of Appendix A.
Atmosphere 2025,16, 337 10 of 30
3. Results
3.1. Temperatures
The ability of air to hold water vapor depends entirely on its temperature. As air
temperature changes, the ability to hold moisture increases or decreases, thus affecting
relative humidity. Table 1is a comparative chart, by measurement periods, of temperature
and relative humidity.
Table 1. Average total temperature and relative humidity by commune and periods. La Florida,
EML, masl: 784 m. Las Condes, EMM, masl: 709 m. Santiago Parque O’Higgins, EMN, masl: 570 m.
Pudahuel, EMO, masl: 469 m. Puente Alto, EMS, masl: 698 m. Quilicura, EMV, masl: 485 m.
EML EMM EMV EMN EMS EMO Average by
Commune
2010–2013
T (◦C) 15.4 15.86 15.80 15.34 14.70 16.80 15.65
RH (%) 58.20 58.13 57.34 60.22 60.07 57.52 58.58
2017–2020
T (◦C) 16.12 15.57 16.85 16.17 15.53 16.78 16.17
RH (%) 55.31 55.00 58.95 57.31 56.07 59.22 56.98
2019–2022
T (◦C) 16.10 14.70 15.50 16.05 15.42 15.31 15.51
RH (%) 56.20 57.83 61.20 60.84 56.96 61.32 59.10
According to Table 1, two periods, 2010–2013 and 2017–2020, show an increase in the
average temperature and a decrease in relative humidity. This is consistent with what was
previously stated: if the temperature of saturated air increases, its capacity to contain water
vapor will increase and relative humidity will decrease. The period of 2019–2022 contains
the most complex moment of the coronavirus pandemic and a strong confinement of the
population with a drastic decrease in activity (transport, industries, etc.). This favored a
decrease in temperature and an improvement in relative humidity, as shown in Table 1.
It is possible to interpret this period, considering that in 2023 Santiago de Chile fell to
sixth place as the warmest since 1950 (2023 was the warmest year since 1961), with a large
accumulation of water vapor in the boundary layer, with a decrease in relative humidity
and discounting phenomena such as El Niño, as a preamble to a relatively rainier 2024.
Based on the Hurst coefficients (H) and the fractal dimension (D) (Appendix A), H
decreases for T and for HR, and D increases for T and HR, indicating a greater complexity
and less predictability of future climatic events. They suggest that a system sensitive to
initial conditions is being created for the coming years, which may lead to a rainier year
(as occurred in Chile in 2024). Table 2allows us to visualize the decline in persistence
(H = Hurst exponent) of both the polluting system and the urban meteorology system with
an increase (2019–2022), in general, of the fractal dimension (D).
Table 2. Values of the Hurst exponent (H) and fractal dimension (D) for all the variables of interest,
the study commons, and the three periods (2010–2013, 2017–2020, and 2019–2022).
PM10 PM2.5 CO T HR WV
EML 2010–2013
H 0.967 0.973 0.959 0.989 0.991 0.976
D 1.033 1.027 1.041 1.011 1.009 1.024
2017–2020
H 0.922 0.963 0.933 0.915 0.942 0.975
D 1.078 1.037 1.067 1.085 1.058 1.025
Atmosphere 2025,16, 337 11 of 30
Table 2. Cont.
PM10 PM2.5 CO T HR WV
2019–2022
H 0.928 0.946 0.933 0.920 0.934 0.942
D 1.072 1.054 1.067 1.080 1.066 1.058
EMM 2010–2013
H 0.972 0.977 0.981 0.991 0.990 0.980
D 1.028 1.023 1.019 1.009 1.010 1.02
2017–2020
H 0.906 0.983 0.933 0.917 0.941 0.976
D 1.094 1.017 1.067 1.083 1.059 1.024
2019–2022
H 0.914 0.969 0.933 0.916 0.935 0.942
D 1.086 1.031 1.067 1.084 1.065 1.058
EMN 2010–2013
H 0.972 0.974 0.953 0.989 0.991 0.968
D 1.028 1.026 1.047 1.011 1.009 1.032
2017–2020
H 0.929 0.960 0.933 0.916 0.942 0.973
D 1.071 1.040 1.067 1.084 1.058 1.027
2019–2022
H 0.934 0.947 0.933 0.921 0.908 0.941
D 1.066 1.053 1.067 1.079 1.092 1.059
EMO 2010–2013
H 0.965 0.955 0.937 0.992 0.989 0.968
D 1.035 1.045 1.063 1.008 1.011 1.032
2017–2020
H 0.936 0.925 0.933 0.919 0.942 0.974
D 1.064 1.075 1.067 1.081 1.058 1.026
2019–2022
H 0.938 0.915 0.933 0.918 0.936 0.941
D 1.062 1.085 1.067 1.082 1.064 1.059
EMS 2010–2013
H 0.969 0.973 0.953 0.990 0.992 0.957
D 1.031 1.027 1.047 1.010 1.008 1.043
2017–2020
H 0.921 0.975 0.933 0.915 0.942 0.976
D 1.079 1.025 1.067 1.085 1.058 1.024
2019–2022
H 0.930 0.964 0.933 0.919 0.927 0.942
D 1.070 1.036 1.067 1.081 1.073 1.058
EMV 2010–2013
H 0.967 0.970 0.952 0.989 0.989 0.956
D 1.033 1.03 1.048 1.011 1.011 1.044
2017–2020
H 0.931 0.966 0.933 0.919 0.942 0.975
D 1.069 1.034 1.067 1.081 1.058 1.025
2019–2022
H 0.930 0.938 0.933 0.920 0.934 0.940
D 1.070 1.062 1.067 1.080 1.066 1.060
Persistence decreases, and complexity increases; thus, the system moves away from
regularity and becomes more complex and unpredictable. The average temperature of each
of the six study communes in each of the measurement periods is as follows: 2010/2013:
T = 15.64 ◦C, 2017/2020: T = 16.20 ◦C, 2019/2022: T =15.51 ◦C.
Atmosphere 2025,16, 337 12 of 30
Using Equation (23) from [
48
] allows us to calculate the effect of chemical reactions
(
Sc
), thermal radiation (R), and heat dissipation through shear stress (
∅
ρcp
), summarized in
the general formula
γ=∅
ρcp+R+Sc
, which applies to each commune and is therefore
subindicated with a C:
γC=(Tl+1
i−Tl
i) + (−1.53 ×10−5Tl
i−1−2Tl
i+Tl
i+1
9+wTl
i−1−Tl
i+1
6)×3600◦C
h(5)
where w is the vertical component of the wind. The average of
γC
for each series of
28,463 data measured by commune as follows:
Av =Average (◦C/h) = γC=∅
ρcp+R+ScC
for the period 2019–2022 (Table 3):
Table 3. Variation in γCin 2019/2022.
Station EMS EML EMN EMO EMV EMM
Average (◦C/h) −0.09113202 −0.0680180 −0.0092004 0.0206236 0. 00662203 −0.10181307
Addition (◦C/h) −2593.89069 −1935.99887 −261.870346 587.009455 453.866082 −2897.90535
Average (K/h) 273.06 K/h 273.08 K/h 273.14 K/h 273.17 K/h 273.16 K/h 273.08 K/h
Height (masl) 485 784 520 469 698 709
From Table 4and its eighth column, the average temperature value is obtained,
15.47
◦
C, which is in the order of the average temperature values arising from the time
series (TTS) of the six communes (2019/2022 period), comparing them as follows:
∈r%=
TTS −15.47◦C
TTS
×100 ∼0.26% <5%
Table 4. Summary of
γC
according to locality (commune), height (referred to sea level), average
temperature, and the increase in temperature per period. The values in columns 6, 7, and 8 were
approximated. The values in columns 1–4 and 6–7 were extracted from [48].
Station H
(masl)
γC
(K/h)
S1,2010–2013
γC
(K/h)
S2,2017–2020
γC
(K/h)
S3,2019–2022
T + Av
(◦C)2010–2013
T + Av
(◦C)2017–2020 T+Av(◦C)2019–2022
EML 784 273.50 273.80 273.08 15.40 + 0.36 = 15.8 16.12 + 0.93 = 17.1 16.09 −0.0680 = 16.02
EMM * 709 273.54 274.00 273.08 15.86 + 0.40 = 16.3 15.57 + 0.62 = 16.2 14.69 −0.1018 = 14.60
EMV 698 273.60 274.10 273.16 15.77 + 0.50 = 16.3 16.85 + 0.66 = 17.5 15.50 + 0.0662 = 15.50
EMN 520 273.50 273.81 273.14 15.34 + 0.36 = 15.7 16.17 + 0.96 = 17.1 16.047 −0.092 = 16.04
EMS 485 273.60 274.10 273.06 14.69 + 0.42 = 15.1 15.53 + 0.94 = 16.5 15.42 −0.0911 = 15.33
EMO 469 273.80 274.10 273.17 16.77 + 0.60 = 17.4 16.80 + 0.80 = 17.6 15.31 + 0.0206 = 15.33
* The commune of Las Condes (EMM, belongs to the small group of communes with the highest income in Chile),
which has made mitigation investments.
The analysis according to
γC
, an indicator of human and natural activities, shows that
human activities decrease significantly if there is a drop in temperature.
3.2. Entropies
Table 5contains the calculation of the entropy for time series of relative humidity and
pollutants (PM
10
, PM
2.5
, and CO). There are 72 series, each containing 28,463 hourly data,
spread over three measurement periods: 2010/2013, 2017/2020, and 2019/2022.
Atmosphere 2025,16, 337 13 of 30
Table 5. S
K
Summary for pollutants and relative humidity in Santiago, Chile, for 2010/2013,
2017/20230, and 2019/2022 periods (Appendix A).
Parameters
Station
SK,P
2010/2013
SK,HR
2010/2013
SK,P
2017/2020
SK,HR
2017/2020
SK,P
2019/2022
SK,HR
2019/2022
EML
SK(1/h) 1.542 0.425 1.577 0.414 1.006 0.229
EMM
SK(1/h) 1.550 0.427 1.406 0.309 0.940 0.180
EMN
SK(1/h) 1.531 0.366 1.479 0.308 1.127 0.148
EMO
SK(1/h) 1.210 0.382 1.630 0.330 1.117 0.205
EMS
SK(1/h) 1.377 0.416 1.702 0.404 1.030 0.149
EMV
SK(1/h) 1.431 0.370 1.220 0.428 0.952 0.100
Table 5allows the construction of Table 6, which contains the quotients between the
maximum entropies of relative humidity and the sum of the maximum entropies of the
pollutants, by commune, for the 3 measurement periods:
Table 6. Santiago, Chile, comparative table of entropies quotients S
K,RH
/S
K,P
, 2010/2013, 2017/20230,
2019/2022 Period).
Parameters
Station
SK,HR/SK,P
2010/2013
SK,HR/SK,P
2017/2020
SK,HR/SK,P
2019/2022
EML 0.28 0.26 0.23
EMM 0.28 0.22 0.19
EMN 0.24 0.21 0.13
EMO 0.32 0.20 0.18
EMS 0.30 0.24 0.15
EMV 0.26 0.35 0.11
From Table 6, Figure 5is constructed. Figure 5shows the decay, according to the
periods studied, of the quotient between the maximum entropy of relative humidity and
the maximum entropies of the pollutants symbolically, SK,HR/SK,P.
Atmosphere 2025, 16, x FOR PEER REVIEW 15 of 33
Figure 5. Decay of the SK, HR ratio with SK, P.
The influence of the maximum entropy of pollutants on the maximum entropy of the
ambient temperature time series is increasing as the periods progress toward the present.
This thermodynamically affects the atmosphere, making it warmer and giving it a greater
capacity to contain water because, if T>T (heating of the atmosphere due to the effect
of pollutants), fractions of the atmosphere in the boundary layer expand, V>V, accord-
ing to the simplified model in Figure 6:
Figure 6. Simplified representation of the increase in water vapor retention by the atmosphere as
the temperature rises (T2 > T1)).
There is dependence on the initial conditions of the systems (HR, T, WS, and pollu-
tants) due to the high variability of human activity that makes them unpredictable. The
warming of the atmosphere becomes chaotic in its location, and it is persistent and in-
creasing. If, at some point, the appropriate temperature and pressure scenario occurs, the
Figure 5. Decay of the SK,HR ratio with SK,P .
Atmosphere 2025,16, 337 14 of 30
The influence of the maximum entropy of pollutants on the maximum entropy of the
ambient temperature time series is increasing as the periods progress toward the present.
This thermodynamically affects the atmosphere, making it warmer and giving it a greater
capacity to contain water because, if
T2>T1
(heating of the atmosphere due to the effect of
pollutants), fractions of the atmosphere in the boundary layer expand,
V2>V1
, according
to the simplified model in Figure 6:
Atmosphere 2025, 16, x FOR PEER REVIEW 15 of 33
Figure 5. Decay of the SK, HR ratio with SK, P.
The influence of the maximum entropy of pollutants on the maximum entropy of the
ambient temperature time series is increasing as the periods progress toward the present.
This thermodynamically affects the atmosphere, making it warmer and giving it a greater
capacity to contain water because, if T>T (heating of the atmosphere due to the effect
of pollutants), fractions of the atmosphere in the boundary layer expand, V>V, accord-
ing to the simplified model in Figure 6:
Figure 6. Simplified representation of the increase in water vapor retention by the atmosphere as
the temperature rises (T2 > T1)).
There is dependence on the initial conditions of the systems (HR, T, WS, and pollu-
tants) due to the high variability of human activity that makes them unpredictable. The
warming of the atmosphere becomes chaotic in its location, and it is persistent and in-
creasing. If, at some point, the appropriate temperature and pressure scenario occurs, the
Figure 6. Simplified representation of the increase in water vapor retention by the atmosphere as the
temperature rises (T2> T1).
There is dependence on the initial conditions of the systems (HR, T, WS, and pollutants)
due to the high variability of human activity that makes them unpredictable. The warming
of the atmosphere becomes chaotic in its location, and it is persistent and increasing. If, at
some point, the appropriate temperature and pressure scenario occurs, the localized water
discharge from the atmosphere can reach very high and risky levels. Figure 6shows how
relative humidity decreases as the measurement periods progress toward the present time
due to a polluted environment that promotes environmental dryness.
Table 7allows the construction of Table 8, which contains the quotients between
the maximum entropies of Temperature and the sum of the maximum entropies of the
pollutants, by commune, for the 3 measurement periods:
Table 7. S
K
summary for pollutants and temperature for Santiago, Chile, for 2010/2013, 2017/20230,
and 2019/2022 periods (Appendix A).
Parameters
Station
SK,P
2010/2013
SK,T
2010/2013
SK,P
2017/2020
SK,T
2017/2020
SK,P
2019/2022
SK,T
2019/2022
EML
SK(1/h) 1.542 0.409 1.577 0.355 1.006 0.175
EMM
SK(1/h) 1.550 0.409 1.406 0.377 0.940 0.182
EMN
SK(1/h) 1.531 0.426 1.479 0.366 1.127 0.172
EMO
SK(1/h) 1.210 0.375 1.630 0.184 1.117 0.180
EMS
SK(1/h) 1.377 0.395 1.702 0.357 1.030 0.168
EMV
SK(1/h) 1.431 0.384 1.220 0.171 0.952 0.153
Atmosphere 2025,16, 337 15 of 30
Table 8. Santiago, Chile, comparative table of entropies, S
K,T
/S
K,P
, for 2010/2013, and 2017/2020,
2019/2022 periods).
Parameters
Station
SK,T/SK,P
2010/2013
SK,T/SK,P
2017/2020
SK,T/SK,P
2019/2022
EML 0.265 0.225 0.174
EMM 0.264 0.268 0.194
EMN 0.278 0.247 0.153
EMO 0.310 0.113 0.161
EMS 0.287 0.210 0.163
EMV 0.268 0.140 0.161
Figure 7shows the decay, according to the periods studied, of the quotient between
the maximum entropy of the temperature and the maximum entropies of the pollutants
symbolically, S
K,T
/S
K,P
. Pollutants generate the highest thermal flows, affecting the ther-
modynamics of the boundary layer.
Atmosphere 2025, 16, x FOR PEER REVIEW 17 of 33
Figure 7. Decay of maximum temperature entropies due to the effect of maximum entropies of pol-
lutants; the dominant thermal effect is carried by the laer.
From Table 6, Figures 8–10 were constructed, showing the areas of influence of the
pollutants with respect to relative humidity: red indicates the strong dominance of pollu-
tion, and green indicates the enhanced activity of relative humidity. The data were rec-
orded in the city of Santiago de Chile, which is located in a basin geomorphology and
correspond to measurements carried out in six communes, located at different heights, in
the periods of 2010/2013, 2017/2020, and 2019/2022.
Figure 7. Decay of maximum temperature entropies due to the effect of maximum entropies of
pollutants; the dominant thermal effect is carried by the latter.
From Table 6, Figures 8–10 were constructed, showing the areas of influence of the
pollutants with respect to relative humidity: red indicates the strong dominance of pollution,
and green indicates the enhanced activity of relative humidity. The data were recorded in
the city of Santiago de Chile, which is located in a basin geomorphology and correspond to
measurements carried out in six communes, located at different heights, in the periods of
2010/2013, 2017/2020, and 2019/2022.
Atmosphere 2025,16, 337 16 of 30
Atmosphere 2025, 16, x FOR PEER REVIEW 18 of 33
Figure 8. Depiction for the period 2010/2013 showing that there is a higher percentage presence,
according to geographic area, of relative humidity (green color). Santiago de Chile is a city that is in
a basin geomorphology that exacerbates the problem of pollution.
Figure 8. Depiction for the period 2010/2013 showing that there is a higher percentage presence,
according to geographic area, of relative humidity (green color). Santiago de Chile is a city that is in a
basin geomorphology that exacerbates the problem of pollution.
Atmosphere 2025,16, 337 17 of 30
Atmosphere 2025, 16, x FOR PEER REVIEW 19 of 33
Figure 9. Depictions show, for the period 2017/2020, a large percentage decline in relative humidity
(green color) in the geographic area studied and which corresponds to the basin geomorphology in
which the city of Santiago is located. The presence of atmospheric pollution is dominant.
Figure 9. Depictions show, for the period 2017/2020, a large percentage decline in relative humidity
(green color) in the geographic area studied and which corresponds to the basin geomorphology in
which the city of Santiago is located. The presence of atmospheric pollution is dominant.
Atmosphere 2025,16, 337 18 of 30
Atmosphere 2025, 16, x FOR PEER REVIEW 20 of 33
Figure 10. Depictions of a certain return of relative humidity in a wide geographic sector (green
color) in the basin geomorphology of the city of Santiago for the period of 2019/2022. This return
occurs during the reduction of activities, in all orders of things, and the confinement of the popula-
tion due to the coronavirus pandemic.
Figure 10. Depictions of a certain return of relative humidity in a wide geographic sector (green color)
in the basin geomorphology of the city of Santiago for the period of 2019/2022. This return occurs
during the reduction of activities, in all orders of things, and the confinement of the population due
to the coronavirus pandemic.
Atmosphere 2025,16, 337 19 of 30
Gibbs free energy under thermodynamic equilibrium condition [49]:
dG = dH −TdS = 0, (6)
dH = TdS/:dt, (7)
dH
dt =.
H=TdS
dt =TSK(8)
For each species, MV represents the meteorological variables, P represents the pollu-
tants, and T is the average temperature of each commune for the measurement period.
dH
dt MV
=TSK,MV (9)
dH
dt P
=TSK,P (10)
dH
dt MV
dH
dt P
∼∆HMV
∆HP=SK,MV
SK,P =CK=∑n
i=1SK,MV
∑n
i=1SK,P COMMUNE
(11)
The enthalpy ratio between each of the species of the participating systems is propor-
tional to
CK
(which is a parameter for each recording location (commune)), following the
trends in Figures 5and 6.
3.3. Thermal Fluxes
From the perspective, in a first approximation, of the results extracted from Tables 6
and 7 and Equations (29) and (30) of [
48
], the total flux in K/h weighting for both urban
meteorology and pollutants gives 788.2 K/h (
∆
= (
δ
Q/dt)
P−
(
δ
Q/dt)
MV
(K/h)) 43.54 K/h)
for the 2010/2013 period, 765.9 K/h (
∆
= 103.3572 K/h) for the 2017/2020 period, and
478.1 K/h (∆= 112.3636 K/h) for the 2019/2022 period, according to Table 9.
Figure 11 shows the trend of the average value
∆
according to the three study periods:
Atmosphere 2025, 16, x FOR PEER REVIEW 22 of 33
Figure 11. The curve represents the difference between the thermal fluxes of pollutants and meteor-
ological variables for the three study periods, 1:2010/2013, 2:2017/2020, and 3:2019/2022. It shows a
break in the upward trend of Δ toward 2023, (A fiing function between Δ and P is given by Δ =
−25.405 P2 + 136.03 P − 67.088; R2 = 1).
The Meteorological Directorate of Chile indicates that 2003–2013 was the driest dec-
ade of the last 150 years, with the inclusion of period 1 of Figure 11.
The year 2020 was considerably rainier than 2019; however, it maintained the deficit
trend of around 40% compared to a normal year. As of June 2021, the precipitation trend
looks similar to that of 2019. It includes period 2.
Despite winter precipitation, 2022 closed as one of the 10 driest years ever recorded;
it includes period 3.
Period 3 (until 2022) announces a break in the upward trend of Δ beyond period 3.
The Meteorological Directorate of Chile confirmed that 2023 was one of the rainiest in the
last 15 years, where some stations far exceeded the amounts expected for a normal year.
At the end of 2024, precipitation is more normal. Even so, the precipitation deficit in Chile
takes over many localities in the country. According to the data released by the Meteoro-
logical Directorate of Chile, only three stations have a surplus greater than 20%.
According to Table 9 and Figure 11, the effect of heat flux on the boundary layer of
each species (pollutants and meteorological variables) decays according to periods. Tables
6 and 7 of [48] show that, for each species, the magnitudes of their thermal fluxes are in
high orders, resulting in narrow differences that result from a more polluted environmen-
tal context and that favors higher temperatures due to greater thermal transfers. In Table
9, the magnitudes of the thermal fluxes (δQ/dt) are in the lower orders, less excited, com-
pared to those of the periods 2010/2013 and 2017/2020, but result in higher differences (Δ).
These fluxes are referred to as less excited levels that are compatible with a downward
shift of the thermal fluxes of urban meteorology, a less excited atmosphere. There is less
energy per species to perform work. Thus, for the period 2019/2022, urban meteorology
once again gains some importance (EMS and EMO localities) compared to the period of
2017/2020, which may be due to the severe lockdown due to the coronavirus pandemic.
This confirms the relevance of the initial conditions and the context in which the thermal
flows develop, which is characteristic of a chaotic system.
3.4. Anomalous Diffusion
Physical and biological systems have now been discovered in which the mean square
displacement of the diffusing substance grows with time in the form < r2 (t) > ∝ t α, where
Figure 11. The curve represents the difference between the thermal fluxes of pollutants and me-
teorological variables for the three study periods, 1:2010/2013, 2:2017/2020, and 3:2019/2022. It
shows a break in the upward trend of
∆
toward 2023, (A fitting function between
∆
and P is given by
∆=−25.405 P2+ 136.03 P −67.088; R2= 1).
Atmosphere 2025,16, 337 20 of 30
Table 9. Average temperature for the measurement period 2019–2022 and the difference between
temporary heat variation between pollutants and meteorological variables using equation (30 of [
48
]).
The last row are average values.
h (masl) T(◦C) SKP (1/h) SKMV (1/h) (δQ/dt)P(K/h)
(
δ
Q/dt)
MV
(K/h)
∆(K/h)
784 (EML) 16.10 1.006 0.679 291.0000 196.4001 94.6000
709 (EMM) 14.70 0.904 0.640 260.2164 184.2240 76.0000
698 (EMV) 15.50 1.030 0.607 297.3095 175.2106 122.1000
520 (EMN) 16.05 1.127 0.603 325.9284 174.3876 151.5408
485 (EMS) 15.42 0.952 0.604 274.7186 174.2963 100.4223
469 (EMO) 15.31 1.117 0.668 322.2098 192.6913 129.5185
15.51 295.2305 182.8683 112.3636
295.2305 + 182.8683∼478.1 K/h.
The Meteorological Directorate of Chile indicates that 2003–2013 was the driest decade
of the last 150 years, with the inclusion of period 1 of Figure 11.
The year 2020 was considerably rainier than 2019; however, it maintained the deficit
trend of around 40% compared to a normal year. As of June 2021, the precipitation trend
looks similar to that of 2019. It includes period 2.
Despite winter precipitation, 2022 closed as one of the 10 driest years ever recorded; it
includes period 3.
Period 3 (until 2022) announces a break in the upward trend of
∆
beyond period 3. The
Meteorological Directorate of Chile confirmed that 2023 was one of the rainiest in the last
15 years, where some stations far exceeded the amounts expected for a normal year. At the
end of 2024, precipitation is more normal. Even so, the precipitation deficit in Chile takes
over many localities in the country. According to the data released by the Meteorological
Directorate of Chile, only three stations have a surplus greater than 20%.
According to Table 9and Figure 11, the effect of heat flux on the boundary layer
of each species (pollutants and meteorological variables) decays according to periods.
Tables 6 and 7
of [
48
] show that, for each species, the magnitudes of their thermal fluxes
are in high orders, resulting in narrow differences that result from a more polluted environ-
mental context and that favors higher temperatures due to greater thermal transfers. In
Table 9, the magnitudes of the thermal fluxes (δQ/dt) are in the lower orders, less excited,
compared to those of the periods 2010/2013 and 2017/2020, but result in higher differences
(
∆
). These fluxes are referred to as less excited levels that are compatible with a downward
shift of the thermal fluxes of urban meteorology, a less excited atmosphere. There is less
energy per species to perform work. Thus, for the period 2019/2022, urban meteorology
once again gains some importance (EMS and EMO localities) compared to the period of
2017/2020, which may be due to the severe lockdown due to the coronavirus pandemic.
This confirms the relevance of the initial conditions and the context in which the thermal
flows develop, which is characteristic of a chaotic system.
3.4. Anomalous Diffusion
Physical and biological systems have now been discovered in which the mean square
displacement of the diffusing substance grows with time in the form < r
2
(t) >
∝
t
α
,
where the value of the exponent divides the processes diffusive in two different regimes:
superdiffusion for
α
> 1 and subdiffusion for
α
< 1, particular cases of the so-called
anomalous diffusion, Figure 12 [17]:
Atmosphere 2025,16, 337 21 of 30
Atmosphere 2025, 16, x FOR PEER REVIEW 23 of 33
the value of the exponent divides the processes diffusive in two different regimes: super-
diffusion for α > 1 and subdiffusion for α < 1, particular cases of the so-called anomalous
diffusion, Figure 12 [17]:
Figure 12. Mean square displacement of anomalous and normal diffusion.
Anomalous diffusion, like normal diffusion, is studied and applied not only in the
physical sciences. It is used in various complex systems, such as the internal structure of
living cells, the process carried out by various species to search for and find food, etc.
The variance of the quadratic displacement is as follows [22] (Appendix B):
<r> ∝ t ~t,
, =t (12)
The quadratic diffusive variance associated with the displacement of the urban me-
teorology variables from the atmosphere to the interior of the boundary layer, close to the
ground, is dependent on the interaction between the entropies of the urban meteorology
variables and the entropies of the pollutants according to the CK quotient. In basin geo-
morphology, the entropic forces of the pollutants are dominant and hinder diffusion
(anomalous sub diffusion with α < 1), according to Tables 7 and 8, with MV = RH or MV
= T.
The Fréchet distribution is a special case of extreme value distribution or heavy-tailed
distribution [50,51]. The distribution function that represents it is as follows:
Pr(X≤x)=e. i
f
x >0,β∈ (0, ∞) (13)
where β > 0 is the shape parameter. The generalization includes a location parameter, n,
and a scale parameter, s > 0, leaving us with the following:
Pr(X≤x)=e
i
f
x >n (14)
In heavy-tailed distributions, there is a higher probability of extreme events com-
pared to more commonly observed distributions, such as the Gaussian or normal distri-
bution. One of the fundamental concepts in heavy-tailed distributions is power law be-
havior, as occurs with the mean value of the squared variance of the position. Power law
distributions are present between the probability density function and the variable of in-
terest:
Figure 12. Mean square displacement of anomalous and normal diffusion.
Anomalous diffusion, like normal diffusion, is studied and applied not only in the
physical sciences. It is used in various complex systems, such as the internal structure of
living cells, the process carried out by various species to search for and find food, etc.
The variance of the quadratic displacement is as follows [22] (Appendix B):
<r2>∝tα∼t
SK,MV
SK,P =tCK(12)
The quadratic diffusive variance associated with the displacement of the urban mete-
orology variables from the atmosphere to the interior of the boundary layer, close to the
ground, is dependent on the interaction between the entropies of the urban meteorology
variables and the entropies of the pollutants according to the C
K
quotient. In basin geomor-
phology, the entropic forces of the pollutants are dominant and hinder diffusion (anomalous
sub diffusion with α< 1), according to Tables 7and 8, with MV = RH or MV = T.
The Fréchet distribution is a special case of extreme value distribution or heavy-tailed
distribution [50,51]. The distribution function that represents it is as follows:
Pr(X≤x)=e−x−βif x >0, β∈(0, ∞)(13)
where
β
> 0 is the shape parameter. The generalization includes a location parameter, n,
and a scale parameter, s > 0, leaving us with the following:
Pr(X≤x)=e−(x−n
s)−βif x >n (14)
In heavy-tailed distributions, there is a higher probability of extreme events compared
to more commonly observed distributions, such as the Gaussian or normal distribution.
One of the fundamental concepts in heavy-tailed distributions is power law behavior, as
occurs with the mean value of the squared variance of the position. Power law distributions
are present between the probability density function and the variable of interest:
f(x)=αx−1−βe−x−β, x >0, with x >0 (15)
We will now separately consider one of the six monitoring stations of the study,
EML, since the treatment for the others is similar. The approach can be carried out from
two perspectives:
Atmosphere 2025,16, 337 22 of 30
1. 0 < c′K= SK,RH/SK,P <∞
Figure 13 is the heavy-tailed probability distribution with a domain of variables such
that it assumes that the greatest influence on the system, according to the different periods
of the study, is exerted by the entropy of relative humidity, the lowest probability is in the
period 2019/2022 and then 2017/2020, and, finally, the highest probability is in the period
of 2010/2013:
Atmosphere 2025, 16, x FOR PEER REVIEW 24 of 33
f
(x)=α xe,x>0,with x>0 (15)
We will now separately consider one of the six monitoring stations of the study, EML,
since the treatment for the others is similar. The approach can be carried out from two
perspectives:
1. 0 < c’K = S K,RH/S K,P < ∞
Figure 13 is the heavy-tailed probability distribution with a domain of variables such
that it assumes that the greatest influence on the system, according to the different periods
of the study, is exerted by the entropy of relative humidity, the lowest probability is in the
period 2019/2022 and then 2017/2020, and, finally, the highest probability is in the period
of 2010/2013:
Figure 13. Declining probability by period: X:2019/2022, Δ:2017/2020, and □:2010/2013.
2. 0 ≤ 1/c’K = S K,P/S K, RH < ∞
Figure 14 is the heavy-tail probability distribution with a domain of variables such
that it assumes that the greatest influence on the system, according to the different periods
of the study, is exerted by the entropy of the pollutants; the lowest probability is in the
period 2010/2013, and then 2017/2020 and 2019/2022.
Figure 14. Ascending probability according to periods Δ:2010/2013, □:2017/2020, and ◊:2019/2022.
Figure 15 is the heavy-tailed probability density function (15) with a domain of vari-
ables such that it assumes that the least influence on the system, according to the different
Figure 13. Declining probability by period: X: 2019/2022, ∆: 2017/2020, and □: 2010/2013.
2. 0 ≤1/c′K= SK,P/SK,RH <∞
Figure 14 is the heavy-tail probability distribution with a domain of variables such
that it assumes that the greatest influence on the system, according to the different periods
of the study, is exerted by the entropy of the pollutants; the lowest probability is in the
period 2010/2013, and then 2017/2020 and 2019/2022.
Atmosphere 2025, 16, x FOR PEER REVIEW 24 of 33
f
(x)=α xe,x>0,with x>0 (15)
We will now separately consider one of the six monitoring stations of the study, EML,
since the treatment for the others is similar. The approach can be carried out from two
perspectives:
1. 0 < c’K = S K,RH/S K,P < ∞
Figure 13 is the heavy-tailed probability distribution with a domain of variables such
that it assumes that the greatest influence on the system, according to the different periods
of the study, is exerted by the entropy of relative humidity, the lowest probability is in the
period 2019/2022 and then 2017/2020, and, finally, the highest probability is in the period
of 2010/2013:
Figure 13. Declining probability by period: X:2019/2022, Δ:2017/2020, and □:2010/2013.
2. 0 ≤ 1/c’K = S K,P/S K, RH < ∞
Figure 14 is the heavy-tail probability distribution with a domain of variables such
that it assumes that the greatest influence on the system, according to the different periods
of the study, is exerted by the entropy of the pollutants; the lowest probability is in the
period 2010/2013, and then 2017/2020 and 2019/2022.
Figure 14. Ascending probability according to periods Δ:2010/2013, □:2017/2020, and ◊:2019/2022.
Figure 15 is the heavy-tailed probability density function (15) with a domain of vari-
ables such that it assumes that the least influence on the system, according to the different
Figure 14. Ascending probability according to periods
∆
: 2010/2013,
□
: 2017/2020, and
♢
: 2019/2022.
Figure 15 is the heavy-tailed probability density function (15) with a domain of vari-
ables such that it assumes that the least influence on the system, according to the different
periods of the study, is exerted by relative humidity; and the lowest probability is in the
period 2019/2022, and then 2017/2020 and 2010/2013.
Atmosphere 2025,16, 337 23 of 30
Atmosphere 2025, 16, x FOR PEER REVIEW 25 of 33
periods of the study, is exerted by relative humidity; and the lowest probability is in the
period 2019/2022, and then 2017/2020 and 2010/2013
Figure 15. Probability density function (Equation (15), with x = C
, Δ: 2010/2013; o: 2017/2020; x:
2019/2022; C
= S K,RH/S K, P. It denotes the lower domain in the current periods of relative humidity
due to the influence of pollutants.
4. Discussion
The first three methods applied (Sections 3.1, 3.2, and 3.3) have a common denomi-
nator and that is that their fundamental concepts are in thermodynamics: temperature,
heat, thermal flows, entropy, and enthalpy. The study of heavy-tail probabilities is in-
cluded in Section 3.4 because the domain of definition of the probability function makes
use of the entropies ratio of relative humidity and the pollutants in this study, a ratio that
is directly related to anomalous subdiffusion (α < 1).
Everyone agrees that a warmer atmosphere, basically due to human activity, allows
the boundary layer to retain more water, breaking historical regularities [49,52,53], as fol-
lows from Figures 6 and 7. The heavy-tail probability study establishes that this extreme
condition has the highest probability of occurrence. Likewise, extreme health events such
as the coronavirus pandemic, which forced a very strong reduction in human activities
and a forced confinement of the population, produced an improvement in the thermal
condition of the boundary layer.
From the perspective of a heat wave, this corresponds to a continuous period of ex-
tremely high temperature for a specific region. The heat wave is measured in relation to
the average temperature considered in the study region. As there is no standardized def-
inition of a heat wave [54–56], the meteorological agencies of each country have their own
definitions in relation to what they estimate is a heat wave. The initial conditions gener-
ated by human activity in a basin geomorphology include the following [22,48]:
Air pollution,
Urban densification,
High albedo construction materials,
Changing of soil roughness due to high-rise buildings,
Thermal islands,
Expansion of cities into agricultural areas,
Concrete piping or channeling of natural waterways and river tributaries, thus re-
ducing the contribution to relative humidity,
Consistent and permanent reduction of tree vegetation, etc.
From this perspective, an atmosphere with low relative humidity and strong air pol-
lution dominance affects the C
indicator, which can be indicated in the geographic areas
Figure 15. Probability density function (Equation (15), with x =
C′
K
,
∆
: 2010/2013; o: 2017/2020;
x: 2019/2022;
C′
K
= S
K,RH
/S
K,P
. It denotes the lower domain in the current periods of relative
humidity due to the influence of pollutants.
4. Discussion
The first three methods applied (Sections 3.1–3.3) have a common denominator and
that is that their fundamental concepts are in thermodynamics: temperature, heat, thermal
flows, entropy, and enthalpy. The study of heavy-tail probabilities is included in Section 3.4
because the domain of definition of the probability function makes use of the entropies
ratio of relative humidity and the pollutants in this study, a ratio that is directly related to
anomalous subdiffusion (α< 1).
Everyone agrees that a warmer atmosphere, basically due to human activity, allows
the boundary layer to retain more water, breaking historical regularities [
49
,
52
,
53
], as
follows from Figures 6and 7. The heavy-tail probability study establishes that this extreme
condition has the highest probability of occurrence. Likewise, extreme health events such
as the coronavirus pandemic, which forced a very strong reduction in human activities
and a forced confinement of the population, produced an improvement in the thermal
condition of the boundary layer.
From the perspective of a heat wave, this corresponds to a continuous period of
extremely high temperature for a specific region. The heat wave is measured in relation
to the average temperature considered in the study region. As there is no standardized
definition of a heat wave [
54
–
56
], the meteorological agencies of each country have their
own definitions in relation to what they estimate is a heat wave. The initial conditions
generated by human activity in a basin geomorphology include the following [22,48]:
Air pollution,
Urban densification,
High albedo construction materials,
Changing of soil roughness due to high-rise buildings,
Thermal islands,
Expansion of cities into agricultural areas,
Concrete piping or channeling of natural waterways and river tributaries, thus reduc-
ing the contribution to relative humidity,
Consistent and permanent reduction of tree vegetation, etc.
From this perspective, an atmosphere with low relative humidity and strong air
pollution dominance affects the
C’
K
indicator, which can be indicated in the geographic
areas where the initial conditions that would mark the effect of a heat wave are found.
Furthermore, heat waves can influence epidemiological events [
57
]. From the perspective
of hydroclimatic volatility (sudden, large, and/or frequent transitions between very dry
and very humid conditions) [
58
], its evolution is expected with anthropogenic warming.
Atmosphere 2025,16, 337 24 of 30
According to the standardized precipitation evapotranspiration index, sudden transitions
at a global level have increased on average since the mid-twentieth century.
Evidence links these increases primarily to thermodynamics, namely the increasing
water vapor holding capacity and potential evaporative demand of the atmosphere. In-
creased hydroclimatic volatility will amplify hazards associated with rapid shifts between
wet and dry states (including flash floods, wildfires, landslides, and disease outbreaks) and
could accelerate a shift in water management toward joint management of drought and
flood risks.
5. Conclusions
After processing 108 time series of urban meteorology (T, WS, and RH) and atmo-
spheric pollutants (PM
10
, PM
2.5
, CO), with each series consisting of 28,463 hourly data,
using chaos theory software, it was shown that all series had their fundamental chaotic
parameters in the appropriate ranges, confirming the chaotic nature of all series. From
one of the parameters of each time series, the Kolmogorov entropy, the quotient between
the sum of the Kolmogorov entropies of the relative humidity (and another quotient with
the temperature, of each commune and for each period), and the sum of the Kolmogorov
entropies of the pollutants (of each commune and for each period), S
K,RH/
S
K,P
, (for the case
of temperature S
K,T/
S
K,P
) was constructed, which allows us to discover the following in a
relatively simple way:
1.
The interactive decay, as the periods advance toward the present, of the quotient.
This decay is compatible with what has been observed (of the relative humidity with
respect to the pollutants and of the temperature with respect to the thermal flow of
the pollutants).
2.
The presence of entropic dynamics associated with relative humidity and temperature,
of lesser magnitude than those caused by pollutants.
3.
The manifestation and permanence of a subdiffusive phenomenon in the stud-
ied basin.
4. The presence of statistics of extreme events through the heavy-tail analysis.
In the context of the boundary layer and in measurements with instruments whose
location does not exceed 10 m from the ground surface, the natural place of most human
activities, the limitations of the applied procedure are as follows:
1.
In the number of pollutants considered, although they are most significant in the
geomorphology of the studied basin;
2.
In which it is necessary to consider a fourth period of measurements of 3.25 years
(2025/2028), which is currently being carried out, together with new measurements,
in the same studied communities, using ultrasonic anemometers with the purpose of
measuring the vertical component of the wind and the sonic temperature and carrying
out comparative studies, during the interperiod, of thermal flows;
3.
Analyze the mountainous and coastal geomorphologies where the authors [
22
] have
already carried out a follow-up showing that, in a first approximation, they seem to
favor anomalous superdiffusion and heavy-tailed extreme events to the benefit of
urban meteorology.
Author Contributions: Conceptualization, P.P.; methodology, P.P.; software, P.P., E.M., G.N., and
S.P.; validation, P.P., E.M., and G.N.; formal analysis, P.P.; investigation, P.P.; resources, P.P., E.M.,
G.N., and S.P.; data curation, P.P., E.M., and G.N.; writing—original draft preparation, P.P. and E.M.;
writing—review and editing, P.P.; visualization, P.P., E.M., G.N., and S.P.; supervision, P.P.; project
administration, P.P.; funding acquisition, P.P. All authors have read and agreed to the published
version of the manuscript.
Atmosphere 2025,16, 337 25 of 30
Funding: Project supported by the Competition for Research Regular Projects, year 2024, code
LPR23-15, Universidad Tecnológica Metropolitana. This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data were obtained from the public network for online monitoring
of air pollutant concentration and meteorological variables. The network is distributed throughout
all of Chile, without access restrictions. It is the responsibility of SINCA, the National Air Quality
Information System, dependent on the Environment Ministry of Chile. The data for the two study
periods are available for free use on the WEB page: URL: https://sinca.mma.gob.cl, accessed on
11 January 2024.
Acknowledgments: To the Research Directorate of the Universidad Tecnológica Metropolitana that
made possible the progress of this study.
Conflicts of Interest: The authors declare no conflicts of interest.
Appendix A
The appendix presents the three tables containing the values of the chaotic parameters
calculated in six communes in the three measurement periods.
Table A1. Chaotic parameters of three pollution variables and three meteorological variables in
six monitoring stations (Santiago, Chile, 2010/2013 period) [48].
Parameters
Station PM10 (µg/m3)
PM
2.5
(
µ
g/m
3
)
CO (ppm)
Temperature (
◦
C)
RH (%) WS (m/s)
EML
λ0.491 0.603 0.514 0.440 0.613 0.777
Dc4.149 4.226 3.950 2.683 3.135 4.470
H 0.967 0.973 0.959 0.989 0.991 0.976
SK(1/h) 0.520 0.465 0.557 0.409 0.425 0.500
EMM
λ0.302 0.585 0.630 0.460 0.713 0.892
Dc3.877 3.966 4.719 3.128 3.507 4.285
H 0.972 0.977 0.981 0.991 0.990 0.980
SK(1/h) 0.441 0.528 0.581 0.409 0.427 0.616
EMN
λ0.576 0.467 0.323 0.501 0.679 0.734
Dc3.952 4.019 4.332 2.951 3.496 4.239
H 0.972 0.974 0.953 0.989 0.991 0.968
SK(1/h) 0.463 0.545 0.523 0.426 0.366 0.470
EMO
λ0.467 0.289 0.229 0.453 0.689 0.855
Dc3.785 4.085 4.640 2.801 3.194 4.053
H 0.965 0.955 0.937 0.992 0.989 0.968
SK(1/h) 0.522 0.428 0.260 0.375 0.382 0.440
EMS
λ0.421 0.542 0.439 0.489 0.725 0.880
Dc4.133 4.012 4.686 3.171 3.697 4.250
H 0.969 0.973 0.953 0.990 0.992 0.957
SK(1/h) 0.452 0.531 0.394 0.395 0.416 0.478
EMV
λ0.561 0.295 0.296 0.495 0.746 0.836
Dc3.788 3.788 4.631 3.155 3.249 3.584
H 0.967 0.970 0.952 0.989 0.989 0.956
SK(1/h) 0.552 0.538 0.341 0.384 0.370 0.448
Atmosphere 2025,16, 337 26 of 30
Table A2. Parameters for chaos study of three pollution variables and three meteorological variables
in six monitoring stations (Santiago, Chile, 2017–2020 period) [48].
Parameters
Station PM10 (µg/m3)
PM
2.5
(
µ
g/m
3
)
CO (ppm)
Temperature (
◦
C)
HR (%) WV (m/s)
EML
λ0.550 0.235 0.026 0.205 0.064 0.935
Dc3.451 1.364 0.580 2.290 2.029 3.697
H 0.922 0.963 0.933 0.915 0.942 0.975
SK(1/h) 0.295 0.596 0.686 0.355 0.414 0.515
EMM
λ0.383 0.614 0.013 0.184 0.067 0.937
Dc2.530 1.215 1.254 2.102 2.203 3.729
H 0.906 0.983 0.933 0.917 0.941 0.976
SK(1/h) 0.514 0.400 0.492 0.377 0.309 0.519
EMN
λ0.621 0.292 0.033 0.223 0.092 0.917
Dc2.948 1.276 2.277 2.280 2.095 3.735
H 0.929 0.960 0.933 0.916 0.942 0.973
SK(1/h) 0.242 0.825 0.412 0.366 0.308 0.471
EMO
λ0.550 0.332 0.046 0.189 0.081 0.928
Dc2.659 1.284 2.334 1.611 2.010 2.755
H 0.936 0.925 0.933 0.919 0.942 0.974
SK(1/h) 0.819 0.424 0.387 0.184 0.330 0.479
EMS
λ0.597 0.279 0.030 0.228 0.063 0.933
Dc3.535 1.396 3.302 2.300 2.306 3.004
H 0.921 0.975 0.933 0.915 0.942 0.976
SK(1/h) 0.898 0.422 0.382 0.357 0.404 0.489
EMV
λ0.516 0.304 0.031 0.170 0.065 0.915
Dc1.148 1.419 2.149 1.577 1.947 2.355
H 0.931 0.966 0.933 0.919 0.942 0.975
SK(1/h) 0.267 0.463 0.490 0.171 0.428 0.395
Table A3. Parameters for chaos study of three pollution variables and three meteorological ones in
six monitoring stations (Santiago, Chile, 2019/2022 period) [52].
Parameters
Station PM10 (µg/m3)
PM
2.5
(
µ
g/m
3
)
CO (ppm)
Temperature (
◦
C)
HR (%) WV(m/s)
EML
λ0.716 0.246 0.025 0.191 0.167 0.314
Dc1.067 1.306 2.089 1.632 2.465 1.991
H 0.930 0.946 0.933 0.920 0.934 0.942
SK(1/h) 0.257 0.367 0.382 0.175 0.229 0.275
EMM
λ0.561 0.345 0.011 0.220 0.203 0.339
Dc0.984 1.531 2.156 1.923 2.691 1.986
H 0.914 0.969 0.933 0.916 0.935 0.942
SK(1/h) 0.247 0.360 0.333 0.182 0.180 0.278
EMN
λ0.727 0.242 0.026 0.184 0.060 0.328
Dc0.978 1.421 2.053 1.626 2.752 1.997
H 0.934 0.947 0.933 0.921 0.908 0.941
SK(1/h) 0.409 0.385 0.333 0.172 0.148 0.283
EMO
λ0.540 0.332 0.015 0.222 0.256 0.347
Dc0.973 1.354 2.095 1.821 2.499 2.019
H 0.940 0.920 0.933 0.918 0.936 0.941
SK(1/h) 0.388 0.400 0.329 0.180 0.205 0.283
Atmosphere 2025,16, 337 27 of 30
Table A3. Cont.
Parameters
Station PM10 (µg/m3)
PM
2.5
(
µ
g/m
3
)
CO (ppm)
Temperature (
◦
C)
HR (%) WV(m/s)
EMS
λ0.747 0.257 0.021 0.194 0.104 0.349
Dc0.940 1.232 1.883 1.662 2.770 2.005
H 0.930 0.964 0.933 0.919 0.930 0.942
SK(1/h) 0.280 0.346 0.404 0.168 0.149 0.290
EMV
λ0.574 0.241 0.580 0.161 0.714 0.080
Dc0.945 1.432 2.127 1.559 2.704 1.939
H 0.930 0.938 0.933 0.920 0.934 0.940
SK(1/h) 0.252 0.415 0.285 0.153 0.100 0.351
Appendix B. [17]
Suppose there is a Kolmogorov entropy that fluctuates, but in infinitesimal amounts
at different times for which, approximately, there is a Lyapunov exponent such that the
separation between two phase space trajectories is maximum (maximum diffusion), as the
figure shows:
SK=λ
Atmosphere 2025, 16, x FOR PEER REVIEW 29 of 33
H 0.930 0.946 0.933 0.920 0.934 0.942
S
K
(1/h) 0.257 0.367 0.382 0.175 0.229 0.275
EMM
λ 0.561 0.345 0.011 0.220 0.203 0.339
D
c
0.984 1.531 2.156 1.923 2.691 1.986
H 0.914 0.969 0.933 0.916 0.935 0.942
S
K
(1/h) 0.247 0.360 0.333 0.182 0.180 0.278
EMN
λ 0.727 0.242 0.026 0.184 0.060 0.328
D
c
0.978 1.421 2.053 1.626 2.752 1.997
H 0.934 0.947 0.933 0.921 0.908 0.941
S
K
(1/h) 0.409 0.385 0.333 0.172 0.148 0.283
EMO
λ 0.540 0.332 0.015 0.222 0.256 0.347
D
c
0.973 1.354 2.095 1.821 2.499 2.019
H 0.940 0.920 0.933 0.918 0.936 0.941
S
K
(1/h) 0.388 0.400 0.329 0.180 0.205 0.283
EMS
λ 0.747 0.257 0.021 0.194 0.104 0.349
D
c
0.940 1.232 1.883 1.662 2.770 2.005
H 0.930 0.964 0.933 0.919 0.930 0.942
S
K
(1/h) 0.280 0.346 0.404 0.168 0.149 0.290
EMV
λ 0.574 0.241 0.580 0.161 0.714 0.080
D
c
0.945 1.432 2.127 1.559 2.704 1.939
H 0.930 0.938 0.933 0.920 0.934 0.940
S
K
(1/h) 0.252 0.415 0.285 0.153 0.100 0.351
Appendix B. [17]
Suppose there is a Kolmogorov entropy that fluctuates, but in infinitesimal amounts
at different times for which, approximately, there is a Lyapunov exponent such that the
separation between two phase space trajectories is maximum (maximum diffusion), as the
figure shows: S=
λ
If x(t) y x(t), they are separations between two trajectories of pollutants (P) and
urban meteorology (MV) in very short times.
a(t)=x(t)
x, =e
=e
,
a(t)=x(t)
x, =e
=e
,
If
xP(t)
y
xMV(t)
, they are separations between two trajectories of pollutants (P) and
urban meteorology (MV) in very short times.
a(t1) = xP(t1)
x0,P =eλPt1=eSK,Pt1
a(t2) = xP(t2)
x0,P =eλPt2=eSK,Pt2
a(t1)a(t2)∼a2=eSK,P(t1+t2)=eSK,P t′→a=eSK,P
2t′
and
b(t1) = xMV(t1)
x0,MV =eλMVt1=eSK,MV t1
b(t2) = xMV(t2)
x0,MV =eλMVt2=eSK,MV t2
b(t1)b(t2)∼b2=eSK,MV(t1+t2)=eSK,MV t′→b=eSK,MV
2t′
Considering a and b in a time, t′′, we obtain
a(t′′ )=xP(t′′ )
x0,P =eSK,Pt′′ →t′ ′ =lna(t′′ )
SK,P
b(t′′ )=xK,MV(t′′ )
x0,MV =eSK,MVt′′ →t′ ′ =lnb(t′′ )
SK,MV
Atmosphere 2025,16, 337 28 of 30
According to
ln bt′=SK,VM
2t′=SK,VM
2
lna(t′′ )
SK,P =CK
2lna(t′′ )
using the assumption of very small times, such that t′∼t′′ =t, we obtain the following:
b(t)=aCK
2
xMV(t)
x0,MV =xP(t)
x0,P CK
2→xMV(t)=x0,MV
(x0,P)CK
2
(xP(t)) CK
2
If the maximum spacing between phase space trajectories for the contaminant system
is assumed to be linear in time, xP(t)=vPt:
xMV(t)=x0,MV
(x0,P)CK
2
(vPt)CK
2=x0,MVv
CK
2
P
(x0,P)CK
2
tCK
2=Kt
SK,MV
2SK,P ∝t
SK,MV
2SK,P
Then, the square root of the arithmetic mean of the squares of displacements in the
direction x is as follows:
<x2>∝t
SK,MV
SK,P ∝tα
In basin geomorphology, the entropic forces of the pollutants are dominant and hinder
diffusion (anomalous sub diffusion with
α
< 1). Data can be extracted from
Tables A1–A3.
In
the studied basin geomorphology, the anomalous subdiffusion of contamination manifests
itself as a ponding.
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