Fractal Dimension of Pollutants and Urban Meteorology of a Basin Geomorphology: Study of Its Relationship with Entropic Dynamics and Anomalous Diffusion

Abstract

A total of 108 maximum Kolmogorov entropy (S_K) values, calculated by means of chaos theory, are obtained from 108 time series (TSs) (each consisting of 28,463 hourly data points). The total TSs are divided into 54 urban meteorological (temperature (T), relative humidity (RH) and wind speed magnitude (WS)) and 54 pollutants (PM₁₀, PM_2.5 and CO). The measurement locations (6) are located at different heights and the data recording was carried out in three periods, 2010–2013, 2017–2020 and 2019–2022, which determines a total of 3,074,004 data points. For each location, the sum of the maximum entropies of urban meteorology and the sum of maximum entropies of pollutants, S_{K, MV} and S_{K, P}, are calculated and plotted against h, generating six different curves for each of the three data-recording periods. The tangent of each figure is determined and multiplied by the average temperature value of each location according to the period, obtaining, in a first approximation, the magnitude of the entropic forces associated with urban meteorology (F_{K, MV}) and pollutants (F_{K, P}), respectively. It is verified that all the time series have a fractal dimension, and that the fractal dimension of the pollutants shows growth towards the most recent period. The entropic dynamics of pollutants is more dominant with respect to the dynamics of urban meteorology. It is found that this greater influence favors subdiffusion processes (α < 1), which is consistent with a geographic basin with lower atmospheric resilience. By applying a heavy-tailed probability density analysis, it is shown that atmospheric pollution states are more likely, generating an extreme environment that favors the growth of respiratory diseases and low relative humidity, makes heat islands more stable over time, and strengthens heat waves.

1. Introduction

Urban meteorology can be characterized by variables that, in a first approximation, have a high impact on the dynamics of the boundary layer of the atmosphere: temperature (T), relative humidity (RH) and the magnitude of wind speed (WS) [1,2,3,4,5]. Its behavior, measured in six communes of Santiago de Chile in three different periods of 3.25 years (2010/2013, 2017/2020 and 2019/2022), is recorded as time series with 28,463 specific data points for each variable, yielding a total of 1,537,002 data points of urban meteorology, which gives greater robustness to the analysis of the data and their trends [6,7,8,9,10,11]. The same number of measurements were carried out for three pollutants, with a high presence in urban environments and with a strong impact on human health, which are 10 µm particulate matter (PM₁₀), 2.5 µm particulate matter (PM_2.5) and carbon monoxide (CO) [12,13,14]. These pollutants were also monitored for 3.25 years in the three periods already mentioned and in the same six communes where urban meteorology was measured. The data-recording instruments for the six variables, all with similar technical characteristics (normed and certified according to EPA), were located in the six locations within a basin geomorphology, with a very rough natural bottom, which determines that the recording of measurements will be performed at different heights (between 450 and 800 m) [15] with respect to sea level.

The behavior of urban meteorology and pollutants near the Earth’s surface is turbulent and involves interactive processes of an irreversible nature. These characteristics make it appropriate to analyze time series from the perspective of chaos theory [16], proving that all of them are chaotic since the characteristic parameters calculated are in the appropriate ranges for this type of processes—a Lyapunov exponent (λ) greater than 0, a correlation dimension (D_C) less than 5, the Kolmogorov entropy (S_K) must be greater than 0, the Hurst exponent (H) must be greater than 0.5 and less than 1, the Lempel–Ziv complexity (LZ) must be greater than 0, the information loss (<ΔI>) must be less than 0, and the series must have fractal dimension (D) [17].

Entropy is a variable that allows for studying the disorder associated with long-term processes, which adjusts very well to the chaotic analysis that requires series with a very large amount of data (over 5000 data points due to the requirement of stability in the calculation of the Lyapunov coefficients). Entropy is a fundamental variable of nature that has been investigated and applied in many different areas. But its application to the study of communications, urban dimensioning, fluids, the Earth–atmosphere system, medicine, biology, etc., is relatively recent [18,19,20,21]. That said, its application to pollutants and the interaction with the atmosphere in the boundary layer is much more current [22,23].

Chaotic and predictable systems disagree in that their trajectories do and do not generate, respectively, new information [24,25,26]. The Kolmogorov–Sinai entropy KS (or metric), symbolized by S_K, places an upper bound on the data gain process; KS was developed by Shannon [27] as the rate of information creation when a chaotic system evolves (Shannon entropy), but not in the field of dynamical systems. KS has been applied to dynamical systems, and Kolmogorov [28] and Sinai [29] proved that it does not change under continuous deformations of space (a topological invariant). If the positive Lyapunov exponents [30,31] are added, KS is obtained as a lower bound [32]; they become equal under continuous measurements along unstable directions, which is usual in chaotic flows. KS is connected with traditional thermodynamic entropy, an indicator of the disorder of the system, by measuring the dispersion of neighboring trajectories towards new domains of the spatial state; since the trajectories depend on the initial conditions (IC), two very close points in a spatial state that are separated in time, the unimportant digits involved in the IC become essential to understanding more about the IC, and influence the evolution of the system. KS is measured as the reciprocal of time (inverse iterations) indicating the average rate of detriment of predictability, its inverse being the time expectancy of a prediction. Entropy allows us to classify systems, as follows: (i) ∞, completely random; (ii) 0, periodic or regular; (iii) greater than 0 and (iv) less than ∞, chaotic. For Shannon, information provides what is distinguishable from a configuration, with entropy being the information not yet available. The process can be observed as a reduction of information, given that predictions from an initial state become less precise over time.

A Markov process is like a data source, where one of l symbols arises randomly in a discrete time sequences [29,33]. The writing of a newspaper, for example, conforms to a Markov process. The symbols can be viewed as serial measurements. Assuming that the l possible symbols are integers from 0 to l − 1, the occurrence of a given symbol p depends on n − 1 previous symbols, and the Markov process is of order n. The Markov process implies the independence of time averages from the starting time, and they are equal to the ensemble (ergodic) averages. For a process of order n whose symbol is p_n, the sequence including the previous n − 1 symbols (p₁ p₂ … p_n) can be symbolized as the base l portion of n numbers p₁ p₂ … p_n, synthesized as P_n. This number specifies the state of the source (this is not the state of a dynamical system) [25]. In the Markovian source, in the sense described, numbers are the product of a series of measurements and are symbolized differently in the partition. For a measurement at time t = 0, it indicates that the dynamical state of the system is located within a component of the partition. For time Δt > 0, a new measurement can yield another result of the component of the partition. The sequence of measurements produces a series of symbols. A sequence of numbers obtained by performing a series of measurements can be considered as the symbols emitted by a Markov source. A different symbol is assigned to each of the n elements of the partition induced by the measuring instrument. Therefore, a series of measurements generates a sequence of symbols, whose temporal information rate is determined according to the limit n→∞ of ΔI_n/nΔt = Δl/Δt. KS can be defined as [29]:

$${\mathrm{h}}_{\mathsf{\mu }}=\underset{\mathsf{\beta },∆\mathrm{t}}{\underbrace{\mathrm{s}\mathrm{u}\mathrm{p}}}\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }{\mathrm{I}}_{\mathrm{n}}{\left(\mathrm{n}∆\mathrm{t}\right)}^{-1}$$

(1)

With suitable measuring instruments, recording samples at the best rate, KS is the average amount of new data (information) acquired per sample.

Given the trajectories x(t) = [x₁(t), x₂(t)…x_d(t)] (Figure 1), the d-dimensional phase space of a dynamical system is divided into boxes of size l^d, with measurements of the state of the system at uniformly spaced times τ. P_i is the cumulative probability that at time t = 0 the system is in box i₀, at t = τ in i₁… and at mτ in i_m. The magnitude K_m [27] is

$${\mathrm{K}}_{\mathrm{m}}=\sum _{\mathrm{i}}^{\mathrm{m}}{\left({\displaystyle \frac{1}{\mathrm{l}\mathrm{o}\mathrm{g}{\mathrm{P}}_{\mathrm{i}}}}\right)}^{\mathrm{k}{\mathrm{P}}_{\mathrm{i}}}$$

(2)

This is related to the information necessary to place the system on a specific path that passes through boxes i₀ … i_m with a certain precision. The coefficient k is related to a measurement unit.

The variation K_m+1 − K_m corresponds to the information in cell i_m+1 in which the system will be given was previously in i₀…i_m, and measures the loss of information on the system as time mτ varies to (m + 1) τ. S_K, the Kolmogorov entropy, is the average loss of information as l and τ → 0,

$${\mathrm{S}}_{\mathrm{K}}=\underset{\mathsf{\tau }\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}} \underset{\mathrm{l}\to 0}{\lim } \underset{\mathrm{m}\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}{\left(\mathrm{m}\mathsf{\tau }\right)}^{-1}\sum _{\mathrm{i}=1}^{\mathrm{m}}{\left({\displaystyle \frac{1}{\mathrm{l}\mathrm{o}\mathrm{g}{\mathrm{P}}_{\mathrm{i}}}}\right)}^{{\mathrm{P}}_{\mathrm{i}}}$$

(3)

S_K has units of information of bits per second and bits per iteration in the case of a discrete system [34,35,36]. The order of the limit process of (3) is (i) m → ∞, (ii) l → 0 (removing dependency on selected partition (m = number of cells (partitions))) and (iii) τ → 0 for continuous systems.

For two interacting systems, such as the atmospheric system (represented by meteorological variables) and the hourly concentration of pollutants, the speed over time at which the systems make information obsolete can be evaluated. From KS, the predictability horizon of each system is calculated [25].

The forces of entropic nature have been under discussion and had some applications for some time. In 2009, Verlinde [37] argued that gravity is an entropic force. Entropic forces are not exotic. For example, when stretching an elastic band, why does it pull back when stretched? A first argument could point out that a stretched elastic band has more energy than an unstretched one. This is an explanation that fits a metal spring. But this is not how rubber works, which is a natural polymer of isoprene (polyisoprene) and an elastomer (elastic polymer). A stretched elastic band has less entropy than an unstretched one, and this can also cause a force (see Figure 2).

Rubber molecules look like long chains. If not stretched, these chains can coil in many random ways, which means a lot of entropy. By stretching one of these chains, the number of shapes that can be given to it decreases, since it is tensioned in one way. Once that point is reached, a lot of energy is required to stretch the molecule; but everything points to a decrease in its entropy. Thus, the force that originates from a stretched elastic band is an entropic force. But how can changes in energy or entropy generate forces? Various approaches have agreed on the formal structure of the entropic force: one uses entropic pressure, the first and second laws of thermodynamics, and another uses Helmholtz free energy and canonical coordinates.

In the first approximation, using the differential form of the first and second principles of thermodynamics,

$$\mathrm{d}\mathrm{U}=\mathsf{\delta }\mathrm{Q}-\mathsf{\delta }\mathrm{W}$$

(4)

Heat (Q) is not a state function, and δQ is used instead of dQ. Physical entropy, S, in its classical form, is defined by the following equation:

$$\mathrm{d}\mathrm{S}={\displaystyle \frac{\mathsf{\delta }\mathrm{Q}}{\mathrm{T}}}$$

(5)

Applying δW = PdV, we obtain

$$\mathrm{d}\mathrm{U}=\mathrm{T}\mathrm{d}\mathrm{S}-\mathrm{P}\mathrm{d}\mathrm{V}$$

(6)

This result is independent of the coordinates used, with S depending on the temperature and volume. Differentiating U (V, T) and S (V, T), and using (6), we obtain (Appendix A),

$$\mathrm{P}=\mathrm{T}{\left.{\displaystyle \frac{\partial \mathrm{S}}{\partial \mathrm{V}}}\right|}_{\mathrm{T}}-{\left.{\displaystyle \frac{\partial \mathrm{U}}{\partial \mathrm{V}}}\right|}_{\mathrm{T}}$$

(7)

The other approach considers that one can calculate the pressure of an ideal gas and show that its origin is completely entropic; only the first term on the right-hand side of (7) is nonzero (Appendix B). For this purpose, it uses the force in canonical coordinates and the Helmholtz energy [38,39].

Geometrically, the entropic force, in general, is the tangent to the entropic surface S,

$$\overrightarrow{\mathrm{F}}\left({\overrightarrow{\mathrm{X}}}_0\right)=\mathrm{T}\times \nabla \mathrm{S}\left(\overrightarrow{\mathrm{X}}\right){|}_{\overrightarrow{\mathrm{X}}={\overrightarrow{\mathrm{X}}}_0}$$

(8)

This force is directly proportional to the variation of entropy with position, which indicates that its origin is associated with thermodynamic processes. Figure 3 presents the tangents to the entropic surface according to the variables x, y and z.

Each value of the Kolmogorov entropy, S_K, is the result of the sum of the positive values of the Lyapunov exponents (bits/h), of each of the time series (each with 28,463 data points) that were used in this research, and which were obtained from measurements of urban meteorology (MV) and pollutants (P) according to the different heights of the measurement locations. They represent an hourly entropic flow, which is also extended to the position derivatives:

$${\mathrm{S}}_{\mathrm{K}}=\dot{\mathrm{S}}={\displaystyle \frac{\mathrm{d}\mathrm{S}}{\mathrm{d}\mathrm{t}}},({\left.{\displaystyle \frac{\partial \dot{\mathrm{S}}}{\partial \mathrm{x}}}\right|}_{{\overrightarrow{\mathrm{x}}}_0},{\left.{\displaystyle \frac{\partial \dot{\mathrm{S}}}{\partial \mathrm{y}}}\right|}_{{\overrightarrow{\mathrm{x}}}_0},{\left.{\displaystyle \frac{\partial \dot{\mathrm{S}}}{\partial \mathrm{z}}}\right|}_{{\overrightarrow{\mathrm{x}}}_0})$$

(9)

The resulting S_K, for each period of 3.25 years, shows that the processes are chaotic, both for urban meteorology and for pollutants. The hourly entropic force can be estimated according to the Kolmogorov entropy, which is the result of the irreversible processes of urban meteorology, S_K,MV ($\overrightarrow{\mathrm{x}}$), and from the Kolmogorov entropy, which is the result of the irreversible processes of the pollutants, S_K,P ($\overrightarrow{\mathrm{x}}$), where the subscript K indicates its origin in the entropic flow. Entropy is a state function that accounts for the outcome of a process. In this research, S_K is the entropy in the unit of time and represents the processes experienced by the chain of data measured at a given height, and it is shown that this entropic flow (localized and hourly maximum) has an associated entropic force that varies according to the height (z = h).

In the case of a curve in the S_K versus h plane, the entropic force is the tangent to the curve, which is differentiated according to the MV and P cases:

$${\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(\mathrm{h}\right)\propto \mathrm{T}{\displaystyle \frac{\partial {\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(\mathrm{h}\right)}{\partial \mathrm{h}}}\to {\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(\mathrm{h}\right)={\mathrm{K}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\mathrm{T}{\displaystyle \frac{\partial {\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(\mathrm{h}\right)}{\partial \mathrm{h}}}\to {\mathrm{K}}_{\mathrm{M}\mathrm{V}}\overline{\mathrm{T}}{\displaystyle \frac{∆{\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(\mathrm{h}\right)}{∆\mathrm{h}}},$$

(10)

$${\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(\mathrm{h}\right)\propto \mathrm{T}{\displaystyle \frac{\partial {\mathrm{S}}_{\mathrm{K},\mathrm{P}}\left(\mathrm{h}\right)}{\partial \mathrm{h}}}\to {\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(\mathrm{h}\right)={\mathrm{K}}_{\mathrm{P}}\mathrm{T}{\displaystyle \frac{\partial {\mathrm{S}}_{\mathrm{K},\mathrm{P}}\left(\mathrm{h}\right)}{\partial \mathrm{h}}}\to {\mathrm{K}}_{\mathrm{P}}\overline{\mathrm{T}}{\displaystyle \frac{∆{\mathrm{S}}_{\mathrm{K},\mathrm{P}}\left(\mathrm{h}\right)}{∆\mathrm{h}}},$$

(11)

K_MV and K_P are constants of proportionality according to meteorological variables and the pollutant.

2. Materials and Methods

Santiago is the main city in Chile and the economic and administrative center of the country. It is located in a geographical depression on the southwestern edge of the continent, surrounded by the snow-capped mountains of the Andes and the Coastal Ranges (Figure 4). Its coordinates are approximately 33°26′16″ S 70°39′01″ W (which coincide with the latitude of Cape Town and Sydney) and it is at an average altitude of 567 m above sea level. It has an area of 837.89 km² and, in 2024, had 8,420,000 inhabitants, which is equivalent to 41.9% of the country’s total population. The climate corresponds to a temperate climate with winter rains and a long dry season, better known as a warm-summer Mediterranean climate, varying to a semi-arid climate, according to the Köppen climate classification. The capital of Chile is the seventh most populated metropolitan area in Latin America, and is estimated to be one of the 50 most populated urban agglomerations in the world.

The methods used in the measurements, the statistical treatment of the time series of the data and their chaotic analysis (Appendix C) have all been developed in [40,41], and

Table 1. The Kolmogorov entropies of urban meteorology, pollutants, and the ratio between both for the periods 2010/2013, 2017/2020 and 2019/2022 of the localities EML (La Florida), EMM (Las Condes), EMO (Pudahuel), EMS (Puente Alto), EMV (Quilicura) and EMN (Santiago-Parque O’Higgins).

		2010/2013	2017/2020	2019/2022	2010/2013	2017/2020	2019/2022	2010/2013	2017/2020	2019/2022
Station	masl (m)	SK,MV (1/h)	SK,MV (1/h)	SK,MV (1/h)	SK,P (1/h)	SK,P (1/h)	SK,P (1/h)	CK = SK,VM/SK,P	CK = SK,VM/SK,P	CK = SK,VM/SK,P
EML	784	1.334	1.284	0.679	1.542	1.577	1.006	0.865	0.814	0.675
EMM	709	1.452	1.205	0.640	1.55	1.406	0.940	0.937	0.857	0.681
EMO	469	1.197	0.993	0.668	1.21	1.63	1.117	0.989	0.609	0.598
EMS	698	1.289	1.25	0.607	1.377	1.702	1.030	0.936	0.734	0.589
EMV	485	1.202	0.994	0.604	1.431	1.22	0.952	0.840	0.815	0.635
EMN	570	1.262	1.145	0.603	1.531	1.479	1.127	0.824	0.774	0.535

is a summary of the calculations by the same authors. His exposition seeks to provide continuity with the presentation, where C_K is

$${\mathrm{C}}_{\mathrm{K},\mathrm{C}\mathrm{O}\mathrm{M}\mathrm{M}\mathrm{U}\mathrm{N}\mathrm{E}}={\left({\displaystyle \frac{\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{S}}_{\mathrm{K},,\mathrm{V}\mathrm{M}}}{\sum _1^{\mathrm{M}}{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}}\right)}_{\mathrm{C}\mathrm{O}\mathrm{M}\mathrm{M}\mathrm{U}\mathrm{N}\mathrm{E}}$$

(12)

Figure 5 shows the trend of the sums of entropies of the meteorological variables (T, HR, WS) for the five locations in the three measurement periods.

Figure 6 shows the trend in the sums of the entropies of the concentrations of the pollutants in this study (PM₁₀, PM_2.5, CO) for the six locations in the three measurement periods.

Figure 7 shows the quotient between the sum of the entropies of the meteorological variables and the sum of the entropies of the concentration of the pollutants of this study (PM₁₀, PM_2.5, CO) for the six locations in the three measurement periods.

Table 2. Average total temperature and relative humidity by commune and periods.

	EML	EMM	EMV	EMN	EMS	EMO	Average by Commune
2010–2013
$\overline{\mathrm{T}}$ (°C)	15.4	15.86	15.80	15.34	14.70	16.80	15.65
$\overline{\mathrm{R}\mathrm{H}}$ (%)	58.20	58.13	57.34	60.22	60.07	57.52	58.58
2017–2020
$\overline{\mathrm{T}}$ (°C)	16.12	15.57	16.85	16.17	15.53	16.78	16.17
$\overline{\mathrm{R}\mathrm{H}}$ (%)	55.31	55.00	58.95	57.31	56.07	59.22	56.98
2019–2022
$\overline{\mathrm{T}}$ (°C)	16.10	14.70	15.50	16.05	15.42	15.31	15.51
$\overline{\mathrm{R}\mathrm{H}}$ (%)	56.20	57.83	61.20	60.84	56.96	61.32	59.10

is a summary of the average values for each study period for each of the six locations, as well as of the temperatures and relative humidities.

3. Results

This section is divided into four parts, based on Table 1 and Figure 5, Figure 6 and Figure 7:

3.1. Determination of the Entropic Force Caused by Urban Meteorology and Pollutants

The figures of the Kolmogorov entropy (S_K) have been constructed according to the height (h), and this has been calculated from the rate of change of the entropy, S_K, with the height, h. The calculated entropy is representative of a process, for each of the periods studied (3.25 years each), associated with each of the meteorological variables and pollutant time series whose measurements were performed at the six aforementioned monitoring stations (according to the locations presented in Figure 4).

3.1.1. Dynamic Behavior of the Meteorological Variables in the Three Periods, the Information Has Been Extracted from Table 1

Figure 8 presents the behavior of the Kolmogorov entropy of the meteorological variables according to the different heights in the six locations for the period 2010–2013.

Table 3. Contains the tangent to the curve S versus h of the meteorological variables for the period 2010–2013.

2010–2013	h (m)	SMV	ΔSMV/Δh	$\overline{\mathbf{T}}$ (K)	$\overline{\mathbf{T}} \times$ SMV/Δh	FMV
	784	1.334	−0.00200	288.50	−0.5771000	−0.58 KMV
	709	1.452	−0.00066	289.01	−0.1907466	−0.19 KMV
	485	1.202	0.00025	288.95	0.0722375	0.07 KMV
	570	1.262	0.00060	288.49	0.1730940	0.17 KMV
	698	1.289	0.00350	287.85	1.0074750	1.01 KMV
	469	1.197	0.00100	289.95	0.2899500	0.29 KMV

shows the result of the calculation of the entropic force, F_MV, due to the meteorological variables of the period 2010–2013.

Figure 9 presents the behavior of the entropic force obtained from the Kolmogorov entropy versus the height of each location for the period 2010–2013.

Figure 10 presents the behavior of the Kolmogorov entropy of the meteorological variables according to the different heights of the six locations for the period 2017–2020.

Table 4. Contains the tangent of the S curve versus h for the meteorological variables for the period 2017–2020.

2017–2020	h (m)	SMV	ΔSMV/Δh	$\overline{\mathbf{T}}$ (K)	$\overline{\mathbf{T}}$ × ΔSMV/Δh	FMV
	784	1.284	0.0004	289.27	0.115708	0.12 KMV
	709	1.205	−0.0015	288.72	−0.433080	−0.43 KMV
	485	0.994	0.0150	290.00	4.350000	4.35 KMV
	570	1.145	0.0020	289.32	0.578640	0.58 KMV
	698	1.250	−0.0005	288.68	−0.144340	−0.14 KMV
	469	0.993	0.0005	289.93	0.144965	0.15 KMV

shows the result of the calculation of the force due to the meteorological variables for the period 2017–2020.

Figure 11 presents the entropic force derived from the Kolmogorov entropic flow versus the height of each location for the period 2017–2020.

Figure 12 presents the behavior of the Kolmogorov entropy of the meteorological variables according to the different heights of the six locations for the period 2019–2022.

Table 5. Contains the tangent, ΔS_MV/Δh, to the S versus h curve of the meteorological variables for the period 2019–2022 and the entropic force F_MV.

2019–2022	h(m)	SMV	ΔSMV/Δh	$\overline{\mathbf{T}}\left(\mathbf{K}\right)$	$\overline{\mathbf{T}}$ × ΔSMV/Δh	FMV
	784	0.679	−0.00015	289.25	−0.0433875	−0.04 KMV
	709	0.640	0.00115	287.85	0.3310275	0.33 KMV
	485	0.604	−0.00025	288.65	−0.0721625	−0.07 KMV
	570	0.603	0.000066	289.20	0.0190872	0.02 KMV
	698	0.607	0.001400	288.57	0.4039980	0.40 KMV
	469	0.668	−0.010000	288.46	−2.8846000	−2.90 KMV

shows the result of the calculation of the force due to the meteorological variables for the period 2019–2022.

Figure 13 presents the entropic force derived from the Kolmogorov entropic flow versus the height of each location for the period 2019–2022.

3.1.2. Dynamic Behavior of the Pollutant in the Three Periods, Data Extracted from Table 1

Figure 14 presents the behavior of the Kolmogorov entropy of the pollutants according to the heights of six locations for the period 2010–2013.

Table 6. Contains the tangent to the curve S versus h of the pollutants for the period 2010–2013.

2010–2013	h (m)	SP	ΔSP/Δh	$\overline{\mathbf{T}}\left(\mathbf{K}\right)$	$\overline{\mathbf{T}}$ × ΔSP/Δh	FP
	784	1.542	−0.00070	288.55	−0.190443	−0.19 KP
	709	1.550	0.00070	289.01	0.190747	0.19 KP
	485	1.431	0.00316	288.95	0.913082	0.91 KP
	570	1.531	−0.00020	288.49	−0.057698	−0.06 KP
	698	1.377	0.00050	287.85	0.143925	0.14 KP
	469	1.210	0.02500	289.95	7.248750	7.25 KP

shows the result of the calculation of the force due to pollutants for the period 2010–2013.

Figure 15 presents the entropic force derived from the Kolmogorov entropic flow versus the height of each location for the period 2010–2013.

Figure 16 presents the behavior of the Kolmogorov entropy of pollutants according to height in six locations at different heights for the period 2017–2020.

Table 7. Contains the tangent to the curve S versus h of the pollutants for the period 2017–2020.

2017–2020	h (m)	SP	ΔSP/Δh	$\overline{\mathbf{T}}\left(\mathbf{K}\right)$	$\overline{\mathbf{T}}$ × ΔSP/Δh	FP
	784	1.577	0.00200	289.27	0.578540	0.58 KP
	709	1.406	0.00130	288.72	0.375336	0.38 KP
	485	1.220	0.00075	290.00	0.217500	0.22 KP
	570	1.479	0.00300	289.32	0.867960	0.87 KP
	698	1.702	0.00050	288.68	0.144340	0.14 KP
	469	1.630	−0.04000	289.93	−11.597200	−11.60 KP

shows the result of the calculation of the force due to pollutants for the period 2017–2020.

Figure 17 presents the entropic force from the Kolmogorov entropic flow versus the height of each location for the period 2017–2020.

Figure 18 presents the behaviors of the Kolmogorov entropy of pollutants according to height in six locations at different heights for the period 2019–2022.

Table 8. Contains the tangent to the curve of S versus h of the pollutants for the period 2019–2022.

2019–2022	h (m)	SP	ΔSP/Δh	$\overline{\mathbf{T}}\left(\mathbf{K}\right)$	$\overline{\mathbf{T}}$ × ΔSP/Δh	FP
	784	1.006	−0.000500	289.25	−0.1446250	−0.14 KP
	709	0.940	0.000100	287.85	0.0287850	0.03 KP
	485	0.952	0.000230	288.65	0.0663895	0.07 KP
	570	1.127	−0.000330	289.20	−0.0954360	−0.10 KP
	698	1.030	−0.001500	288.57	−0.432855	−0.43 KP
	469	1.117	−0.048000	288.46	−13.846080	−13.85 KP

shows the results of the calculation of the force due to pollutants for the period 2019–2022.

Figure 19 presents the entropic force derived from the Kolmogorov entropic flow versus the height of each location for the period 2019–2022.

3.1.3. The Summary Is Presented in

Table 9. The F_MV/F_P ratios according to the three study periods.

	2010–2013	2017–2020	2019–2022
h (m)	FMV/FP	FMV/FP	FMV/FP
784	3.105	0.207	0.286
709	−1.000	−1.132	11.000
485	13.000	19.772	−1.000
570	−2.833	0.666	−0.200
698	7.214	−1.000	−0.930
469	0.040	−0.013	0.209

and Compares F_MV and F_P According to Period and Height of the Measurement Locations

Next, each figure represents, by period (2010–2013, 2017–2020 and 2019–2022), the ratio of entropic forces F_MV/F_P and the detrimental effect of the pollutants on the present. Furthermore, it was assumed that K_MV → K_P. The comparative figures of the entropic forces, according to height, indicate the increasingly dominant effects of the dynamics of pollutants on urban meteorology over the years.

The dynamics associated with atmospheric pollution processes (which have a high thermal presence) are becoming dominant, which is confirmed by Figure 20, Figure 21 and Figure 22. The magnitude of this process has been increasing since the last measuring time. This has an impact on the behavior of the climatology located inside the boundary layer in the geomorphology of the studied basin; it affects the seasons of the year, increases the effects of urban densification (thermal islands), accentuates heat waves, affects the relative humidity, etc., all being conditions that favor climate change [40,41,42].

3.2. Relationship Between Entropic Dynamics and Anomalous Diffusion

The equations developed below arise from the chaotic analysis of measurements, in the form of time series, of urban meteorological variables and the concentrations of pollutants, and within the chaotic parameters obtained is the Kolmogorov entropy; all the consequences described arise from measurements. Using (12),

$${\mathrm{C}}_{\mathrm{K},\mathrm{C}\mathrm{O}\mathrm{M}\mathrm{M}\mathrm{U}\mathrm{N}\mathrm{E}}={\left({\displaystyle \frac{\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{C}}_{\mathrm{K},\mathrm{M}\mathrm{V},,\mathrm{i}}}{\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{C}}_{\mathrm{K},\mathrm{P},\mathrm{i}}}}\right)}_{\mathrm{C}\mathrm{O}\mathrm{M}\mathrm{M}\mathrm{U}\mathrm{N}\mathrm{E}}={\left({\displaystyle \frac{{\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V}}(x,y,z)}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}(x,y,z)}}\right)}_{\mathrm{C}\mathrm{O}\mathrm{M}\mathrm{M}\mathrm{U}\mathrm{N}\mathrm{E}}$$

(13)

Considering, in general, the entropy quotient (omitting the Commune) and dependence on the vertical direction,

$${\displaystyle \frac{\partial }{\partial z}}{\mathrm{C}}_{\mathrm{K}}={\displaystyle \frac{\partial }{\partial z}}{\displaystyle \frac{{\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}}={\displaystyle \frac{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}\frac{\partial }{\partial z}{\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V}}-{\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\frac{\partial }{\partial z}{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}^2}}={\displaystyle \frac{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}\frac{F_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(\mathrm{z}\right)}{\overline{\mathrm{T}}}-{\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\frac{F_{\mathrm{K},\mathrm{P}}\left(\mathrm{z}\right)}{\overline{\mathrm{T}}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}^2}}$$

(14)

Reducing this, we derive

$${\displaystyle \frac{\partial }{\partial z}}{\mathrm{C}}_{\mathrm{K}}={\displaystyle \frac{{\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(\mathrm{z}\right)-{\mathrm{C}}_{\mathrm{K}}{\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(\mathrm{z}\right)}{\overline{\mathrm{T}}{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}}={\displaystyle \frac{∆}{\overline{\mathrm{T}}{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}}$$

(15)

Considering the average temperature ($\overline{\mathrm{T}}$) of the measurement period, by location, and determining that the entropy is greater than zero and that the range of variation of C_K is greater than 0 and less than +∞, an expression can be constructed that allows for relating the dynamics associated with urban meteorology with the dynamics associated with pollutants, represented by ${\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(\mathrm{z}\right)-{\mathrm{C}}_{\mathrm{K}}{\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(\mathrm{z}\right)$ $=∆$:

• If ${∆}_{\mathrm{B}\mathrm{A}\mathrm{S}\mathrm{I}\mathrm{N}}={\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(z\right)-{\mathrm{C}}_{\mathrm{K}}{\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(z\right)<0\to {\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(z\right)<{\mathrm{C}}_{\mathrm{K}}{\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(z\right)\to {\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(z\right)~$large values if 0 < C_K < $+1$;
• If $∆={\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(z\right)-{\mathrm{C}}_{\mathrm{K}}{\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(z\right)=0\to {\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(z\right)={\mathrm{C}}_{\mathrm{K}}{\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(z\right)\to {\displaystyle \frac{{\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}}{{\mathrm{F}}_{\mathrm{K},\mathrm{P}}}}={\mathrm{C}}_{\mathrm{K}}\text{>}0, \mathrm{by} \mathrm{definition},$ the action of C_K on ${\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(z\right)$ moderates the force ${\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(z\right)$ approximating it to ${\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(z\right)$;
• If ${∆}_{\mathrm{M}\mathrm{O}\mathrm{U}\mathrm{N}\mathrm{T}\mathrm{A}\mathrm{I}\mathrm{N}/\mathrm{C}\mathrm{O}\mathrm{A}\mathrm{S}\mathrm{T}}={\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(z\right)-{\mathrm{C}}_{\mathrm{K}}{\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(z\right)>0\to {\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(z\right)>{\mathrm{C}}_{\mathrm{K}}{\mathrm{F}}_{\mathrm{K},\mathrm{P}}\left(z\right)\to$ domain of ${\mathrm{F}}_{\mathrm{K},\mathrm{M}\mathrm{V}}\left(z\right)$ when 1 < C_K < $+\mathrm{\infty }$.

Case 1: A basin geomorphology, where the entropic forces of the contaminants are sufficient to hinder diffusion, making it anomalous with α < 1 [43].

Case 2: This is a condition that was not found in the processes studied.

According to data from [43], it is confirmed for the following.

Case 3: Coastal and mountain geomorphology [43], where the entropic forces of the meteorological variables favor the diffusion of the pollutants in the form of anomalous superdiffusion with α > 1.

According to [43], it is possible to assume the existence of a quantity of Kolmogorov entropy that experiences infinitesimal changes in very short times, and that the Lyapunov exponent is related to two neighboring trajectories whose distance in phase space is maximum (maximum diffusion), as shown in Figure 23.

If ${\mathrm{x}}_{\mathrm{P}}\left(\mathrm{t}\right)$ and ${\mathrm{x}}_{\mathrm{M}\mathrm{V}}\left(\mathrm{t}\right)$ are separations between two trajectories of pollutants (P) and urban meteorology (MV) for two very small times (t₁ and t₂), then the variance of the quadratic displacement is as follows (Appendix E, [43]):

$$\text{<}{\mathrm{x}}^2\text{>} \text{∝}{\mathrm{t}}^{{\displaystyle \frac{{\mathrm{S}}_{\mathrm{K},\mathrm{MV}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}}}={\mathrm{t}}^{{\mathrm{C}}_{\mathrm{K}}}\text{∝}{ \mathrm{t}}^{\mathsf{\alpha }}$$

(16)

This confirms the relationship, from the point of view of the measured data, between the effects of the entropic force on anomalous diffusion.

3.3. Fractal Dimension of the Time Series for the Three Study Periods

The six monitoring stations, the same for the three periods, measure six time series each. This determines a total of 108 time series, all chaotic. For each time series, its fractal dimension (D_F) is calculated, which is presented in

Table 10. The calculated values of the Hurst exponent (H) and the fractal dimension (D_F = 2 − H) for the 108 time series of the six study communes in the three periods (2010–2013, 2017–2020 and 2019–2022) [42]. The table includes, for each commune and period, the average value of the fractal dimension of pollutants and urban meteorology, and both quantities are compared.

		PM10	PM2.5	CO	T	HR	WV	${\overline{\mathbf{D}}}_{\mathbf{F},\mathbf{P}}$	${\overline{\mathbf{D}}}_{\mathbf{F},\mathbf{M}\mathbf{V}}$	${\overline{\mathbf{D}}}_{\mathbf{F},\mathbf{M}\mathbf{V}}/{\overline{\mathbf{D}}}_{\mathbf{F},\mathbf{P}}$
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	1.034	1.015	0.982
	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	1.060	1.056	0.996
	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	1.064	1.068	1.004
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	1.023	1.013	0.990
	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	1.059	1.055	0.996
	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	1.061	1.069	1.008
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	1.034	1.017	0.984
	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	1.059	1.056	0.997
	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	1.062	1.076	1.013
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	1.047	1.017	0.971
	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	1.069	1.055	0.987
	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	1.071	1.068	0.997
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	1.035	1.020	0.986
	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	1.057	1.056	0.999
	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	1.057	1.071	1.013
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	1.037	1.022	0.986
	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	1.056	1.055	0.999
	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	1.066	1.069	1.003

. This indicates that the urban meteorology and pollutant systems are fractal in nature, which should be confirmed in their interactive behavior. If the entropic dynamics are used as a measure of interaction, which is directly related to the entropy, calculated for each fractal series, and to the variation of entropy with height (extracted from Figure 8, Figure 10, Figure 12, Figure 14, Figure 16 and Figure 18), it can be observed that there is also a growth over time of the fractal dimension, which is aligned with an increase in the dynamic effect. From the point of view of this study, it is demonstrated that it is the entropic dynamics associated with pollutants that produce the greatest effect (Figure 24).

The construction of Figure 24 uses data corresponding to the 784 masl height of the EML locality from the first row of Table 9, and the column of ${\overline{\mathrm{D}}}_{\mathrm{F},\mathrm{P}}$ is from Table 10.

3.4. Heavy Tail Probability

In six communes of the city of Santiago de Chile, diffusive processes are studied through the Fréchet probability density function (heavy-tailed probability):

$$\mathrm{f}\left(\mathrm{x}\right)=\mathsf{\alpha }{\mathrm{x}}^{-1-\mathsf{\beta }}{\mathrm{e}}^{-{\mathrm{x}}^{-\mathsf{\beta }}},\mathrm{x}>0 \mathrm{with} \mathrm{x}={\mathrm{C}}_{\mathrm{K}}>0$$

(17)

From Equation (13), it is possible to generate a domain for f(x), Equation (17), which indicates the effect of the entropy of the pollutants on the entropies of urban meteorology. This effect can be classified as extreme when the C_K range tends to small values due to the high influence of the entropy of the pollutants on the entropy of urban meteorology, which is indicated by the probability value obtained from f(x).

3.4.1. 2010/2013 Period

Figure 25 presents the probability calculation according to f(x). This period can be considered as one of relatively low urban densification, numbers of vehicles and polluting sources, even reaching non-intensive margins.

3.4.2. 2017/2020 Period

Figure 26 presents the probability calculation according to f(x). This period corresponds to a growth in urban densification, the number of vehicles and new polluting sources.

3.4.3. 2019/2022 Period

Figure 27 presents the probability calculation according to f(x). This period corresponds to an intensive and persistent historical growth in urban densification, the number of vehicles and polluting sources. The period was also affected by the coronavirus pandemic, the confinement of the population and the reduction in activities.

Figure 25, Figure 26 and Figure 27 show that, in the line showing time periods leading up to the present, the highest probabilities are obtained for the presence of extreme events associated with pollutants where the dominance of the entropy of the pollutants prevails over the entropy of meteorological variables.

3.5. Corollary

Section 3.1, Section 3.2, Section 3.3 and Section 3.4 offer propositions intertwined by the entropies and the entropy quotients of different systems. All propositions are demonstrated experimentally and lead to the formulation of what could be called a theorem.

A system (which may include subsystems, such as different pollutants) of maximum average fractal dimension for a given time, compared to another system (which may include subsystems, such as those of the different urban meteorological variables) and with which it interacts, maximizes its entropic dynamics, minimizing the properties of the other system. The heavy-tailed probability functions are at their maximum for this type of process if it is maintained over time.

By applying the experimental results, it is possible to develop a proof.

Proof. 

Considering Equations (10) and (11) and forming the quotient, we derive

$${\displaystyle \frac{{\mathrm{F}}_{\mathrm{M}\mathrm{V}}}{{\mathrm{F}}_{\mathrm{P}}}}={\displaystyle \frac{\left(\overline{\mathrm{T}}\frac{\mathrm{d}{\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V}}}{\mathrm{d}\mathrm{h}}\right)}{\left(\overline{\mathrm{T}}\frac{\mathrm{d}{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}{\mathrm{d}\mathrm{h}}\right)}}~{\displaystyle \frac{∆{\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V}}}{∆{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}}={\displaystyle \frac{{\mathrm{S}}_{\mathrm{K}.\mathrm{M}\mathrm{V}}-{\mathrm{S}}_{\mathrm{K}.\mathrm{M}\mathrm{V},0}}{{\mathrm{S}}_{\mathrm{K}.\mathrm{P}}-{\mathrm{S}}_{\mathrm{K}.\mathrm{P},0}}}~\left({\mathrm{C}}_{\mathrm{K}}-{\displaystyle \frac{{\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V},0}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}}\right)\times {\left(1-{\displaystyle \frac{{\mathrm{S}}_{\mathrm{K},\mathrm{P},0}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}}\right)}^{-1}$$

(18)

where S_K,MV,0 and S_K,P,0 represent the initial entropies and S_{K, MV} and S_{K, P} represent the final entropies (MV, meteorological variables; P, pollutants). Following Figure 24, we derive

$${\displaystyle \frac{{\mathrm{F}}_{\mathrm{M}\mathrm{V}}}{{\mathrm{F}}_{\mathrm{P}}}}~\mathrm{f}\left({\overline{\mathrm{D}}}_{\mathrm{F},\mathrm{P}}\right), \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{f} \mathrm{i}\mathrm{s} \mathrm{n}\mathrm{o}\mathrm{t} \mathrm{a} \mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r} \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$

f decreases as ${\overline{\mathrm{D}}}_{\mathrm{F},\mathrm{P}}$ increases and the periods advance towards the most recent. The heavy-tailed probability function (17), for extreme events, is f(x) = f(C_K) with x = ${\mathrm{C}}_{\mathrm{K}}$ $>$ 0, where C_K is defined according to (12), which is the quotient between the entropies of urban meteorology and the entropies of the pollutants. □

If a system’s average fractal dimension and, consequently, its entropic dynamics grow over time when compared with another system, its probability of occurrence increases, since the heavy tail function, for extreme events, shows dependence on the average fractal dimension of the most dominant entropic system.

4. Discussion

From Figure 8, Figure 10, Figure 12, Figure 14, Figure 16 and Figure 18, the variation in S (in general) with height is

$${\displaystyle \frac{dS}{dh}}=\left\{\begin{array}{c}<0, \mathrm{tendency} \mathrm{towards} \mathrm{lower} \mathrm{dynamic} \mathrm{equilibrium}\\ =0, \mathrm{unstable} \mathrm{equilibrium} \mathrm{condition}\\ >0, \mathrm{tendency} \mathrm{towards} \mathrm{greater} \mathrm{dynamics} \mathrm{equilibrium}\end{array}\right.$$

(19)

Accordingly

$${\displaystyle \frac{dS}{dh}}<0\to \mathrm{the} \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e} \mathrm{F}<0,\mathrm{t}\mathrm{h}\mathrm{e} \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m} \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}$$

$${\displaystyle \frac{dS}{dh}}=0\to \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e} \mathrm{F}=0, \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y} \mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e} \mathrm{h}\mathrm{a}\mathrm{s} \mathrm{a}\mathrm{n} \mathrm{i}\mathrm{n}\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}$$

$${\displaystyle \frac{dS}{dh}}>0\to \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{c}\mathrm{e} \mathrm{F}>0,\mathrm{t}\mathrm{h}\mathrm{e} \mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{m} \mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{s}$$

The following thus pertains:

(i) If the entropic forces F < 0, which can be very fast and turbulent (which is compatible with high Kolmogorov turbulence at the surface level) and show energy loss, then it can be restrictive for the diffusion (more difficult or content);

(ii) If the entropic forces F > 0, which can be very fast and turbulent (which is compatible with high Kolmogorov turbulence at the surface level) and show energy gain, then it can be more permissive and contribute to the diffusion.

In both cases, the diffusion may be anomalous [43,44,45,46,47,48,49].

Figure 5, which depicts the entropies of the meteorological variables versus the height, together with Figure 8, Figure 10 and Figure 12, shows that the entropic forces due to the meteorological variables present a regime that decays as we advance towards the current period, leading to weaker dynamics. Figure 6, showing the entropies of the contaminants versus the height, together with Figure 14, Figure 16 and Figure 18, shows that the entropic forces due to the contaminants present a regime that favors the persistence of the studied contaminants. This is corroborated by Figure 20, Figure 21 and Figure 22, which compare the dynamics of the meteorological variables and pollutants of the three periods, from the past to the present, considered in the study. These figures, together with Table 9, show the growing dominance of pollutant dynamics over urban meteorology dynamics, with more negative proportions (Table 9, period 2017/2020 and 2019/2022), generating a sink-like effect for pollutants.

The nature of entropic forces has been analyzed for many years [37,49,50,51], but this application to the interactive problem between urban meteorology and pollutants in a basin geomorphology is not common, so it is not easy to find specialized literature for comparative purposes. Atmospheric pollution caused by human activity has acquired an influential role in urban meteorology, altering its historical cycles and behaviors, favoring the formation of extreme events that end up harming human beings themselves [42,52]. Air pollution is a direct indicator of urban densification, the use of concrete in buildings, changes in soil roughness due to high-rise buildings, the still-persistent increase in internal combustion vehicles, the elimination and/or piping with concrete of the many natural networks of water courses present in a geographic basin, etc. Among the varied consequences is the presence of irregular thermal flows over time, and of a heterogeneous magnitude that lead to events with a low horizon of predictability but that are repetitive, such as pandemics, heat waves, thermal islands, droughts, declines in relative humidity, etc.

With the methodology applied, these actions can be classified, definitively, as extreme, and imply varied consequences, such as the presence of irregular thermal flows over time and a heterogeneous magnitude, that favor events with a low horizon of predictability but that are repetitive, such as pandemics, waves of heat, thermal islands, droughts, decline in relative humidity, excessive rain in very short periods, etc.

If the squared variance is <x²> $~{\mathrm{t}}^{\mathsf{\alpha }}~{\mathrm{t}}^{{\mathrm{C}}_{\mathrm{K}}}~{\mathrm{t}}^{{\displaystyle \frac{{\mathrm{S}}_{\mathrm{K},\mathrm{M}\mathrm{V}}}{{\mathrm{S}}_{\mathrm{K},\mathrm{P}}}}}$ [43], given the calculated value of C_K (Figure 7) for the six communes, the diffusion is anomalous and of a sub-diffusive nature. Heavy tail probability studies carried out on a basin geomorphology (Figure 25, Figure 26 and Figure 27) confirm the lower entropic influence of urban meteorology compared to that of pollutants, and this type of probability is also adapted to the study of the propagation of a pandemic (Figure 8, [52]), showing its durability in areas of high urban densification, high pollution and low ventilation, where pollution stagnates for long periods of time.

The magnitude of the average fractal dimensions of the polluting system and the urban meteorology system is lower for all measurement locations in the 2010/2013 period compared to the other two data recording periods for the same locations. The minor fractality of the polluting system favors the entropic dynamics of urban meteorology, which makes it more dominant, as is reflected in Figure 20. By increasing the average fractal dimension of the polluting system in the 2017/2020 period, its dynamics are mostly seen to be “flattening” the dynamics of the urban meteorology system (Figure 21). This trend was broken, at a certain level, in the 2019/2022 period that coincided with the coronavirus pandemic and the strong confinement applied in the city of Santiago de Chile, with the consequent drop in the influence of the polluting system. Even so, this polluting dynamic is strong with respect to urban meteorology (Figure 22). This is confirmed by the average value of the Hurst coefficient for the polluting system compared to that of the urban meteorology system; Hurst is a long-range measure of dependence within a time series. That is, the events of the past influence the events of the future, and there is a “base state” of the polluting system, which, even when applying confinement and reducing activities to a minimum (2019/2022), prevails.

5. Conclusions

This research is based on measurements and comparative studies, performed in three periods of 3.25 years, of magnitudes that are derived from measurements in basin geomorphology. The entropy curves are very variable, depending on the height and time, according to the measurement periods of 3.25 years. The tangents to the curves allow connection with the entropic forces (F), demonstrating that these dynamics have their origin in thermal processes. The data and their derived magnitudes show that evolutionarily, from the past to the future, the greatest influence on local climatology is exerted by the entropic dynamics of pollutants; this makes a quantifiable contribution to climate change. This dynamic can contain or promote diffusion within each of the periods studied. When it contains F < 0, the entropy decreases and the phenomenon of anomalous subdiffusion occurs (C_K < 1), and when it drives diffusion F > 0, the entropy increases, which can favor diffusion. Entropic forces in the vertical direction have polarity (positive–negative), which affects the diffusion process. The data also indicate two relevant aspects related to a timeline towards the present—the growth of the fractal dimension of the pollutants, compared to the fractal dimension of urban meteorology, and the growth of the entropic force due to the polluting system. This trend can be moderated or, perhaps, reversed very slowly if an event with extreme global characteristics, such as the coronavirus pandemic in 2019/2022, favors an improvement in the urban meteorology system. The probability analysis of extreme phenomena, using heavy-tailed probability density, whose domain of variables is C_K (Equation (13)), demonstrates that the pollutants, their growth and the disturbance they cause to the urban meteorology system are extreme phenomena. The applied heavy-tailed probability assigns the new perturbed states the highest probability.