Open AccessArticle

Symmetry and the Nanoscale: Advances in Analytical Modeling in the Perspective of Holistic Unification

Paolo Di Sia

^1,2

School of Science & School of Engineering, University of Padova, 35131 Padova, Italy

School of Medicine, Applied Technical Sciences, University of Verona, 37134 Verona, Italy

Symmetry 2023, 15(8), 1611; https://doi.org/10.3390/sym15081611

Submission received: 20 July 2023 / Revised: 12 August 2023 / Accepted: 18 August 2023 / Published: 21 August 2023

(This article belongs to the Special Issue Symmetry and the Nanoscale: Matter and Mind - Mathematical Modeling and Phenomenology)

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Abstract

Analytical modeling presents symmetries and aesthetic-mathematical characteristics which are not catchable in numerical computation for science and technology; nanoscience plays a significant role in unification attempts, considering also models including holistic aspects of reality. In this paper we present new discovered results about the complete analytical quantum-relativistic form of the mean square deviation of position

R^{2} (t)

related to a recently introduced Drude–Lorentz-like model (DS model), already performed at classical, quantum and relativistic level. The function

R^{2} (t)

gives precise information about the distance crossed by carriers (electrons, ions, etc.) inside a nanostructure, considering both quantum effects and relativistic velocities. The model has a wide scale range of applicability; the nanoscale is considered in this paper, but it holds application from sub-pico-level to macro-level because of the existence of a gauge factor, making it applicable to every oscillating process in nature. Examples of application and suggestions supplement this paper, as well as interesting developments to be studied related to the model and to one of the basic elements of a current unified holistic approach based on vacuum energy.

Keywords:

symmetry; nanoscale; mean square deviation of position; quantum-relativistic effects; analytical modeling; unification; holism; vacuum energy

1. Introduction

The reflection on many contemporary theories dealing with techno-science has underlined how the scientific theory normally precedes subjective and social conditions; the techno-scientific theory does not tend to deal with reality in a global way, but only with the relationship between man and reality, considering the concept of reality as limited by a scientific-phenomenological and non-holistic anthropocentrism.

A holistic view of reality is such if it goes beyond the study and imitation of some characteristics of a particular organism, involving the incorporation of the general characteristics of life in its design and the application of these characteristics on multiple spatial, temporal and organizational scales of global influence.

Science and technology continue to advance, tending towards a complete understanding of the structure and behavior of matter; the nanoscale plays an important role in this evolution, through the study of complex systems (multiscale space-time structures) up to the most complex system, the human brain [1].

In the first decades of the twenty-first century, a concentrated effort has brought together nanotechnology, biotechnology, information technology and new technologies based on cognitive science. With proper attention to ethical issues and societal needs, the result can be a huge improvement in human capabilities, new products, social outcomes and improved quality of life.

The correlation between different scales is one of the focal points of complexity science, where the correlation between scales is established by analyzing the compromise among the dominant mechanisms [2,3,4]. Multiscale analytical methods present mathematical characteristics and properties not visible in numerical approaches and propose interesting approaches to complex systems linking reductionism and holism.

Nanomaterials are extremely interesting as they provide the building blocks for the construction of nanobiomaterials, which can be both biocompatible and non-toxic. This trend has emerged in nanomedicine and is commonly associated with engineered nanoparticles in the context of bioimaging, drug delivery systems, diagnostic tools and therapeutic modalities [5].

A fruitful combination concerns nanomaterials with biophotonics, i.e., nanobiophotonics (the aspects of electromagnetic nature are related to quantum reality) and approaches of holistic nature. Therefore, the question concerns how complete our global understanding of reality without the need of a holistic view of science is [6].

The accurate control and handling of individual atoms has recently made the production of artificial structures of nanometer size possible, with new high interesting properties. These nanostructures represent the frontier of progress in materials innovative technology.

It is of primary significance to obtain small objects, but to also have the possibility of checking the nanometric dimensions with careful construction processes, in order to reveal special and peculiar properties.

Physical low-dimensional systems (one or two dimensions) have attracted significant attention in recent years, both for the great number of experimental realizations (thanks to the use of nanotechnology and to techniques of laser light), and for the available theoretical techniques. To this class belong the mesoscopic systems, materials with strong electronic correlations and systems of atoms confined in optical lattices.

The reduction in at least one of the dimensions of physical systems to dimensionality in the of order or less than micron (

μ m

) raised the fame of mesoscopic physics, in which microscopic quantum effects and macroscopic effects play an equal role. The great interest in this field is due to the recent experimental accessibility to sub-micro-dimensions, that brought around the birth of nanotechnology and nanoelectronics [7].

Regarding electronics at the mesoscopic scale, a new captivating class of phenomena has been brought to the forefront by the use of quantum physics as the electron spins for the information transport (spintronics) and the realization of macroscopic quantum states, opening interesting prospects in the technology of quantum computation.

Concerning materials with strong electronic correlations, the concourse of different interactions and of different microscopic degrees of freedom originates interesting complex macroscopic behaviors [8].

The deep understanding of the conduct of these systems represents a challenge for fundamental physics, which has a huge technological potential and has already produced significant results. The research activity in this area will improve our understanding of the basic mechanisms underlying the specified materials and systems, thereby suggesting new features which lead to the creation of “new concept devices” and highlighting the importance of the study of new materials.

In addition, artificial intelligence (AI) is one of the trends for science. The resonance phenomenon is helpful in modern optics and photonics; considering that numerical simulations normally require long calculation time and considerable computational resources, there is a limit of applicability for structures with desired resonances. The use of AI brings faster spectra predictions of resonance. To overcome intrinsic problems of non-linearity and complexity, solutions are studied as the resonance-informed deep learning (RIDL) strategy for fast and precise prediction of the optical response for ultra-high-Q-factor resonances [9].

These efforts are oriented in the control of the nanoscale light-matter interaction, also through the use of superfocusing hybrid photonic-plasmonic devices. The demand for efficient photonic–plasmonic input–output interfaces to improve plasmonic device performance is one of the current mainstream areas involving nanoscience.

Also, machine learning methods can significantly reduce the cost of computation compared to numerical simulations, paving the way for super-resolution optical imaging; however, analytical methods remain one of the best solutions in this direction [10].

Therefore, the theoretical study and the search for mathematical modeling is pivotal, in particular analytical modeling, and will hopefully lead to integration of the characteristics of the two pillars of modern physics, namely quantum mechanics and relativity. This paper deals with the technical quantum-relativistic extension of a model, which analytically describes the charge transport, that has appeared in the literature in classical, quantum and relativistic versions [11,12,13]. In detail, the study of the mean square deviation of position

R^{2} (t)

will be done.

The following paragraphs summarize the starting elements of this new model (Section 2), the new calculated analytical expressions of

R^{2} (t)

in the quantum-relativistic case (Section 3), the results and applications (Section 4) and the conclusions (Section 5).

2. A Recently Appeared Drude–Lorentz-Type Model

A recent theoretical analytical formulation is showing to fit very well with observed scientific data and also provides stimulating new predictions of several characteristics in nanostructures. It contains a gauge factor which allows its use for the study of dynamics processes from sub-pico level to macro level.

The model is based on the total Fourier transform of the frequency-dependent complex conductivity

σ (ω)

, which can be inferred by the linear response theory [14,15,16,17,18].

The presence of a (0, +∞) integration in the Green-Kubo formula is an obstacle for the analytical inversion, but in this model it is overcome by evaluation of the integral on the whole-time axis (−∞, +∞), thus considering the real part of the complex conductivity. The new introduced key idea is the integration on the entire time axis (−∞, +∞), not on the half time axis (0, +∞), as it is ordinarily considered in the literature [19].

The model expands the Drude–Lorentz relation entailing the complex conductivity and rests on the reversal of the total Fourier transform in the complex plane considering the whole-time axis:

< {\vec{v}}^{α} (0) {\vec{v}}^{β} (t) >_{T} = \frac{k_{B} T V}{π e^{2}} \int_{- \infty}^{+ \infty} d ω Re σ_{β α} (ω) e^{i ω t}

(1)

The mean square deviation of position of particles at equilibrium

R^{2} (t)

is defined as:

R^{2} (t) = 〈 {[\vec{R} (t) - \vec{R} (0)]}^{2} 〉

(2)

where

R^{2} (t)

is the position vector at the time t. Considering that it holds:

\vec{R} (t) - \vec{R} (0) = \int_{0}^{t} \vec{v} (t ’) d t ’

(3)

where

\vec{v} (t)

is the velocity, and that for a homogeneous system in equilibrium the invariance property for time translation holds, it is possible to rewrite Equation (2) considering a transformation of coordinates relative to the region of integration. We get:

R^{2} (t) = 2 \int_{0}^{t} d t ’ (t - t ’) 〈 \vec{v} (t ’) \cdot \vec{v} (0) 〉

(4)

R^{2} (t)

can then be evaluated using the velocities correlation function.

Considering the Cauchy integration in the complex plane,

R^{2} (t)

is precisely evaluated by the residue theorem. The analytical expressions of

{〈 \vec{v} (t) \cdot \vec{v} (0) 〉}_{T}

R^{2} (t)

and the diffusion coefficient

D (t)

have been obtained for the classical, quantum and relativistic case [11,12,13].

3. Analytical Expression of the Mean Square Deviation of Position in the Quantum-Relativistic Case

(a) Regarding the quantum behaviour, we considered the consequence of a frequency-dependent electric field of the form

\vec{E} = {\vec{E}}_{0} e^{- i ω t}

; going after the time-dependent perturbation theory, we considered the factor

e \vec{E} \cdot \vec{r}

as perturbing potential (in this context

\vec{r}

is the position vector of the particle).

The matrix elements of the dipole moment of the charge in the direction of the electric field, between the initial

Φ_{0}

and the excited

Φ_{j}

states, are given by:

x_{j 0} = \int Φ_{j}^{*} e x Φ_{0} d r .

(5)

Defining the oscillator strength of the j-th transition as:

f_{j} = \frac{2 m}{ℏ^{2}} \sum_{j} ℏ ω_{j} {| x_{0 j} |}^{2},

(6)

and keeping into account of the relation between permittivity and conductivity of the system:

ε (ω) = 1 + 4 π i \frac{σ (ω)}{ω} = 1 + \frac{4 π N e^{2}}{m} \sum_{j} \frac{f_{j}}{ω_{j}^{2} - ω^{2} + i ω Γ_{j}},

(7)

we obtain the relation:

\frac{i σ (ω)}{ω} = \frac{1}{4 π} \sum_{i}^{} \frac{ω_{p_{i}}^{2}}{(ω_{i}^{2} - ω^{2}) - i ω Γ_{i}},

(8)

with:

ω_{p_{i}}^{2} = \frac{4 π N e^{2}}{m} f_{i} .

(9)

In previous equations it is:

-: $ω_{i} = (E_{i} - E_{0}) / ℏ;$
-: $E_{i}$ , $E_{0}$ energies of the excited and the ground states, respectively;
-: $Γ_{i} = 1 / τ_{i}$ inverse of the decay time of every mode;
-: N density of carriers.

As in the classical case, the analytical calculation formally brings us to Equation (1), but here the fundamental difference is that the real part of conductivity inside the integral is calculated at quantum level. The key factors, including the quantum behaviour, are the weights

f_{i}

. The relaxation times can be obtained by

τ_{i} = 1 / Γ_{i}

and weights

f_{i}

(Equation (9)). The calculation of N can be exactly obtained by Equation (9), considering that it holds:

\sum_{i} f_{i} = 1

[7].

(b) Concerning the relativistic behaviour, starting by the dynamics law:

\frac{d}{d t} (m_{p a r t} \vec{v}) = \sum_{i} {\vec{F}}_{i},

(10)

we studied the relativistic variation of the mass along an x-axis in the fixed ground reference frame. Regarding the forces acting on the generic carrier, we considered an outer passive elastic-type force of the type

F_{e l} = K x

, a passive friction-type force of the type

F_{f r} = λ \dot{x}

, depending by the velocity and with

λ = m_{p a r t} / τ

, and an outer oscillating electric field

E = e E_{0} e^{- i ω t}

, considering solutions of the type

x = x_{0} e^{- i ω t}

. The analytical calculation brings to three sets of results, in relation to the value of the quantity

Δ

, as in the classical and quantum case [7].

The expressions of

R^{2} (t)

in the quantum (Q) and relativistic (R) cases are, respectively [20]:

(Q) Quantum case:

(Q₁) Case

Δ_{R_{q u a n t}} > 0

R^{2} (t) = 2 (\frac{k_{B} T}{m^{*}}) \sum_{i} (\frac{f_{i}}{ω_{i}^{2}} (- \frac{1}{α_{i R}} \sin (\frac{α_{i R}}{2} \frac{t}{τ_{i}}) \exp (- \frac{t}{2 τ_{i}}) - \cos (\frac{α_{i R}}{2} \frac{t}{τ_{i}}) \exp (- \frac{t}{2 τ_{i}}) + 1)),

(11)

with:

α_{i R}^{} = \sqrt{4 τ_{i}^{2} ω_{i}^{2} - 1}

(12)

\in ℜ^{+}

(positive real numbers);

(Q₂) Case

Δ_{I_{q u a n t}} < 0

R^{2} (t) = 4 (\frac{k_{B} T}{m^{*}}) \sum_{i} (f_{i} τ_{i}^{2} (\frac{1}{α_{i I} (1 + α_{i I})} \exp (- \frac{(1 + α_{i I})}{2} \frac{t}{τ_{i}}) - \frac{1}{α_{i I} (1 - α_{i I})} \exp (- \frac{(1 - α_{i I})}{2} \frac{t}{τ_{i}}) + \frac{2}{1 - α_{i I}^{2}}),

(13)

with:

α_{i I}^{} = \sqrt{1 - 4 τ_{i}^{2} ω_{i}^{2}}

(14)

\in (0, 1) \subset ℜ

m₀ is the rest mass, m* is the effective mass, K is the Boltzmann’s constant, T is the system’s temperature,

ω_{i}

and

τ_{i}

are the frequencies and decaying times of each mode.

(R) Relativistic case:

(R₁) Case

Δ_{R_{r e l}} > 0

R^{2} (t) = 2 (\frac{k_{B} T}{m_{0}}) (\frac{1}{ω_{0}^{2}}) [- \frac{1}{α_{R_{r e l}}} \sin (\frac{α_{R_{r e l}}}{2 ρ} \frac{t}{τ}) \exp (- \frac{t}{2 τ ρ}) - \cos (\frac{α_{R_{r e l}}}{2 ρ} \frac{t}{τ}) \exp (- \frac{t}{2 τ ρ}) + 1],

(15)

with:

α_{R_{r e l}}^{} = \sqrt{4 γ ω_{0}^{2} τ^{2} - 1}

(16)

\in ℜ^{+}

(positive real numbers);

(R₂) Case

Δ_{I_{r e l}} < 0

R^{2} (t) = 4 (\frac{k_{B} T}{m_{0}}) (τ^{2}) (γ) [\frac{1}{α_{I_{r e l}} (1 + α_{I_{r e l}})} \exp (- \frac{(1 + α_{I_{r e l}})}{2 ρ} \frac{t}{τ}) - \frac{1}{α_{I_{r e l}} (1 - α_{I_{r e l}})} \exp (- \frac{(1 - α_{I_{r e l}})}{2 ρ} \frac{t}{τ}) + \frac{2}{(1 - α_{I_{r e l}}^{2})}],

(17)

with:

α_{I_{r e l}}^{} = \sqrt{1 - 4 γ ω_{0}^{2} τ^{2}}

(18)

\in (0, 1) \subset ℜ

It holds:

α_{I_{r e l}}^{} = \sqrt{Δ_{I_{r e l}}}

α_{R_{r e l}}^{} = \sqrt{Δ_{R_{r e l}}}

γ = 1 / \sqrt{1 - β^{2}}

β = v / c

ρ = 1 + β^{2} γ^{2} = γ^{2}

With the procedure used for obtaining the expressions of the velocities correlation function in the quantum-relativistic case [20], the new results for the mean square deviation of position

R^{2} (t) = 〈 {[\vec{R} (t) - \vec{R} (0)]}^{2} 〉

have the following analytical form:

(Q-R) Quantum-Relativistic case:

(Q-R₁) Case

Δ_{i R_{Q - R}} > 0

R^{2} (t) = 2 (\frac{k_{B} T}{m_{0}}) \sum_{i} [(\frac{f_{i}}{ω_{i}^{2}}) (- \frac{1}{α_{i R_{Q - R}}} \sin (\frac{α_{i R_{Q - R}}}{2 ρ} \frac{t}{τ_{i}}) \exp (- \frac{t}{2 τ_{i} ρ}) - \cos (\frac{α_{i R_{Q - R}}}{2 ρ} \frac{t}{τ_{i}}) \exp (- \frac{t}{2 τ_{i} ρ}) + 1)],

(19)

with:

α_{{i R}_{Q - R}} = \sqrt{4 γ ω_{i}^{2} τ_{i}^{2} - 1}

(20)

\in ℜ^{+}

(positive real numbers);

(Q-R₂) Case

Δ_{{i I}_{Q - R}} < 0

R^{2} (t) = 4 (\frac{k_{B} T}{m_{0}}) (γ) \sum_{i} {(f_{i} τ_{i}^{2}) [\frac{1}{α_{i I_{Q - R}} (1 + α_{i I_{Q - R}})} \exp (- \frac{(1 + α_{i I_{Q - R}})}{2 ρ} \frac{t}{τ_{i}}) - \frac{1}{α_{i I_{Q - R}} (1 - α_{i I_{Q - R}})} \exp (- \frac{(1 - α_{i I_{Q - R}})}{2 ρ} \frac{t}{τ_{i}}) + \frac{2}{(1 - α^{2}_{i I_{Q - R}})}]},

(21)

with:

α_{i I_{Q - R}}^{} = \sqrt{1 - 4 γ ω_{i}^{2} τ_{i}^{2}}

(22)

\in (0, 1) \subset ℜ

It holds:

α_{i I_{Q - R}}^{} = \sqrt{Δ_{i I_{Q - R}}}

α_{i R_{Q - R}}^{} = \sqrt{Δ_{i R_{Q - R}}}

γ = 1 / \sqrt{1 - β^{2}}

β = v / c

ρ = 1 + β^{2} γ^{2} = γ^{2}

4. Results and Applications

The achieved results in the classical case explain the extreme short times and elevated mobilities with which charges diffuse in mesoporous systems, and are of broad interest in all systems involving photocatalysis and photovoltaics. The small times of few

τ

show facile charge diffusion inside the nanoparticles. The undescribed experimental fact of ultrashort injection of charge carriers, particularly in Grätzel’s cells, can be referred to in this phenomenon. Deviations by the Drude model become powerful in nanostructured materials, such as photoexcited TiO₂ nanoparticles, ZnO films, InP nanoparticles, semiconducting polymer molecules and carbon NTs [21,22,23].

The quantum and relativistic models provided newsworthy additional features and added novel ones. The quantum model considers the weight of varied modes with the own

ω_{i}

and

τ_{i}

values, the relativistic model meets the extreme high velocities of carriers inside nanostructures. The quantum-relativistic case connects both characters.

As examples of application, Figure 1 illustrates the evolution of

R^{2} (t)

vs. time for the fixed value

α_{R_{Q - R}}^{} = 10

(

Δ > 0

) in the case of one only mode (not the quantum behaviour) for ZnO (

m^{*} = 0.24 m_{e}

;

τ = 0.84 \cdot 10^{- 13} s

) [22]. As velocities of carriers, we considered

v = 10^{7} c m / s

(blue solid line),

v = 10^{10} c m / s

(red dashed line) and

v = 2.5 \cdot 10^{10} c m / s

(green dot-dashed line) (Table 1). From Equation (20) this follows:

ω = 5.98 \cdot 10^{13} s^{- 1}

In Figure 2, we considered the case

Δ < 0

with

α_{I_{Q - R}}^{} = 0.5

in relation to one only mode (not the quantum behaviour). From Equation (22) this follows:

ω = 0.52 \cdot 10^{13} s^{- 1}

[22].

Then, we considered experimental data from the literature, by which three states were extracted. Table 2 summarizes the calculated used values. Data are related to single-walled carbon nanotube films at the temperature of 300 K [24,25,26,27].

Thanks to Equations (20) and (22), the values of

α

for the three states result:

α_{I_{1}} = 0.998

;

α_{R_{2}} = 8.57

;

α_{R_{3}} = 5.51

. These values must be calculated for each considered velocity.

Figure 3, Figure 4 and Figure 5 show the behaviour of

R^{2} (t)

vs. time considering the data in Table 2.

Figure 6 reproduces what we obtained when considering the global state as sum of the previous three states.

In Figure 7 we see the general quantum-relativistic case, i.e., considering the quantum behaviour related to the presence of more than one state and the variation of velocity (which was increasing in our case).

From these examples we conclude that it is possible to perform a careful and general quantum-relativistic study of the carrier transport in nanostructures.

The assumption of relativistic speeds and quantum features inside a nanostructure is an attractive present topic; quantum processes with extreme high speeds of carriers are pertinent for current science and technology and are elected by a key point for the future in theoretical and phenomenological (nano)physics.

What is debated here can be thought of as the starting point of a research pathway referred to physical suppositions, namely how scattering, ballistic transport, influence of temperature [26], size effect, etc., can impact on the studied phenomenon. A challenge arises for phenomenological scientists whose insight could fruitfully match these theoretical efforts.

Regarding the present phenomenological investigation, two methods are suggested:

(1): Photon-Induced Near-Field Electron Microscopy: this inspection technique connects the spatial resolution at the nanoscale of the electron microscopy with the femto-second temporal resolution of extreme fast light impulses; it can be used to check very fast occurrences present at very small length scales. A way for raising the electron-light interactions in very short intervals consists of enlarging the light field through two synchronized femto-second light impulses. Variations of the time delay among the exciting light impulses and the electronic imaging ones allow one to obtain snapshots of the evanescent field as it evolves on femto-second intervals. The application of still shorter pulses can allow us to keep trace of the extreme fast processes happening in photonic and plasmonic devices [28,29].
(2): Graphene based plasmonics: the non-linear optical properties of a plasma expected in the relativistic movement of electrons subjected to a high laser field are of central significance in the present research. Recently herein showed fast progress in the sector of graphene plasmonics, especially considering graphene’s special global properties. The application of graphene plasmonics will give stimulating results in the little-exploited terahertz to mid-infrared regime.

Graphene and plasmonics strongly overlap, both for the inherent plasmons of graphene and for their mixture with noble metal nanostructures. Graphene based plasmonics can allow for the creation of novel optical devices that work in various frequency ranges, from terahertz to visible light, associated with ultra-high speed [30,31,32,33].

5. Conclusions

In this paper we introduced new analytical formulas related to a quantum-relativistic model for the study of transport dynamics at the nanoscale. The obtained results allow the accurate study of transport dynamics contemplating quantum and relativistic effects. In particular, the quantum-relativistic formulas for the function

R^{2} (t)

have been introduced.

At a phenomenological level, the extreme fast carrier injection is suggested, in relation to manageable technical needs related to a raise of the wavelength of oscillations and a reduction in the amplitude of

R^{2} (t)

The quantum-relativistic version of the model is mathematically very elegant, because of its analytical formulation, and it provides an attractive novelty which can be fruitfully objected in experimental investigation through time-resolved techniques, like TRTS, PINEM and Graphene based plasmonics [34,35,36].

The model considers all parameters which can influence the system at chemical, physical and structural level, as the intrinsic ones in the model, like the system’s temperature T, the values of

α_{i I_{Q - R}}^{}

α_{i R_{Q - R}}^{}

τ_{i}

and

ω_{i}

, possible variations of the effective mass

m *

, changes of the chiral vector, the quantum weights of modes, the carrier density N, the speed of carriers; managing them, it is possible to carry out a careful tuning of

{〈 \vec{v} (t) \cdot \vec{v} (0) 〉}_{T}

R^{2} (t)

and

D (t)

From quantum physics, we know that everything is energy and vibration, and this is connected with the energy of the vacuum. The attempts to define the structure and the global intrinsic properties of this space constitute a rigorous modern line of research.

In this direction, interesting results were obtained in relation to the ontological interpretation of quantum theory [37,38,39], with the Fourier transform playing a very important role. This is one of the last core elements of the described analytical model of transport dynamics at the nanometric level, which has a gauge factor allowing for its use in a wider range than nanoscale only [40].

Other related concepts connected to this project concern how particular symmetries of

σ (ω)

are reflected on the solutions of the model, the possible involvement of advanced and delayed waves, the “quantum self-ergonomics”, the concept of “extended self” and other peculiarities involving aspects of holistic nature. These last fields go beyond the science officially accepted by the scientific community but deserve deep attention and insight as they could solve many current puzzles, as well as expand the current global knowledge. The history of science teaches that situations like the one presented here are certainly not new and have paved the way for fruitful progress in the knowledge of human beings and the reality surrounding her/him [41,42,43].

Funding

This research received no external funding.

Data Availability Statement

The new obtained data are included in the paper; this project is in progress.

Acknowledgments

The author wishes to thank Teresa Yu, Chief-Section Managing Editor, and all those who are supporting the success of the Special Issue “Symmetry and the Nanoscale: Matter and Mind—Mathematical Modeling and Phenomenology”.

Conflicts of Interest

The author declares no conflict of interest.

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Figure 1.

R^{2} (t)

vs. t with

α_{R_{Q - R}}^{} = 10

T = 300 K

;

v = 10^{7} c m / s

(blue solid line),

v = 10^{10} c m / s

(red dashed line) and

v = 2.5 \cdot 10^{10} c m / s

(green dot-dashed line).

Figure 1.

R^{2} (t)

vs. t with

α_{R_{Q - R}}^{} = 10

T = 300 K

;

v = 10^{7} c m / s

(blue solid line),

v = 10^{10} c m / s

(red dashed line) and

v = 2.5 \cdot 10^{10} c m / s

(green dot-dashed line).

Figure 2.

R^{2} (t)

vs. t with

α_{I_{Q - R}}^{} = 0.5

T = 300 K

;

v = 10^{7} c m / s

(blue solid line),

v = 10^{10} c m / s

(red dashed line) and

v = 2.5 \cdot 10^{10} c m / s

(green dot-dashed line).

Figure 2.

R^{2} (t)

vs. t with

α_{I_{Q - R}}^{} = 0.5

T = 300 K

;

v = 10^{7} c m / s

(blue solid line),

v = 10^{10} c m / s

(red dashed line) and

v = 2.5 \cdot 10^{10} c m / s

(green dot-dashed line).

Figure 3.

R^{2} (t)

vs. t for state 1 with data from Table 2.

Figure 3.

R^{2} (t)

vs. t for state 1 with data from Table 2.

Figure 4.

R^{2} (t)

vs. t for state 2 with data from Table 2.

Figure 4.

R^{2} (t)

vs. t for state 2 with data from Table 2.

Figure 5.

R^{2} (t)

vs. t for state 3 with data from Table 2.

Figure 5.

R^{2} (t)

vs. t for state 3 with data from Table 2.

Figure 6.

R^{2} (t)

vs. t for the sum of the previous three states (data from Table 2).

Figure 6.

R^{2} (t)

vs. t for the sum of the previous three states (data from Table 2).

Figure 7.

R^{2} (t)

vs. t for a state given by the sum of the previous three states (data from Table 2), considering three different velocities (

v = 10^{7} c m / s

(blue solid line),

v = 10^{10} c m / s

(red dashed line) and

v = 2.5 \cdot 10^{10} c m / s

(green dot-dashed line)).

Figure 7.

R^{2} (t)

vs. t for a state given by the sum of the previous three states (data from Table 2), considering three different velocities (

v = 10^{7} c m / s

(blue solid line),

v = 10^{10} c m / s

(red dashed line) and

v = 2.5 \cdot 10^{10} c m / s

(green dot-dashed line)).

Table 1. Computed data for the three examined velocities of carriers.

v (cm/s)	β²	1/ρ	γ
10⁷	0.11 × 10⁻⁶	0.998	1.001
10¹⁰	0.11	0.888	1.061
2.5 × 10¹⁰	0.69	0.31	1.796

Table 2. Values of each examined state.

States	ω_i (×10⁻¹² Hz)	τ_i (×10¹² Hz)	f_i
1	6.59	0.0042	0.312
2	1166.01	0.0037	0.176
3	2000.05	0.0014	0.512

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Di Sia, P. Symmetry and the Nanoscale: Advances in Analytical Modeling in the Perspective of Holistic Unification. Symmetry 2023, 15, 1611. https://doi.org/10.3390/sym15081611

AMA Style

Di Sia P. Symmetry and the Nanoscale: Advances in Analytical Modeling in the Perspective of Holistic Unification. Symmetry. 2023; 15(8):1611. https://doi.org/10.3390/sym15081611

Chicago/Turabian Style

Di Sia, Paolo. 2023. "Symmetry and the Nanoscale: Advances in Analytical Modeling in the Perspective of Holistic Unification" Symmetry 15, no. 8: 1611. https://doi.org/10.3390/sym15081611

APA Style

Di Sia, P. (2023). Symmetry and the Nanoscale: Advances in Analytical Modeling in the Perspective of Holistic Unification. Symmetry, 15(8), 1611. https://doi.org/10.3390/sym15081611

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Symmetry and the Nanoscale: Advances in Analytical Modeling in the Perspective of Holistic Unification

Abstract

1. Introduction

2. A Recently Appeared Drude–Lorentz-Type Model

3. Analytical Expression of the Mean Square Deviation of Position in the Quantum-Relativistic Case

4. Results and Applications

5. Conclusions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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