An efficient methodology to explore dynamical system configurations with the significance of coherent–chaotic peculiarities

1 Introduction

It is well known that quantum mechanics harnesses interference to explore the pertinence of matter formation in the evolution of partially chaotic dynamical systems. Pioneering mathematical methods investigated particle-emitting systems at various energies and relative momenta regimes [1,2]. The applications within the domains of quantum interferences include descriptions of partial coherence systems and the evolution of complex phenomena for granular, partially chaotic sources. The intrinsic peculiarities explored based on nonlinear dynamics principles have wave nature characteristics at specific momenta domains [3–6]. Multiplicities for the hybrid granular systems are considered as wave packets that exhibit the thermal wavelengths [7,8]. Mathematical investigations examine substantial aspects of the complex systems’ capabilities and can determine the peculiarities to consider system characteristics at various energies [9–11].

In particular, the presented formulation examines system geometries and substantial hallmarks at various temperatures. The fundamental premise of such an approach is based on the origination of partial coherence complex systems [12,13]. Therefore, multi-particle quantum correlations are preferred over those of two basic interferometers for exploring chaotic emissions. Numerous attempts to measure the internal particulars using meaningful interferences have been made previously [14]. These modern and innovative techniques identify the forms of quantum correlations and simplify important complex factors about the granular systems produced [15–17].

Several factors, such as the identification of freshly collected experimental information, the creation of new structures, and advances in mathematical modeling, might cause the approaches to shift from one kind of model to another [18]. For instance, novel methods of research could uncover previously undiscovered facets of system dynamics that qualify for the creation of fresh conceptual frameworks within the nonlinear quantum phenomenon [19,20]. The rudimentary scientific principles have explored the droplet distributions that define the random variables and the consequences of coherence on the numerous particle manifestations. The comparable correlation parameters were evaluated within the harmonic fields to examine the system characteristics [21–24]. The relative dynamical, partially granular systems exhibit various particles that are helpful for the current interferometry, yielding results that are more meaningful than previous attempts [25–27]. Various demonstrations of the structure eccentricities and probes of the various extents of complex systems synchronization have been discussed [28–31].

The system multiplicities exhibit coherent characteristics that influence the nonlinear chaotic phenomenon according to the experimental annotation, and such coherence extents manifest mutual relations within the droplet magnitudes at specified regimes [32,33]. The extent of the system’s internal coherent fractions depends on the system asymmetries and partially chaotic signals. Hence, such system fluctuations arise from resonant dissipation, flux, and source geometry [34]. The aspects of partial chaotic radiations also define the parameters that explored the chaotic measurements scenario at different fractions [35,36]. We evaluate the coherence fraction and the corresponding parameters by implementing interferometry measurements. To intensify the momenta, the degree of system temperatures and the considered coherence on the interference, as well as their corresponding normalization, are explored in the computation for the engineering applications.

This article is structured as follows: Section 2 examines the system dynamics and outlines some of its interesting properties. Section 3 demonstrates the collective behavior of the systems with correlation peculiarities. We explore the formulation of numerical findings in Section 4. In Section 5, we explain and analyze the system with various parameters and describe the coherence influences on key variables. The summary and the conclusion are presented in Section 6.

2 Methodologies and systems peculiarities

We demonstrate the techniques for evaluating the evolution of the hybrid dynamical systems and examine the coherent–chaotic characteristics for the pertinent droplets. The elucidations of the partially coherent mechanisms are described for produced systems that emanate particles that have the particular momenta at specified energies. The research suggests using dimensional configurations to analyze pragmatic scenarios and the coherence aspects that are scientifically important in modern physical processes. Quantum mechanics established the concept of interference within identical particles to explore the regions where emitted particles are chaotic. Such phenomena appear to be complex, and therefore, dynamical measurements can provide high-fidelity databases to collect system information in full-scale applications. This encourages the use of innovative methodologies about the accessible data to capture the entire matter’s peculiarities, and the systems’ computations at certain momenta regimes can be explored at various energies. It is appropriate to investigate the solvable complex problem with the chaotic parameter and its relative correlation within the considered dynamical systems. The corresponding meaningful expression becomeswhere β and B 0 represent the amplitude as well as the Basel function, and t is the interval of time between bounces. The trajectory with the field measures the system characteristics for numerical and analytical study. The matter pattern formation can compute the summation by the meaningful integral

f r , τ = β ∑ μ = − ∞ ⌊ τ / t ⌋ B 0 k f | r − r μ t | e μ t / t − τ , (1)

f r , τ = β t ∫ − ∞ τ B 0 k | r − r τ ′ | e τ ′ − τ / t d τ ′ . (2)

Such an approximation can be justified when the measurements of the horizontal movements appear to be much wider than the considered time scales at a finite temperature. These examinations show that the granular complex systems can be manipulated aswhere the corresponding parameters become

f r , τ = β ∑ μ = 0 ∞ 2 − δ μ , 0 B μ k r × Φ μ τ cos μ θ + Ω μ τ sin μ θ , (3)

Φ μ τ = ∫ − ∞ τ d τ ′ t B μ k r e τ ′ − τ / t ⁡ cos μ θ τ ′ , (4)

Ω μ τ = ∫ − ∞ τ d τ ′ t B μ k r e τ ′ − τ / t ⁡ sin μ θ τ ′ . (5)

In particular, these equations assess the dynamical system sustainability practices of different scientific contributions by evaluating their performance across key chaos indicators. The external factors present long-term structures, geometrical goals, and intermediate milestones in the medical fields. The mathematical formulation was required to investigate the configuration of the chaotic spreading produced, which was held at different temperatures and momentum regimes. Here, we clarify the excitation amplitude of the condensation within quantum mechanics, showing the ground-state regimes [37]. There are two systems in quantum computing, one with coherent sources and one with chaotic sources. In this article, we harness innovative approaches to demonstrate the interferences at higher levels within the domain. Thus, we can also obtain the derivatives of the above equations as

Φ ̇ μ = − Φ μ t m + 1 t B μ k r cos ⁡ μ θ , (6)

Ω ̇ μ = − Ω μ t m + 1 t B μ k r sin ⁡ μ θ . (7)

This is a useful method for computing the interferences among various particles developed under nonlinear dynamics analysis, and the partial derivative of the meaningful equations can be demonstrated in the component form as

∂ x f = cos ⁡ θ ∂ r − 1 r sin ⁡ θ ∂ θ f = β ∑ μ = 0 ∞ 2 − δ μ , 0 × k B μ − 1 k r K μ ⁡ cos ⁡ θ − μ B μ k r r L μ , (8)

∂ y f = sin ⁡ θ ∂ r + 1 r cos ⁡ θ ∂ θ f = β ∑ μ = 0 ∞ 2 − δ μ , 0 × k B μ − 1 k r K μ ⁡ sin ⁡ θ + μ B n k r r M μ . (9)

The derivations for granular dynamical systems within hydrodynamics have peculiar relations with the correlation functions and, therefore, transport matrices in such systems are defined in the form of a response function. The coefficients are introduced to demonstrate granular nonlinear systems and analyze the collisions at different energies and temperatures. Such nonlinear response problems can be expressed with statistical mechanics and hydrodynamics where the hydrodynamics field η α ( r s , t ) (density and temperature of granular and flow velocity) is formulated as functions of time and position with small deviations δ η α ( r s , t ). The intriguing formulation can be expressed in terms of R μ , ν ( r s , t )

δ η μ r s , t = ∑ ν ∫ d r s ′ R μ ν r s − r s ′ , t δ η ν r s ′ , 0 . (10)

These functions can be computed to obtain the results of the system coherent expansions and their cooling ratios. Such transparent coefficients with hydrodynamic equations exhibit the flux correlation function containing the thermal limit of volume Ω and particle multiplicity N. Such a correlation can be formulated in terms of an integral as

ξ μ ν ζ s , T g ; t ≡ 1 Ω ∑ σ ∫ d Γ ϕ μ Γ ; ζ s , T g U ν σ t , T g ϒ σ Γ ; ζ s , T g . (11)

The flux ϕ μ is associated with energy, densities, and momentum and, thus, the ϒ σ second flux has a mutual relation with new invariant densities. Hydrodynamics particularly describes fluid peculiarities regarding energy ε r s , t, number ζ s r s , t, and momentum densities ϱ r s , t. Therefore, the granular fluid elucidation presents macroscopic equations for such densities, and it is a typical point for normal fluids to replace the momentum–energy densities by the flow velocity u r s , t as well as temperature T g r s , t with ζ s. Such illustrations become where m represents the mass. Thus, ε 0 ζ s , T g demonstrates the specified function of ζ s and T g. It is very useful to express the required equation to distinguish the hydrodynamic expressions for the response functions in the following formwhich demonstrates the transport matrix at finite temperature and momenta as

ε s r s , t = 1 2 m ζ s r s , t u 2 r s , t + ε 0 ζ s r s , t , T g r s , t , (12)

ϱ r s , t = m ζ s r s , t u r s , t , (13)

∂ t − ζ c ζ s , T g T g ∂ T g + ζ s , T g ; h , t R ̃ ζ s , T g ; h , t = 0 , R ̃ μ ν ζ s , T g ; h , 0 = δ μ ν , (14)

H ζ s , T g ; k , t = − ∂ t − ζ c ζ s , T g T g ∂ T g R ̃ ζ s , T g ; k , t R ̃ − 1 ζ s , T g ; k , t = H h y ζ s , T g ; 0 − ∂ t − ζ c ζ s , T g T g ∂ T g + H h y ζ s , T g ; 0 R ̃ ζ s , T g ; k , t R ̃ − 1 ζ s , T g ; k , t . (15)

It is useful to mention that the particle production from the freeze-out regimes exhibits an original quantum particle emanation phase, like the electromagnetic wave initial phase. Such systems are known to be fully chaotic. Thus, the required expression becomes

L ̄ T g l Ψ = L ̄ T g Ψ H h y − I ζ c T g ∂ T g l − 2 H h y = L ̄ T g Ψ H h y − I ζ c T g ∂ T g l − 2 H h y + Ψ L ̄ T g H h y − I ζ c T g ∂ T g l − 2 H h y = Ψ H h y H h y − I ζ c T g ∂ T g l − 2 H h y + Ψ − ζ 0 T g ∂ ∂ θ H h y − I ζ c T g ∂ T g l − 2 H h y = Ψ H h y − I ζ c T g ∂ T g l − 1 H h y = Ψ H h y − I ζ c T g ∂ T g l . (16)

We examine the trajectory of droplets traveling in an oscillating bath of identical particles in the occurrence of a dominant potential. Such descriptions of hybrid systems use reduced rotating symmetry, which makes it possible to demonstrate the nonlinear systems techniques with a unique formulation. We acquired representations of the structure characteristics with an internal harmonic impulse to demonstrate the usefulness of symmetry simplification. The present expression explains chaotic–coherent phenomena, and their integrating bifurcations result in the formation of a global attractor. The study shows that the droplets’ angular momentum dynamically changes with temperature and momentum at a defined volume.

2.1 Dynamical analysis

The main objective of this section is to investigate the characteristics of the particle production regimes at significant energies that resulted from the collision of hybrid systems. The current analysis describes powerful statistical methods for examining the idiosyncrasies of complex system dynamics. We can classify the resilience of a system across several hundred multiplicities under coherent structures and identify its tangential relationship with temperature variations. The mathematical computation of the precise coefficients is discussed to explore the system condensations with particular parameters. More specifically, the distinctive characteristic length has an inverse relation with the produced systems radii υ g = σ g r ( v e t + R g ), which acquires an extension velocity that declines across time and has the quantifiable typical Gaussian width. Under such conditions, small multiplicities of bosons exist at the ground state corresponding to the higher temperatures, and the remaining particles occupy excess energies to maintain their existence in the excited states.

In particular, the hybrid granular systems exhibit broad distribution in the following extremely energetic states of excitement, and the considered system size becomes wider with the extension of the characteristic length, which diminishes the coherence at higher temperatures. Therefore, we take into account the significance of the temperature–particle energy relationship that plays a key role in approximating the thermodynamic cloud densities using a standard particle distribution and such phenomena expressed with a mathematical equation within the Gaussian distribution width, which increases and eventually reaches a specific broader threshold limit [38,39]. As a result, the considered coherence probabilities have a negligible significant effect on the correlation estimation at higher momenta and temperature, and the density matrices for partially chaotic sources at finite T g exhibit multiplicities. Therefore, α becomes

ζ s 1 p 2 , p 1 = ∑ l = 0 ∞ u l * p 2 u l p 1 g l Φ g e − β E ̃ l 1 − Φ g e − β E ̃ l , (17)

ζ s 1 r 2 , r 1 = ∑ l = 0 ∞ u l * r 2 u l r 1 g l Φ g e − β E ̃ l 1 − Φ g e − β E ̃ l . (18)

We specifically explore the different physical characteristics that determine the dynamics of the entire system dimension Ω at both finite and infinite ranges. In the theory of thermodynamics, the phase transition has quantitative significance at different degrees of Ω. To establish the limit, we must begin with structures with a finite consumption, and the effects of the limiting capacity are crucial in identifying the coherence–chaos properties. We obtain an extension of density using the various identities, and such expressions become

Δ ζ s = 3 π 2 m 3 m − μ c h 3 2 ∫ 0 ∞ z 2 d z c t h m 2 + 2 m m − μ c h z 2 − m 2 T g − c t h m 2 + 2 z 2 m m − μ c h − μ c h 2 T g ≅ 3 π 2 m 3 m − μ c h 3 2 ∫ 0 ∞ z 2 d z c t h z 2 m − μ c h 2 T g − c t h z 2 + 1 m − μ c h 2 T g ≅ 6 T g π 2 m 3 m − μ c h 2 ∫ 0 ∞ − z 2 d z 1 z 2 + 1 − 1 z 2 = 3 T g π m 3 m − μ c h 2 . (19)

Because the intrinsic structures of the considered volume require substantial discussions about the peculiarities of complex systems and, therefore, consider the influences of intermediate fluctuations for the whole concentration, the corresponding expressions can be written as

ζ s ≅ 1 Ω ∑ p , g ∞ ⟨ η p , g ⟩ = 3 1 Ω 1 exp m − μ c h / T g − 1 + 1 2 π 2 ∫ p 1 ∞ p 2 d p 1 exp p 2 + m 2 − μ c h / T g − 1 + 1 Ω 6 exp p 1 2 + m 2 − μ c h / T g − 1 . (20)

These equations demonstrate the density matrices in momentum and spatial spaces, respectively. Such meaningful density matrices provide the corresponding density distributions, which can be used in modern technology at finite temperatures as

ζ s r = ∑ l = 0 ∞ g l Φ f e − β E ̃ l 1 − Φ f e − β E ̃ l | u l r | 2 . (21)

The left and right panels of Figure 1 illustrate the spatial densities ζ s ( r ) vs. r for two different system sizes at various temperature regimes. These panels express the small-large systems that are defined by the characteristic lengths. Such systems of sources become distended and cool with their evolution and show their intrinsic characteristics at various momenta scales. The densities have domination at small r for the fixed temperature, and they increase while decreasing the temperature due to the coherence implications of the emission process. The densities of the small system exhibit their domination over those of the large system at fixed temperature and r. There are various causes for such rising densities due to the coherent particle emissions, which compelled the particles to collapse at a lower temperature, as proposed by Bose and Einstein. However, the system with large geometry, as shown in the right panel, shows more chaotic characteristics at all the temperatures for fixed particle multiplicities than a system with small geometry, as shown in the opposite panel. Specifically, the panels manifest the tiered structure prompted by the coherence emissions of the complex systems within the small temperature regimes. Therefore, the analogous r-dependent congruencies revealed a wider amplification for the small system than that for the large one, which demonstrates the partially chaotic emissions at the condensation maxima.

In Figures 2a,b, we demonstrate that the r-dependent ζ s ( r ) have small multiplicities with two different sources defined by the system size parameter at various temperature regimes.

The systems under consideration exhibit a lower particle abundance with the association of wavelength at reasonably low temperatures. The density distributions show significant dependence on r and decrease progressively with declining r, which demonstrates the remarkable nonlinear aspects. It also determines the density characteristics as r varies with steady temperature, allowing examination of the properties of Bose–Einstein condensation. One can note that density exhibits enlarged peaks with lower temperatures at small domains of r, which has an inverse relationship with the system temperatures. Actually, the significant particle productions associated with moderate condensation fraction, and the corresponding densities, show their dominance at lower temperatures.

We studied the systems of energetic particles within the existence of condensation at various temperatures, along with the expansion of the sources in the hadronic medium, and observed an obvious decrease in the density of the systems in the presence of condensation at lower temperatures. However, the density profile increases with increasing particle multiplicities under specific conditions of temperature and momenta. We also explored the reciprocities of the source size with condensation and quantum correlations at various relative momenta. The higher the number of particles, the higher the chemical potential in the condensate, which suppresses the quantum correlations. It is useful to understand how the density and correlations change if most of the system particles are in the condensate. Such densities can suppress and, therefore, the corresponding correlations are higher when the maximum multiplicities arise from the excited states. The densities approach the maximum values, which suggests that the systems composed of emitted particles in the condensate become more prominent and influence the quantum interferences significantly. Peculiarities of the produced particle systems are some of the most interesting topics in the modern world, and studies about such properties of systems at finite temperatures have meaningful results.

3 Problem formulation and systems peculiarities

In this section, we explore the correlations that provide significant information about the space–time structure of the emanation systems and are crucial to understanding quantum multi-particle chaotic–coherent characteristics. Dynamical complex systems with various meaningful quantum–interference equations are explored at various momenta, and these techniques are used to compute the interferences at a higher order through granular droplets for nonlinear radiated particle production systems. Therefore, we enumerate the formulation within the partial chaotic emissions aswhere Ψ g r ( 1,2 ) represents the single pair interferences at higher energies. The numerical values in such measurements fluctuate between one and zero for chaotic and coherent systems, respectively [38]. In particular, the third-order interference versus relative momenta, which has the radius R g a, can be demonstrated with quantum interferences that show suppression by condensation. Therefore, the higher-order interferences are dependent on the system condensation fractions. Nonlinear dynamics can be used to learn more about the spatial region that emitted the particles. Chaos, coherence, droplet multiplicity, and their corresponding radii are important elements of the current study for hybrid granular chaotic systems. The substantial correlation in the multi-particle interferences is determined by considering the mutual relations that identify droplet multiplicities as independent variables [27,29]. The present study also explores the rudimentary interferences of pairs at higher-order interferences, and the corresponding correlations for hybrid granular systems can be demonstrated with a quantum mechanical expression. The mathematical expression for full quantum interferences at third order can be expressed as

C 2 g p 1 , p 2 = 1 + Ψ g r 1,2 , (22)

C g p 3 , p 2 , p 1 = ∫ D 3 p 3 , p 2 , p 1 ; ψ 3 , ψ 2 , ψ 1 Ψ p 3 , p 2 , p 1 ; ψ 3 , ψ 2 , ψ 1 2 d ψ 3 d ψ 2 d ψ 1 ∫ D 3 p 3 , p 2 , p 1 ; ψ 3 , ψ 2 , ψ 1 d ψ 3 d ψ 2 d ψ 1 . (23)

Here, D ( Ω ) delineates the droplets’ central normal distribution. Intensity interferences of identical bosons reveal the size and shape of the particle-emanated system of the considered particle. The width of the correlations indicates the radii of the system. The main goal of the current exploration is the application of Bose–Einstein statistics to explore the coherent–chaotic peculiarities [34,35]. The correlation function that examines the matter structure includes the existence of condensation at lower temperatures and can be probed in the form of the normalized intercepts for the systems of granular sources.

In addition, when triplets of particles are emitted from different or single-source droplets, then the corresponding additional expressions appear consecutively. The bottom expression of the equation shows the triplet intensity interference under the computational methods when pairs and triplets of three particles are emitted from the same and from different droplets that belong to distinct subsystems. Specifically, such interferences exhibit a peculiar hallmark of the partially chaotic–coherent systems different from the primary correlations at finite momenta. The substantial quantum interferences demonstrate the analysis of the system’s chaotic dynamics and geometries within the principles of interferometry. Hence, the third-order interference with the peculiarities of partial chaotic–coherent components becomeswhere the second and third terms reveal the different and same compartments, respectively. The fourth position shows the emission of triplets when one comes from a different droplet, and the pair emanates from the same droplet. The expressions at the last and second last positions appear in the occurrence of all multiplicities from the same and distinct drops [40–42]. In particular, the more sophisticated and results-oriented task of the present study is to investigate the genuine correlations c g that are obtained with the pairs subtractions from the full correlations in order to examine the geometry of the granular systems. The simplified expression can be written as

Θ g q 3 = 1 + 1 − υ r Π 2 g q 2 + υ r 1 − ω c 2 Π 2 s q 2 + 2 υ r 1 − υ r ω c 1 − ω c 2 Π 3g1 q 3 + 2 1 − υ r 1 − 2 υ r Π 3 g q 3 + 2 υ r 2 1 − ω c 3 Π 3 s q 3 , (24)

c g q 3 = 1 + 2 Λ c 2 Π 3 s q 3 + 2 Λ c − 1 Λ c − 2 Λ c 2 Π 3 g q 3 + 2 Λ c − 1 Λ c 2 Π 3g1 q 3 . (25)

Numerous methods have been developed from several converging ideas, all of which may be thought of as the generalizations of the particle-emitting process. The principal early contributions were made by researchers working on the analysis of particle-emanated nonlinear systems. Subsequent developments with the granular structure explore the primordial mathematical findings through different fields of implementation from the core of the matter. Interference versus momentum encourages us to create a trustworthy numerical method for conducting accurate systems investigations. A meaningful and intriguing expression for genuine interference exhibiting partial condensation is

θ g q 3 = 1 + 2 υ r 1 − υ r μ c 1 − ω c 2 Π 3g1 q 3 + 2 1 − υ r 1 − 2 υ r Π 3 g q 3 + 2 υ r 2 1 − ω c 3 Π 3 s q 3 . (26)

The aim of this study is to determine the distinct peculiarities of particle-emanated systems through intensity interferometry. The system characteristics can be probed within the quantum interferences for particular scenarios that examine the geometry and intrinsic aspects of the sources by using correlations under the consequences of coherent emissions derived from the hybrid granular system as a demonstrative tool for quantum chaotic phenomena [39,42]. Hence, we establish a relationship between quantum coherence and chaos to explore the nonlinear dynamical systems. We evaluate these associations within the influences of granularity and the Gaussian paradigm, respectively.

Intensity interferometry provides superior methods for computing the dynamical properties and spatial extent of the region of freeze-out. The emitted particles of the final state stop interacting after that region and determine the correlations between multiple particles, paving a meaningful path that explores the intrinsic peculiarities of the systems. Furthermore, we understand that more advanced relationships contain more details than two-particle interferences. Therefore, we investigate higher-order interference as an objective of invariant momenta to determine the coherent percentage along the data with various dimension levels, which means that the interferences at higher-order performance can be computed using the parameters of the normalized correlators.

C f p 1 , p 2 , p 3 , p 4 = 1 + ω c Ψ 2 S a , e + 1 − ω c Ψ 2 G a , e + ω c 2 Ψ 22 s a e , i o + 1 − ω c 2 Ψ 22 g a e , i o + 2 ω c 2 Ψ 3 S a , e , i + 2 1 − ω c 1 − 2 ω c Ψ 3 G a , e , i + 2 ω c 1 − ω c Ψ 26 G a , e , i + 2 ω c 3 Ψ 4 s a , e , i , o + 2 1 − ω c 1 − 2 ω c 1 − 3 ω c Ψ 4 g a , e , i , o + 2 ω c 2 1 − ω c Ψ 41 g a , e , i , o + 2 ω c 1 − ω c 1 − 2 ω c Ψ 42 g a , e , i , o . (27)

More interestingly, under the particular conditions when minor and major radii have the specific relation s r 2 + R g 2 = R g a, then the parameters for the granular and Gaussian sources acquire the equality Ψ 2 g r = Ψ 2 p. If the whole system maneuvers as a unit droplet, then the combined radius of that droplet is s r = R g a. Hence, the assimilated matter exists as a partially thermal Gaussian source. Numerous factors influence the interferences that demonstrate the system of different phases is comprised of a granular structure. Within these components, if the particles are emitted by the long-lived resonance, then the corresponding possible evaluation of the systems via these quantum correlations is questionable. Such factors can be eliminated by using the nonlinear dynamics and methodology of normalized correlations. We have investigated the structure of the partial chaotic systems in the presence of condensation components that exhibit a relation between chaotic flow concentration and coherence fragments that have peculiar aspects of the nonlinear phenomena.

4 Numerical findings

This study applies nonlinear dynamics to investigate chaotic–coherent peculiarities and their role in dynamical systems. The exploration focuses on identifying complex system characteristics with nonlinear dynamics and exploring the substantial hybrid chaotic system characteristics. Mathematical analysis is used to reveal the nature of the systems under various scenarios, and coherent structures are also probed at finite temperature–momentum regimes. The findings highlight the importance of considering correlations and looking ahead at potential research avenues. This technique finds possible solutions that illustrate how dynamically complex systems transform between cold and hot intermediary multiplicities and provides quantitative information on coherence droplets under the influences of condensation. An important mathematical association is observed with constantly changing phase transitions of the general efficacy predicted by an extremely significant temperature. As a result, the chaotic phase has ideal quantum associations. Such systems consist of partially chaotic droplet transportation, explained by the speed of the system’s evolution. The corresponding wave function Ψ p ( x ̄ a v ) can be expressed as

ϵ p 1 ϵ p 2 d 6 Θ τ υ d 3 p 1 d 3 p 2 ≐ ∫ d 4 x 1 d 4 x 2 Γ x 1 , p 1 Γ x 2 , p 2 + | e τ e υ | 2 − 1 9 Γ c o x 1 , p 1 Γ c o x 2 , p 2 + δ τ υ Γ x 1 , p Γ x 2 , p − 1 9 Γ c o x 1 , p Γ c o x 2 , p cos Q x a v = ∫ d 4 x 1 d 4 x 2 Γ x 1 , p 1 g x 2 , p 2 + | e τ e υ | 2 − 1 9 1 + δ τ υ × Γ c o x 1 , p 1 Γ c o x 2 , p 2 + δ τ υ Γ x 1 , p Γ x 2 , p cos Q x ̄ a v = ∫ d 4 x 1 d 4 x 2 Γ c h x 1 , p 1 Γ c h x 2 , p 2 + 1 3 Γ c h x 1 , p 1 Γ c o x 2 , p 2 + Γ c o x 1 , p 1 Γ c h x 2 , p 2 + | e τ e υ | 2 Γ c o x 1 , p 1 Γ c o x 2 , p 2 + δ τ υ Γ c h x 1 , p Γ c h x 2 , p + 2 3 Γ c h x 1 , p Γ c o x 2 , p cos Q x a v . (28)

The coherent emission behaves monotonically when h = Γ c o = 0 and, thus, the radii of four momentum appear smooth for the chaotic emission Γ c h ( x , p ). Using a specific intensity interferometry approach to evaluate the system adaptability, we define the normalized perturbation index for the systems to investigate the properties of the particulate-propelled regimes. Some mathematical formulations can be used to illustrate the standardized characteristic for the triplets, and the computed findings of such parameters should be two for ideal chaotic emission. The intercept components for ideal coherent systems appeared to be zero, and r g must be zero for completely coherent systems. Therefore, the quantum computed formulation demonstrates the intrinsic peculiarities for multiple subsystems with coherence and for chaotic systems at finite momentum regimes. Thus, the normalized parameter of the third order with identical momenta is

r g α β γ = Π 3 g α , β , γ Ψ g r α , γ Ψ g r α , β Ψ g r β , γ . (29)

The investigation of the hybrid granular matter structures can be used to determine the significance of the phase transformation, provided that such transitions appear to be first order by nature. We explore techniques that reveal the substantial structure intensity interferences and examine the correlations from partially chaotic systems of droplets that have significant fluctuations. The relative momenta minima and maxima at various energies represent the signature of the structure within the phase transitions. We additionally note that the quantum dynamics within the granular structure have maximum results at a relative momentum for the droplet emissions. Therefore, we can express the dynamical illustrations in the droplet framework to explore the system’s geometry within the specific hallmarks. As a result, the intercept for the granular and the partially chaotic systems acquired a more sophisticated form, as shown below

α 3 g r = I 3 g r = Λ c − 2 Λ c Λ c − 1 . (30)

The exact atmosphere of the chaos and coherence systems created immediately under the system expansion at specific energies is depicted in Table 1. It illustrates structures that are made up of miniature, distributed, tiny drops through multiplication. These systems have gained interest due to their distinctive characteristics and extensive applications in technology and engineering. The findings of chaotic intercepts are elevated by increasing Λ c, resulting in a higher probability of particles emanating from independent droplets. Such nonlinear systems exhibit specific temperatures and then establish fragmented droplets with variations in the temperature environment. The individual droplets undergo phase transitions as long as the phase changes. To examine the system’s distinctive aspects, we evaluate the influences on standardized chaotic parameters and the accompanying intercepts. The findings demonstrate that the established fourth-order chaotic factor increases considerably with the combination of the droplets due to the emergence of numerous Λ c.

TABLE 1. Multiple chaotic intercept parameters under the influences of the intrinsic peculiarities of complex dynamical systems droplets.

Moreover, it is obvious that the related parameters α 4 g r = ( 1 − 2 υ r ) ( 1 − 3 υ r ) / ( 1 − υ r ), α 3 g r = ( 1 − 2 υ r ) / ( 1 − υ r ), and α 2 g r = ( 1 − υ r ) fluctuate significantly in various coherence domains with the divergence of υ r for the investigated properties of the hybrid chaotic spectra. The aspects of these distinct types of systems exhibit the amalgamated and abstract specific degree that demonstrates the source proliferation of nonlinear phenomena. As a result, the dynamical systems are made up of partial chaos and coherence structures containing different multiplicities. The significance of coherence in an environment of chaotic emanations is determined by suppressing the intercepting of the fourth-order fragments with the inverse of droplets parameter υ r.

Specifically, the effects of υ r deviation on the numerous intercepts revealed unique information about the fragments, different from the basic chaotic standardization components. These findings of the calibrated higher chaotic aspects intercept α 4 g r and support less statistical analysis than α 2 g r at specified υ r standards, which is consistent with modern research. Such indications represent the standardized chaotic variables at a higher order that comprise more substantial properties than the fundamental order to explain the coherent–chaotic information. As a consequence, it reinforces the shape of the intercepts associated with the considered criteria that can be used to analyze the chaos restraint to understand the source variable expansiveness. Furthermore, we surmised that the fragmented nonlinear systems form the granular structure of numerous droplets at hadronization time, and such fragmentation occurs as a consequence of the system’s violent expansion within the largest fluctuations of the earliest matter dispensation. Thus, the droplets of the system spread chaotically because of surface tension. We also consider that the droplet radii in the local frame have a Gaussian distribution and, hence, the centers of droplets are distributed randomly within the cylinders. It is obvious that the standardization of chaotic parameters exhibits lower values in the existence of significant condensation than the obtained parameters, which are consistent at lower orders. The authors believe that this work contributes to understanding the effects of chaos and condensation on the chaotic parameters of different droplets at different temperatures.

5 Results and discussions

The most important characteristics of quantum mechanics are quantum chaos and coherence, which originate from the principles of superposition and intensity interferometry, respectively. A rigorous relationship between condensation and chaos still needs improvement, even though entanglement as a sign of quantum chaos has been thoroughly explored in both the primary and multi-particle scenarios for system peculiarities. Here, we quantify the contribution of coherence to understand the chaotic quantum particle emissions. The methodology provides an intriguing domain for source exploration. There are several computational methods for which quantum coherence influences the analysis of quantum stochastic processes, but the proposed technique plays an essential role and performs very well with nonlinear dynamics to demonstrate the system’s characteristics. Certain quantum correlation is employed to demonstrate the significance of the system composition, and the findings probe the quantitative data to illustrate the conformation among particles that begin with consistent interphase during the evolution of disseminated nonlinear systems. The evolving systems are partially chaotic with droplets of organization that are made clear by the specific transformation and correlations. We explore the complex hybrid coherent systems employing nonlinear dynamics to evaluate their controllability, probe the dynamics of coherent droplets, and understand the peculiarities of the systems that created intricate matter.

The left panel of Figure 3 illustrates the correlations B r q t for the spreading granular systems with radii R g and reveals the variation at the intercept with fixed droplets Λ c. Quantum statistically, ideal chaotic systems at q t = 0 should exhibit their substantial numerical value of 6 for such quantum interferences. These results describe the influence of source size to explore the correlation widths with finite relative momenta and, thus, the corresponding data show their declination in the wide regions of q t at fixed Λ c. The causes of such suppressions are meaningful because lower q t at zero relative momenta ( q t = 0 ) correlate and have the chaotic limit, which is shown in the top solid lines. However, it is difficult for the particles to correlate at wide regimes of the relative momenta due to the huge difference in the momenta of the emitted particles. Therefore, the source dimensions have an inverse relation with q t, and it appears that the widths of the peaks are suppressed with increasing radii at specific momenta regimes. The contribution from chaotic emission has thermal particles that can remain in excited states. The emission of particles from pure coherent droplets contributes to the zero quantum interference. Thus, there is no correlation. Such pattern formations are represented by a magenta circle that touches the coherent limit one. The results fluctuate between 1 and 6 for the partially chaotic emissions.

The right panel of Figure 3 shows the interferences B r versus q t for the complex systems with the variation of droplet multiplicities Λ c at the fixed radii R g and r s in the finite momenta regimes. Such meaningful correlations increase gradually with the magnitude of the droplets and touch the chaotic limit in the presence of excess coherent droplets. The results indicate that the particles emanate from several disseminated droplets due to chaos peculiarities and demonstrate the absence of coherence characteristics. On the other hand, B r is suppressed by decreasing the droplet multiplicity Λ c as a consequence of coherence emission. One can find that the density of system droplets has an impact on the intercepts of the correlation function in the regimes of zero q t. These results show the decline in the absence of huge droplets due to the system coherence hallmarks. From the diagrams, the contribution of the excessive droplets in the correlations shows the dominance of chaotic emission over that of the coherence emanations. The interferences at modulation droplets decrease at the wide q t due to the nonidentical energy and momenta of the considered nonlinear systems.

Figures 4a,b illustrate the triplet correlations of the nonlinear dynamical systems containing several coherent–chaotic emission droplets that emanated particles at finite size. The source radius of the granular systems has the radius R g = 6 fm, which explores the connection between the system’s intrinsic geometry and the chaotic fractions. When the granular sources containing stochastic particle ejection droplets are compared with the coherence droplets, it is obvious that the intercepts are sensitive to the coherent components and increase with an increase in droplet multiplicity Λ c. Therefore, the reduction of the intercept is considerably smaller, revealing the intrinsic system characteristics. For the excess granular emitters with chaos-coherence droplets, the variation of purely chaotic correlation functions manifests in the chaotic regimes and shows an obvious deviation.

In particular, such results demonstrate that the correlations depend significantly on the droplet multiplicity Λ of the nonlinear dynamical systems with coherence ejection droplets. Note that the computation of droplet-dependent correlations examined the system composition through intercepts at zero relative momentum and shows the dwindling with increasing momentum difference. The figures also illustrate an obvious decline as the magnitude of droplets starts to decrease in specified regimes. Thus, the chaotic emissions and the corresponding interference parameters in general have certain relations with the system components.

The fundamental knowledge of the dynamics and geometries of the systems is explored in quantum correlations. Many higher-order quantum interferences with nonlinear dynamics have various time configurations, and some techniques are used to demonstrate the considered interferences in the current research. The results of the interferences examined for partially chaotic systems and, thus, the corresponding findings, probe the connection between the system aspects and the coherence fractions. Various results about correlations and normalized interferences with chaotic parameters are shown to describe the system characteristics and, therefore, the relevant chaotic–coherence consequences, which are determined as measures of correlation strength as well as the abilities at finite strength. Using such meaningful data, the consequences of coherence on chaos production reveal the system geometry and also have particular relations for systems at different multiplicities of droplets.

For the computations of nonlinear dynamical correlations, we examine the triple momentum difference variable q t 2 = q 23 2 + q 12 2 + q 31 2, where q l m = − ( p l − p m ) 2, which is also known as the absolute triplet momentum difference between l and m particles, respectively. q l m and q t are investigated with the energy and the angles of the emanated multiplicities. Small momentum differences arise from lower energies and the small angles between the emitted particles. Thus, q t, the kinematic variable, is explored to compute the correlations with quantum interferences.

Figure 5 demonstrates the granular interference Θ g for a partially chaotic structure by varying the smallest radii s r and coherent components. The top and bottom panels on the left show the purely chaotic emanations, in which we observe that the findings have the chaotic limit at zero relative momenta. Such correlations decrease gradually with increasing q t due to the absence of interference at wide q t. These results approach the chaos limit more rapidly for the small radius subsystems than for the larger sources at constant q t and ω c h. The reasons for such findings show that the relative momenta determine the size of the systems and exhibit an inverse relation with the radii of the sources. On the other hand, the top and bottom panels on the right explore the interferences for the partial chaos systems with the variation of ω c. Note that the Θ g decreases with the increasing of ω c at constant q t due to the non-existence of chaos phenomena and, thus, an excess of coherent proportion results in immediate repression of correlations. As a result, this increased coherence relationship is prompted by particle production from small single droplets.

The mean density of phase space can be determined as a function of relative momenta using the measured genuine correlation functions. At relatively high momenta, the results reveal an obvious difference from Bose–Einstein dissemination. Therefore, we investigate the genuine quantum interferences and demonstrate that the genuine three-particle interferences make significant contributions to granular source investigations. Such interferences are analyzed, and the corresponding results are extremely sensitive to nonsymmetric mechanisms of matter production, which have not been explored by lower-interference methodologies at high energies. The mix of double and triplet correlations analysis allows investigating the particle production degree of coherence, but it is difficult to probe with pair correlations alone due to the other influences on the quantum interferences. Therefore, three particles participate in this interesting correlation, and the primary correlation is subtracted to explore the nonlinear phenomena. It exhibits chaotic limits of 3 and a coherent limit of 1 at the intercepts. In such comprehensive correlations, the results show considerably more suppression than those of the full interferences and have numerous applications in engineering, technology, and medical fields.

In Figure 6, we plot the genuine correlations G r vs. q t for the nonlinear systems that have the granularity with the variation of radii at fixed droplet multiplicities. Quantum statistically, G r in the left panel contains the chaotic limit and, thus, the upper red solid line represents that limit. The bottom straight green line results represent the coherent limit in the absence of the interferences. Therefore, the results fluctuate between coherent and chaotic limits at all regimes of q t. The observed results explore the various bands at different radii for the chaotic emissions because, in such interferences, three particles take part in the quantum phenomena instead of pair particles. One can observe that the width of correlations increases with decreasing the radii at particular momenta regimes that examine the source geometry and its characteristics.

The right panel of Figure 6 demonstrates the genuine correlations for three particles versus q t relative momenta within the system, which is composed of the various droplets. Thus, correlations increase with droplet multiplicity and attain the chaotic limit of 3 at the intercept for the excessive droplets. The presented spectrum interferences attain unity considerably faster than those for the full interference peculiarities. Such findings appear because the spectra of genuine interferences consist only of triplets, and the corresponding influences of the quantum aspects disappear due to the same droplet emanations.

This appears to be an anomalous performance in the presence of a condensation degree, which significantly influences interferences, and these graphical illustrations exhibit the interesting domains at particular momenta. However, the chaotic intercepts at lower multiplicity decrease substantially due to the occurrence of the condensation mechanism, as demonstrated in the bottom green pattern. The chaotic optimum at negligible relative velocities appears to have a magnitude of 3 with an abundance of Λ c = 60, as shown in the illustration that addresses implications of its triggering defining features. The relationships indicate cancellation by the volume of little droplets Λ c = 3, which at first corresponds with condensation extraction. This results in the emission of particles originating from exactly the same coherent droplets due to extreme variation at massive q t. The associations show a peculiar decline even though there is low q t while the percentage of droplets is relatively small, demonstrating condensate peculiarities and analyzing the system’s distinctive characteristics.

The decrease starts gradually towards the large regimes of q t at fixed radii, which is an indication of noninterference due to large relative momenta. The radii and the relative momenta of the systems have an inverse relation when investigating the fundamental properties of the complex system. Such meaningful interferences attain a numerical value of 1 considerably more quickly than the value of full interferences, as shown by the green circles at the bottom.

Figure 7 explores the genuine granular interference θ g for partially chaotic structure vs. q t and the smallest radii s r within the coherent components. The panels show the interference for ideal chaos emissions, which have zero coherence at different momenta, and the findings demonstrate that the interference reaches its chaotic limit within the zero relative momenta. Such genuine interference decreases in the large regime of q t because the interference disappears at large relative momenta. These results fall to the incoherent limit at all s r for the constant q t and ω c h. The reason for this is that the genuine interferences exhibit their peculiarities more consistently with q t, which determine the geometry of the systems. However, the right top and bottom panels show the significant suppression of θ g with increasing ω c for different s r, due to the coherence phenomenon. Hence, the excess of coherence components influences the quantum interference due to Einstein correlations for the indistinguishable particles at small q t.

The systems have also been investigated by using the normalized interferences to examine whether the considered particles emanated independently. We formulated and made the measurements at high statistics about genuine interferences and computed the standardized correlator to collect the quantitative extent of chaotic emissions from the emitted particle system. We found that the strength of coherent–chaotic increases with droplet multiplicities, and the corresponding slope suppresses substantially more than that of two particles due to the condensing effect, which appears to be more effective in higher-level entanglements than in lower order, as shown in the graphical representations. Bose–Einstein interferences contained information about the density of the state, and it is worth mentioning that the basic correlation of a pair of interferences is silent about the Fourier phase within the investigated systems. Therefore, the correlations between two particles behave independently of the Fourier phase, but such phases occur in the correlations between three particles. Hence, such phases can be isolated by comparing the genuine triplet interferences to the product of the pair interferences.

We investigate the data of such normalized chaos interferences within the various droplets versus relative momentum for a higher order to examine the mutual relations for the partially chaotic system, as well as the peculiarities of partially coherent emitted droplets at finite relative momenta. The condensation appeared to be zero at high droplet multiplicity and suppressed the normalized correlation at a low multiplicity of droplets. The results of r r versus relative momenta q t at fixed Λ c are illustrated in the panels of Figure 8. The red line at the top of the panel shows the chaoticity limit over all the regimes of relative momentum, and the various spectra below this solid line show the remarkable behavior of systems with the variation of radii. The widths of the curves decrease as the radius increasesR g at the whole relative momenta and vice versa due to the inverse relation of the source radius with q t. The results of such correlations decrease significantly with increasing q t and touch zero at large relative momenta due to the absence of interferences.

In addition, the right panel of Figure 8 demonstrates the normalized correlation r r versus q t with the variation of droplets at particular radii. The results show an enhancement plateau with gradual multiplicity Λ c and reach 2 at q t = 0 where the systems have identical energy. Such results demonstrate that in the presence of significant droplet magnitude, the point where such a condition takes place is known as the chaotic limit. These correlations represent the droplet-dependent quantum interference for granular sources, and the spectra vary with the multiplicity due to the presence of condensation. With smaller droplets, the factor Λ c is suppressed, and then the corresponding intercepts start to overlay toward the small relative momenta and show the consequences of condensation at the intercept. r r for the coherence component sources tends to be zero in the wide regimes of the relative momenta. In particular, the slope for the normalized true three-particle correlations declines much faster than that of the full three-particle response curves at the same droplet multiplicity. This finding highlights the source peculiarities in distinct regimes.

In particular, such meaningful parameters of chaoticity increase with rising system temperature and droplet availability. The particles exhibit enormous energy at extreme temperatures, and the condensation influences dissipate significantly. Thus, the standardized third-order chaotic parameters stabilize to their chaotic limit when relative momentum is decreased at particular temperatures. The figures show that the normalized chaotic correlation functions are more appropriate for examining the coherence as well as the dimensions of partial coherence emission systems within nonlinear dynamics. Such correlations appear to be more sensitive to droplet multiplicities for the nonlinear systems that have coherent droplets.

Figure 9 shows the normalized partially chaotic correlation functions Γ g versus q t computed for the granular structure within the partially coherence characteristics under the variation of the small source radii s r and the coherence fraction ω c. The left panels demonstrate the interferences for different s r with the variation of coherence and relative momenta. The results of Γ g show that the standard chaos parameters exhibit a chaotic limit of 2 within the zero relative momenta. Such normalized interferences are suppressed significantly at large regimes of q t caused by the disappearance of the interferences.

The findings are consistent with the chaotic limit at the whole s r for the constant q t and ω c. Such results explored both the genuine normalized interferences that exhibit their peculiarities more consistently with q t and the geometry of the granular systems. The significant suppression of Γ g appeared with increasing ω c for different s r due to the presence of coherence emissions. Therefore, the excess of condensation influences the corresponding chaos coefficients over the whole q t regimes.

We have shown normalized chaotic parameters probe the partially coherent characteristics of dynamical systems at freeze-out, where the granular sources contained minute condensation for large droplet multiplicities. The results provide significant information by substantiating the rational connection between quantum coherence in granular fragment configurations. The quantum chaos examined in the higher-order correlations and findings was remarkably consistent with the chaos–coherent computation at wide momentum regimes. We quantitatively corroborate the coherence of dynamical systems to examine the condensate transitions and, thus, the enhancement of the momentum has meaningful characteristics. The relative strength between the system and the particle emissions performs identically without any fluctuations. The data demonstrate the correlation curves for the nonlinear dynamical systems, which also increases the connection between the corresponding characteristics.

6 Summary and conclusions

This study has successfully investigated comprehensive dynamical systems peculiarities for evaluating the chaotic–coherent fractions through specific techniques. The findings indicate that the droplets effectively prolong the complex dynamical systems’ characteristics to sustain the system temperature. The mathematical formulations provide new techniques that measure the correlation strength to determine the condensation and nonlinear systems implications. The research offers valuable insights for determining complex dynamical system geometry by different interferences with their corresponding chaotic parameters that flourish in the regimes of different momenta. The comparisons of chaotic and coherent influences exhibit peculiar relations with the system evolution. These investigations present a source-structured approach for evaluating both the condensation and the chaotic essence of expanding dynamical systems at higher temperatures. Such demonstrations connect the applications of physics and mathematics to the development of interesting experimental objectives, particularly those associated with the granular production of complex systems under condensation. These hybrid systems assessments are provided with various differential equations that enable exploration of the specific peculiarities via data analysis. The results have pragmatic applications to various fields to probe the potential of chaotic hybrid systems. These investigations could focus on production and technical concepts.

The goal of the present research focuses on ameliorating coherence–chaos fractions, incorporates innovations such as artificial intelligence and machine learning, and introduces the computations for specific industrial applications. These measurements can be integrated to improve hybrid granular systems calibrations by enabling various analyses and chaos consequences. The study primarily probes systems’ intrinsic characteristics; future work could also increase its applicability by incorporating numerical results and recommendations for further investigations. To energize aspects of chaos validation, the innovative techniques play an important role in pure condensation–chaotic prospects through various differential equations. These recommendations reflect the broader objective of supporting the modern world and improving transparency within rapidly evolving dynamical systems sustainability.

In conclusion, the study offers various techniques for evaluating the peculiarities of dynamical systems and probing intrinsic characteristics within the chaos-coherence dominations. The acquired methodologies serve as valuable tools for complex stochastic phenomena, geometrical evolution, and chaos aspects, which help drive the transitions towards the significant periodicity and localization in energy–momenta spectra. These perspectives are influential in the examined phase transitions of the created nonlinear systems. Such dynamical systems are powerful frameworks for addressing and understanding complex medical phenomena across various specialties. The research continues to expand into areas such as medical images for secure aeromedicine and therapeutic applications through chaos control algorithms and pandemic modeling, establishing chaotic dynamics as an indispensable tool for modern precision medicine that simplifies the inherent complexity of living systems. The authors believe that this research plays a key role in the influence of chaotic vibration on the chaotic parameters of finite droplets in different temperature regimes.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

Edited by: Jesus Manuel Munoz-Pacheco, Benemérita Universidad Autónoma de Puebla, Mexico

Reviewed by: Rami Ahmad El-Nabulsi, Czech Education and Scientific Net work, Czechia

Daniel Clemente-Lopez, InnovaBienestar de México, Mexico