Fractal Spacetime Research Articles

Generalized Kelvin–Voigt Creep Model in Fractal Space–Time

Generalized Kelvin–Voigt Creep Model in Fractal Space–Time

Support of the free measure for quantum field on fractal space-time

Support of the free measure for quantum field on fractal space-time

The Nature of Pointer States and Their Role in Macroscopic Quantum Coherence

This article begins with an interdisciplinary review of a hydrodynamic approach to understanding the origins and nature of macroscopic quantum phenomena in high-temperature superconductivity, superfluidity, turbulence and biological systems. Building on this review, we consider new theoretical insights into the origin and nature of pointer states and their role in the emergence of quantum systems. The approach includes a theory of quantum coherence underpinned by turbulence, generated by a field of pointer states, which take the form of recirculating, spin-1/2 vortices (toroids), interconnected via a cascade of spin-1 vortices. Decoherence occurs when the bosonic network connecting pointer states is disrupted, leading to their localisation. Building further on this work, we explore how quantum particles (in the form of different vortex structures) could emerge as the product of a causal dynamic process, within a turbulent (fractal) spacetime. The resulting particle structures offer new insights into intrinsic spin, the probabilistic nature of the wave function and how we might consider pointer states within the standard “point source” representation of a quantum particle, which intuitively requires a more complexed description.

Torricelli’s Law in Fractal Space–Time Continuum

A new formulation of Torricelli’s law in a fractal space–time continuum is developed to compute the water discharge in fractal reservoirs. Fractal Torricelli’s law is obtained by applying fractal continuum calculus concepts using local fractional differential operators. The model obtained can be used to describe the behavior of real flows, considering the losses in non-conventional reservoirs, taking into account two additional fractal parameters α and β in the spatial and temporal fractal continuum derivatives, respectively. This model is applied to the flows in reservoirs with structures of three-dimensional deterministic fractals, such as inverse Menger sponge, Sierpinski cube, and Cantor dust. The results of the level water discharge H(t) are presented as a curve series, showing the impact and influence of fluid flow in naturally fractured reservoirs that posses self-similar properties.

Expansion of the universe on fractal time: A study on the dynamics of cosmic growth

In this paper, fractal space–time, the Hubble horizon and the energy–momentum tensor are examined in relation to the FLRW metric. It offers a Fractal Friedman equation along with its answer. Also included is the scale factor, which includes fractal structures for closed, flat and open universes. They offer fresh insights into the behavior and evolution of the universe through detailed plots that vividly illustrate their potential cosmological implications.

Log-periodic gravitational-wave background beyond Einstein gravity

Periodic patterns of logarithmic oscillations can arise in primordial curvature perturbation spectra and in the associated gravitational-wave background via different mechanisms. We show that, in the presence of log oscillations, the spectral shape of the stochastic background has a unique parametrization independent of its physical origin. We also show that this log-periodic modulation can be generated in any scenario beyond Einstein gravity endowed with an approximate discrete scale invariance, a symmetry typical of deterministic fractal spacetimes that could emerge in quantum gravity under certain conditions. We discuss how a log-oscillatory spectral shape arises from concrete inflationary models beyond Einstein gravity and the prospects for detection in Einstein Telescope and other next-generation gravitational-wave observatories. We find that these instruments will be sensitive to log-periodic features if the detection is made with a high signal-to-noise ratio (SNR) and that the error scales as .

Construction of free quantum fields on nested fractal space-times

Free boson quantum fields on unbounded nested fractal space-times are constructed with mathematical rigour. We verify analogous properties to the Wightman’s axioms except for those involving Poincaré invariance.

Fractal space-time and variations of the hubble constant

Spatial variations of the Hubble constant are considered according to Riess, et al. (2018). It is noted that the values of the Hubble constant form an almost fractal manifold. This fact suggests that the variations may be associated with local gravitational perturbations in the neighborhoods of galaxies, in which there are Cepheids and supernovae selected for measurement. The aim and purpose of the study is to show that the spatial variations of the Hubble constant may be due to the fact that the galaxies belong to outskirts of the Local Supercluster.

Numerical simulation of fractal wave propagation of a multi-dimensional nonlinear fractional-in-space Schrödinger equation

This paper studies a quantum particle traveling in a fractal space-time, which can be modelled by a fractional modification of the Schrödinger equation with variable coefficients. The Fourier spectral method is used to reveal the solution properties numerically, and the fractal properties are illustrated graphically by choosing different coefficients and different fractional orders. Some novel isosurface plots of the dynamics of pattern formation in the fractional Schrödinger equation with variable coefficients are shown.

A variational principle for fractal Klein-Gordon equation

This paper studies the Klein-Gordon equation and two modifications in an infinite Cantor set and a fractal space-time. Their variational formulations are established and discussed, and the spatio-temporal discontinuity requires both spatio-fractal derivative and temporal fractal derivative for practical applications. Some basic properties of the local fractional derivative and the two-scale fractal derivative are elucidated, and the derivation of the Euler-Lagrange equation is illustrated.

Observational constraints on the fractal cosmology

In this paper, we explore a fractal model of the universe proposed by Calcagni (J High Energy Phys 03:120, 2010) for a power-counting renormalizable field theory living in a fractal spacetime. Considering a timelike fractal profile, we derived field equations in fractal cosmology, in order to explore the structure formation and the expansion history in fractal universe. Numerical investigations based on matter power spectra diagrams report higher structure growth in fractal cosmology, being in contrast to local galaxy surveys. Additionally, according to the evolution of Hubble parameter diagrams, it can be understood that Hubble constant would decrease in fractal cosmology, which is also incompatible with low redshift estimations of H_0. So, concerning primary numerical studies, it seems that fractal cosmology is not capable to alleviate the tensions between local and global observational probes. Then, in pursuance of more accurate results, we constrain the fractal cosmology by observational data, including Planck cosmic microwave background (CMB), weak lensing, supernovae, baryon acoustic oscillations (BAO), and redshift-space distortions (RSD) data. The derived constraints on fractal dimension beta indicate that there is no considerable deviation from standard model of cosmology.

Motion of Poly-vectors in Fractal Clifford Spaces

Fractal Clifford Spaces (FCS) may be considered as a challenging approach to the unification of micro-physics and macro-physics. Trajectories of these manifolds are described by different poly-vectors describing paths and their deviation equations for not only to test particles but also for charged, rotating and rotating charged objects. Due to relaxing the condition of differentiability for all extended objects in the manifold. In this approach, a specialized Bazanski Lagrangian will be discussed to describe the fractal version of extended objects in a fractal space-time.

Fractal Modeling of Polymer Plasma Laser Ablation, Plasma Plume Tsallis Entropy and Its q-Statistics Interpretation, Part I: Theory.

Polymer plasma produced by laser ablation is investigated in a theoretical manner. In relation to the fact that the charge carrier circulation is assumed to take place on fractal curves, the so-called fractality type, electrical charge transport can be resolved by an extended scale relativity method. In addition, an elegant mathematical model, utilizing a conjecture of fractal space-time, is elaborated. The complete solution and its graphical representation for temperature distribution in two-dimensional and three-dimensional cases are successfully introduced. The discrete physical behavior and irrevocable transformation of nanoscale microdomain substructures by laser ablation are realistically examined. Further, benefiting from the interpretation of the fractal analysis, each of the experimental results can be fairly explained. On top of that, this paper presents a proof of Tsallis nonextensive q-statistics, especially for the plasma plume studied. Tsallis entropy in direct connection with fractal dynamics and chaotic-type mechanics of the plasma plume and time-series representation of plasma temperature is introduced for the first time in the present publication, and the q-statistics of the plume plasma temperature are also studied, among others.

A fractal model for current generation in porous electrodes

This article aims to investigate the solution of the fractional Troesch's equation that models the mechanism of current generation in porous electrodes by numerical integration of its equivalent power-form representation and by deriving its approximate Taylor series solution. Simulation results compare to experimental data demonstrate that the overpotential distribution and efficiency within a porous electrode, from the bottom to the top of the bed is predicted by using Troesch's fractal overpotential diffusion equation, when the fractal parameter values are α ≥ 1.2618 for a porous electrode that follows the Hausdorff dimension, or α ≥ 1.236 for El Naschie's fractal space-time.

Barrow black hole corrected-entropy model and Tsallis nonextensivity

The quantum scenario concerning Hawking radiation, gives us a precious clue that a black hole has its temperature directly connected to its area gravity and that its entropy is proportional to the horizon area. These results have shown that there exist a deep association between thermodynamics and gravity. The recently introduced Barrow formulation of back holes entropy, influenced by the spacetime geometry, shows the quantum fluctuations effects through Barrow exponent, Δ, where Δ=0 represents the usual spacetime and its maximum value, Δ=1, characterizes a fractal spacetime. The quantum fluctuations are responsible for such fractality. Loop quantum gravity approach provided the logarithmic corrections to the entropy. This correction arises from quantum and thermal equilibrium fluctuations. In this paper we have analyzed the nonextensive thermodynamical effects of the quantum fluctuations upon the geometry of a Barrow black hole. We discussed the Tsallis' formulation of this logarithmically corrected Barrow entropy to construct the equipartition law. Besides, we obtained a master equation that provides the equipartition law for any value of the Tsallis q-parameter and we analyzed several different scenarios. After that, the heat capacity were calculated and the thermal stability analysis was carried out as a function of the main parameters, namely, one of the so-called pre-factors, q and Δ.

A REMARK ON WANG’S FRACTAL VARIATIONAL PRINCIPLE

Wang et al. established successfully a variational principle in a fractal space by the semi-inverse method. This paper argues that in the fractal space, time should be also considered as a fractal, and Wang’s variational principle is extended to fractal space and fractal time.

The Solution Cosmological Constant Problem. Quantum Field Theory in Fractal Space-Time with Negative Hausdorff-Colombeau Dimensions and Dark Matter Nature

The cosmological constant problem arises because the magnitude of vacuum energy density predicted by quantum field theory is about 120 orders of magnitude larger than the value implied by cosmological observations of accelerating cosmic expansion. We pointed out that the fractal nature of the quantum space-time with negative Hausdorff- Colombeau dimensions can resolve this tension. The canonical Quantum Field Theory is widely believed to break down at some fundamental high-energy cutoff Λ* and therefore the quantum fluctuations in the vacuum can be treated classically seriously only up to this high-energy cutoff. In this paper we argue that Quantum Field Theory in fractal space-time with negative Hausdorff-Colombeau dimensions gives high-energy cutoff on natural way. We argue that there exists hidden physical mechanism which cancel divergences in canonical QED4, QCD4, Higher-Derivative - Quantum-Gravity, etc. In fact we argue that corresponding supermassive Pauli-Villars ghost fields really exists. It means that there exists the ghost-driven acceleration of the univers hidden in cosmological constant. In order to obtain desired physical result we apply the canonical Pauli-Villars regularization up to Λ*. This would fit in the observed value of the dark energy needed to explain the accelerated expansion of the universe if we choose highly symmetric masses distribution between standard matter and ghost matter below that scale Λ*,i.e., fs.m (µ) ≈ −fg.m (µ), µ = mc, µ ⩽ µeff, µeff < Λ*/c. The small value of the cosmological constant explaned by tiny violation of the symmetry between standard matter and ghost matter. Dark matter nature also explaned using a common origin of the dark energy and dark matter phenomena.

On non-linear Schrödinger equations for open quantum systems

Recently two generalized nonlinear Schr\"{o}dinger equations have been proposed by Chavanis [Eur. Phys. J. Plus 132 (2017) 286] by applying Nottale's theory of scale relativity relying on a fractal space-time to describe dissipation in quantum systems. Several existing nonlinear equations are then derived and discussed in this context leading to a continuity equation with an extra source/sink term which violates Ehrenfest theorem. An extension to describe stochastic dynamics is also carried out by including thermal fluctuations or noise of the environment. These two generalized nonlinear equations are analyzed within the Bohmian mechanics framework to describe the corresponding dissipative and stochastic dynamics in terms of quantum trajectories. Several applications of this second generalized equation which can be considered as a generalized Kostin equation have been carried out. The first application consists of the study of the position-momentum uncertainty principle in a dissiaptive dynamics. After, the so-called Brownian-Bohmian motion is investigated by calculating classical and quantum diffusion coefficients. And as a third example, transmission through a transient (time dependent) parabolic repeller is studied where the interesting phenomenon of early arrival is observed even in the stochastic dynamics although the magnitude of early arrival is reduced by friction.

New insights into the physical processes that underpin cell division and the emergence of different cellular and multicellular structures

Despite decades of focused research, a detailed understanding of the fundamental physical processes that underpin biological systems (structures and processes) remains an open challenge. Within the present paper we report on biomimetic studies, which offer new insights into the process of cell division and the emergence of different cellular and multicellular structures.Experimental studies specifically investigated the impact of including different concentrations of charged bio-molecules (cytokinin and gibberellic acid) on the growth of BaCO3−SiO2 based structures. Results highlighted the role of charge density on the emergence of long-range order, underpinned by a negentropic process. This included the growth of synthetic cell-like structures, with the intrinsic capacity to divide and change morphology at cellular and multicellular scales.Detailed study of dividing structures supports a hypothesis that cell division is dependent on the establishment of a charge-induced macroscopic quantum potential and cell-scale quantum coherence, which allows a description in terms of a macroscopic Schrödinger-like equation, based on a constant different from the Planck constant. Whilst the system does not reflect full correspondence with standard quantum mechanics, many of the phenomena that we typically associate with such a system are recovered.In addition to phenomena normally associated with the Schrödinger equation, we also unexpectedly report on the emergence of intrinsic spin as a macroscopic quantum phenomena, whose origins we account for within a four-dimensional fractal space-time and a macroscopic Pauli equation, which represents the non-relativistic limit of the Dirac equation.

Quantum Field theory in Fractal Space-Time with Negative Hausdorff-Colombeau Dimensions. The Solution Cosmological Constant Problem.

The cosmological constant problem arises because the magnitude of vacuum energy density predicted by quantum field theory is about 120 orders of magnitude larger than the value implied by cosmological observations of accelerating cosmic expansion. We pointed out that the fractal nature of the quantum space-time with negative Hausdorff- Colombeau dimensions can resolve this tension. The canonical Quantum Field Theory is widely believed to break down at some fundamental high-energy cutoff Ʌ* and therefore the quantum fluctuations in the vacuum can be treated classically seriously only up to this high-energy cutoff. In this paper we argue that Quantum Field Theory in fractal space-time with negative Hausdorff-Colombeau dimensions gives high-energy cutoff on natural way. In order to obtain disered physical result we apply the canonical Pauli-Villars regularization up to Ʌ. It means that there exist the ghost-driven acceleration of the univers hidden in cosmological constant.