On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics

El-Nabulsi, Rami Ahmad

doi:10.1098/rspa.2019.0729

“…More specific models can be developed within this continuum mechanics framework in order to tackle IBVPs of very complex, multiscale fractal materials. This approach has also led to a new tool in quantum mechanics [31].…”

Section: Discussionmentioning

confidence: 99%

Thermo-poromechanics of fractal media

Li

¹

,

Ostoja–Starzewski

²

2020

Phil. Trans. R. Soc. A.

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This article advances continuum-type mechanics of porous media having a generally anisotropic, product-like fractal geometry. Relying on a fractal derivative, the approach leads to global balance laws in terms of fractal integrals based on product measures and, then, converting them to integer-order integrals in conventional (Euclidean) space. Proposed is a new line transformation coefficient that is frame invariant, has no bias with respect to the coordinate origin and captures the differences between two fractal media having the same fractal dimension but different density distributions. A continuum localization procedure then allows the development of local balance laws of fractal media: conservation of mass, microinertia, linear momentum, angular momentum and energy, as well as the second law of thermodynamics. The product measure formulation, together with the angular momentum balance, directly leads to a generally asymmetric Cauchy stress and, hence, to a micropolar (rather than classical) mechanics of fractal media. The resulting micropolar model allowing for conservative and/or dissipative effects is applied to diffusion in fractal thermoelastic media. First, a mechanical formulation of Fick’s Law in fractal media is given. Then, a complete system of equations governing displacement, microrotation, temperature and concentration fields is developed. As a special case, an isothermal model is worked out. This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

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“…It is motivating to obtain a position-dependent charge density and electric field using LOSA. It is notable that in the absence of the magnetic field, an effective position-dependent mass mfalse~∗=(l0k/(lk−xk))1−αkm∗ may also be defined that has a wide range of applications in various areas of physics and mainly solid-state physics [17–19,78–86]. Since the electric field boldE=−normal∇αμp, then the effective electric field is given by right left right left right left right left right left right left3pt0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278em 2em 0.278emE~=−∇αμfalse~p=−∇α((l0klk−xk)αk−1μp)=−(l0klk…”

Section: Fractal Gl Theory Of Superconductivitymentioning

confidence: 99%

“…The concept of fractal has received a particular interest in nearly all fields of sciences and engineering, mainly in determining the macroscopic properties of materials [1–3], in studying elastic and continuum medium [4–10], in analysing some aspects of electromagnetic theory [11–16] and in exploring some aspects of quantum mechanics [17–23], etc. More applications of fractals in sciences are found in [24,25].…”

Section: Introductionmentioning

confidence: 99%

Superconductivity and nucleation from fractal anisotropy and product-like fractal measure

El-Nabulsi

¹

2021

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Superconductivity is analysed based on the product-like fractal measure approach with fractal dimension α introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media. Our study shows the emergence of a massless state at the boundary of the superconductor and the simultaneous occurrence of isothermal and adiabatic processes in the superconductor depending on the position of the electrons. Several physical quantities were found to be position-dependent comparable with those arising in heavy doping and p–n junction. At the boundary of the superconductor, a shrinkage of the magnetic field was observed, leading to a scenario equivalent to the Meissner–Ochsenfeld effect. An enhancement of the London penetration depth is revealed and such an improvement was observed in pnictides at the onset of commensurate spin-density-wave order inside the superconducting phase at zero temperature. The Bardeen–Cooper–Schrieffer theory was also analysed and the appearance of zero-energy states is detected. Nucleation of superconductivity in a bulk was also studied. The system acts as a quantum damped harmonic oscillator and our analysis showed that type-I superconductivity occurs when κ < 2 / ( 1 + α ) , whereas type II occurs for κ > 2 / ( 1 + α ) , where κ is the Ginzburg–Landau parameter. The transition at the passage from the ‘genuine’ to the ‘intermediate’ type-I estimates 0.767767 < α ≤ 1 .

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“…Fractal kinetic theory has been explored in surface chemical reactions [4], spin dynamics [5], adsorption on mesoporous carbons [6], reaction kinetics of proteins and enzymes [7], diffusion-limited reactions [8][9][10], non-extensive thermodynamics [11], rock fractures in fluid flows [12], colloid science [13], fluids dynamics simulations [14], porous media [15] and among others. More generally, fractals have imperative implications in materials sciences [16], fluids flows [17,18], elastic and continuum media [19], electromagnetic theory [20,21], quantum mechanics [22][23][24][25][26], astrophysics [27][28][29][30][31][32][33][34][35][36], fluid mechanics and nanofluid [37,38], transport in complex porous media [39], materials sciences [40], heat transfer [41], etc.…”

Section: Introductionmentioning

confidence: 99%

Fractal Pennes and Cattaneo–Vernotte bioheat equations from product-like fractal geometry and their implications on cells in the presence of tumour growth

El-Nabulsi

¹

2021

J. R. Soc. Interface.

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In this study, the Pennes and Cattaneo–Vernotte bioheat transfer equations in the presence of fractal spatial dimensions are derived based on the product-like fractal geometry. This approach was introduced recently, by Li and Ostoja-Starzewski, in order to explore dynamical properties of anisotropic media. The theory is characterized by a modified gradient operator which depends on two parameters: R which represents the radius of the tumour and R 0 which represents the radius of the spherical living tissue. Both the steady and unsteady states for each fractal bioheat equation were obtained and their implications on living cells in the presence of growth of a large tumour were analysed. Assuming a specific heating/cooling by a constant heat flux equivalent to the metabolic heat generation in the tissue, it was observed that the solutions of the fractal bioheat equations are robustly affected by fractal dimensions, the radius of the tumour growth and the dimensions of the living cell tissue. The ranges of both the fractal dimensions and temperature were obtained, analysed and compared with recent studies. This study confirms the importance of fractals in medicine.

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On a new fractional uncertainty relation and its implications in quantum mechanics and molecular physics

Cited by 47 publications

References 56 publications

Thermo-poromechanics of fractal media

Thermo-poromechanics of fractal media

Superconductivity and nucleation from fractal anisotropy and product-like fractal measure

Fractal Pennes and Cattaneo–Vernotte bioheat equations from product-like fractal geometry and their implications on cells in the presence of tumour growth

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