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

Abstract

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 .

Micropolar mechanics of product fractal media

“…The Carpinteri column [23] is a simple example of an anisotropic product-like fractal porous structure with a microstructure modelled by the Sierpi ński carpet in the cross-section and a Cantor set along the longitudinal axis. Natural examples of product-like fractals are found in [10][11][12][13] with applications in molecular physics, superconductivity, tumour growth and fluid flows. Also, the geometry and mechanics of planetary rings, like those of Saturn, can be handled by fractal products in polar coordinate systems [24].…”

“…Applications range over solid mechanics of fractal media (elasticity, beams, elastic-brittle fracture, viscoelasticity, among others) as well as fluid mechanics, molecular dynamics and quantum mechanics [e.g. 4,[10][11][12][13][14][15][16][17].…”

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

“…In the present study, we would like to construct an anisotropic kinetic theory in fractal dimensions starting from a completely dissimilar fractal perspective known by the concept of 'product-like fractal geometry' which was introduced by Li and Ostoja-Starzewski in [75] and entitled Li and Ostoja-Starzewski approach (LOSA). This approach describes fruitfully the nonlinear dynamics and the physics in anisotropic fractal-dimensional media [76][77][78][79][80][81][82] and has a series of motivating implications in various fields of sciences and engineering [76,[83][84][85]. Given that LOSA is productively used in anisotropic media, it will be of interest to explore its implications in kinetic biology.…”