Time-fractional diffusion equation and its applications in physics

In physics, process involving the phenomena of diffusion and wave propagation have great relevance; these physical process are governed, from a mathematical point of view, by differential equations of order 1 and 2 in time. By introducing a fractional derivatives of order $\alpha$ in time, with $0 < \alpha < 1$ or $1 <= \alpha <= 2$, we lead to process in mathematical physics which we may refer to as fractional phenomena; this is not merely a phenomenological procedure providing an additional fit parameter. The aim of this thesis is to provide a description of such phenomena adopting a mathematical approach to the fractional calculus. The use of Fourier-Laplace transform in the analysis of the problem leads to certain special functions, scilicet transcendental functions of the Wright type, nowadays known as M-Wright functions. We will distinguish slow-diffusion processes ($0 < \alpha < 1$) from intermediate processes ($1 <=\alpha <= 2$), and we point out the attention to the applications of fractional calculus in certain problems of physical interest, such as the Neuronal Cable Theory

The Wright functions of the second kind in Mathematical Physics

In this review paper we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics.We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind. Keywords: Fractional Calculus, Wright Functions, Green's Functions, Diffusion-Wave Equation,Comment: 31 pages, 10 figure

On the asymptotics of wright functions of the second kind

The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book Fractional Calculus and Waves in Linear Viscoelasticity (2010)], Fσ(x)=◀∑▶,Mσ(x)=∞∑n=0(−x)n/n!Γ(−nσ+1−σ)  (0&lt;σ&lt;1)   for x → ± ∞ are presented. The situation corresponding to the limit σ → 1− is considered, where Mσ(x) approaches the Dirac delta function δ(x − 1). Numerical results are given to demonstrate the accuracy of the expansions derived in the paper, together with graphical illustrations that reveal the transition to a Dirac delta function as σ → 1−.</p

Wright functions of the second kind and Whittaker functions

In the framework of higher transcendental functions, the Wright functions of the second kind have increased their relevance resulting from their applications in probability theory and, in particular, in fractional diffusion processes. Here, these functions are compared with the well-known Whittaker functions in some special cases of fractional order. In addition, we point out two erroneous representations in the literature

Theory of nematic charge orders in kagome metals

Kagome metals $A$V$_3$Sb$_5$ ($A=$K, Rb, Cs) exhibit an exotic charge order (CO), involving three order parameters, with broken translation and time-reversal symmetries compatible with the presence of orbital currents. The properties of this phase are still intensely debated, and it is unclear if the origin of the CO is mainly due to electron-electron or electron-phonon interactions. Most of the experimental studies confirm the nematicity of this state, a feature that might be enhanced by electronic correlations. However, it is still unclear whether the nematic CO becomes stable at a temperature equal to ($T_{\text{nem}} = T_\text{C}$) or lower than ($T_{\text{nem}} < T_\text{C}$) the one of the CO itself. Here, we systematically characterize several CO configurations, some proposed for the new member of the family ScV$_6$Sn$_6$, by combining phenomenological Ginzburg-Landau theories, valid irrespective of the specific ordering mechanism, with mean-field analysis. We find a few configurations for the CO that are in agreement with most of the experimental findings to date and that are described by different Ginzburg-Landau potentials. We propose to use resonant ultrasound spectroscopy to experimentally characterize the order parameters of the CO, such as the number of their components and their relative amplitude, and provide an analysis of the corresponding elastic tensors. This might help understand which mean-field configuration found in our study is the most representative for describing the CO state of kagome metals, and it can provide information regarding the nematicity onset temperature $T_\text{nem}$ with respect to $T_\text{C}$.Comment: 16 pages, 4 figures + supplemental material (2 pages

Van Hove tuning of AV3Sb5 kagome metals under pressure and strain

From first-principles calculations, we investigate the structural and electronic properties of the kagome metals AV3Sb5 (A = Cs, K, Rb) under isotropic and anisotropic pressure. Charge-ordering patterns are found to be unanimously suppressed, while there is a significant rearrangement of p-type and m-type Van Hove point energies with respect to the Fermi level. Already for moderate tensile strain along the V plane and compressive strain normal to the V layer, we find that a Van Hove point can be shifted to the Fermi energy. Such a mechanism provides an invaluable tuning knob to alter the correlation profile in the kagome metal, and suggests itself for further experimental investigation. It might allow us to reconcile possible multidome superconductivity in kagome metals not only from phonons but also from the viewpoint of unconventional pairing

Van Hove tuning of AV3Sb5 kagome metals under pressure and strain

From first-principles calculations, we investigate the structural and electronic properties of the kagome metals AV3Sb5 (A=Cs, K, Rb) under isotropic and anisotropic pressure. Charge-ordering patterns are found to be unanimously suppressed, while there is a significant rearrangement of p-type and m-type Van Hove point energies with respect to the Fermi level. Already for moderate tensile strain along the V plane and compressive strain normal to the V layer, we find that a Van Hove point can be shifted to the Fermi energy. Such a mechanism provides an invaluable tuning knob to alter the correlation profile in the kagome metal, and suggests itself for further experimental investigation. It might allow us to reconcile possible multidome superconductivity in kagome metals not only from phonons but also from the viewpoint of unconventional pairing

Van Hove tuning of AV3Sb5 kagome metals under pressure and strain

From first-principles calculations, we investigate the structural and electronic properties of the kagome metals AV3Sb5 (A=Cs, K, Rb) under isotropic and anisotropic pressure. Charge-ordering patterns are found to be unanimously suppressed, while there is a significant rearrangement of p-type and m-type Van Hove point energies with respect to the Fermi level. Already for moderate tensile strain along the V plane and compressive strain normal to the V layer, we find that a Van Hove point can be shifted to the Fermi energy. Such a mechanism provides an invaluable tuning knob to alter the correlation profile in the kagome metal, and suggests itself for further experimental investigation. It might allow us to reconcile possible multidome superconductivity in kagome metals not only from phonons but also from the viewpoint of unconventional pairing