Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder

Abstract

New aspects of electron transport in quantum wires with Lévy-type disorder are described. We study the weak scattering and the incoherent sequential tunneling in one-dimensional quantum systems characterized by a tempered Lévy stable distribution of spacing between scatterers or tunneling barriers. The generalized Dorokhov–Mello–Pereyra–Kumar equation contains the tempered fractional derivative on wire length. The solution describes the evolution from the anomalous conductance distribution to the Dorokhov function for a long wire. For sequential tunneling, average values and relative fluctuations of conductance and resistance are calculated for different parameters of spatial distributions. A tempered Lévy stable distribution of spacing between barriers leads to a transition in conductance scaling.

Fractal Stochastic Processes on Thin Cantor-Like Sets

“…However, it should be noted that the relationship between fractals and fractional calculus is not evident, often artificial and criticized in some works [13]. In any case, the matter is not complete without introducing additional procedures, for example, averaging over realizations of random fractals [14][15][16]. A relation of fractional time derivatives to the continuous time random walk theory with fractal time behavior is presented in [14] and utilized in many works (see, e.g., [17][18][19]).…”

Fractional Schrödinger equation in gravitational optics

“…Further "fractionalization" of optical beams attracts much attention in both theoretical and numerical studies of both linear [20] and nonlinear [21,22,23] Schrödinger equations. Another important achievement relates to the fabrication and exploration of optical [24,25] and quantum fractals [26,27]. Then the fractional Laplacian (1.1) plays important role in both quantum and wavediffusion processes [27] (see Sec 2).…”

“…Another important achievement relates to the fabrication and exploration of optical [24,25] and quantum fractals [26,27]. Then the fractional Laplacian (1.1) plays important role in both quantum and wavediffusion processes [27] (see Sec 2).…”

On solutions of fractal fractional differential equations

“…These operators can be considered as upper classes of differential and integral operators as they can be converted to classical, fractal and fractional differential and integral operators in the limit cases [14,12,22,5,17,11]. For more details see [6,7,15,16,19,4,8,10,18,9,20,21].…”