Fractional Classical and Quantum Transport on Networks
Thomas Michelitsch, Alejandro Pérez Riascos, Bernard Collet, Andrzej Nowakowski, Franck Nicolleau · 2019
This chapter describes classical and quantum dynamics on networks with continuous time and an evolution defined in terms of the fractional Laplacian. It explores the fractional diffusion that allows defining continuous-time random walks with a long-range dynamics providing a general framework for anomalous diffusion and navigation in networks. The chapter provides exact results for the stationary probability distribution, the average fractional probability of return and a global time that quantifies the efficiency to explore the network and coincides with the Kemeny's constant of a stochastic process. It analyzes continuous-time quantum walks that combine a dynamics with long-range displacements, similar to Levy flights, and the quantum superposition of states. The chapter explores, with some analytical results, the fractional quantum dynamics in interacting cycles, and also deals with exact results for finite rings, interacting cycles, complete graphs and infinite rings.