THE SLICER MAP: MOMENTS, CORRELATIONS AND UNIVERSALITY

This thesis concerns the relation of different models of anomalous transport, and the possibility of identifying a corresponding universality class. Investigation of transport of matter in highly confining media is a very active field of research with numerous applications to bio- and nano-technology. We proceed from a model, called Slicer Map (SM), developed by Salari et al. CHAOS 25, 073113 (2015), that captures some features of anomalous transport, while being analytically tractable. The SM is a one-parameter family of non-chaotic, one-dimensional dynamical systems. Different trajectories neither converge nor separate in time, except at discrete instants, when the distance between trajectories jumps discontinuously, because they are separated by a slicer. This is reminiscent to the role of corners in polygonal billiards. The SM shows sub-, super-, and normal diffusion as a function of its control parameter α, that characterises the power-law distribution of the length of ballistic flights. Salari and co-authors analytically expressed the time dependence of the moments of positions as a function of α, and compared it with the meansquare displacement of the Lévy-Lorentz gas (LLg), that also depends on a single parameter β. The LLg is a stochastic process, that is much more complex than the SM. Surprisingly it was found that the moments of the positions distributions of the SM and the LLg have the same asymptotic behaviour when the parameters α and β are chosen in order to match the exponent of the second moment. However, moments only partially characterise transport processes. Hence in this thesis we derive analytic expressions for the position auto-correlations of the SM, and we compare them with the numerically estimated position auto-correlations of the LLg. Remarkably, the same relation that produces the agreement of the moments leads to the agreement of the position auto-correlation functions, at least for the low scatterers density of LLg. In the search of a universality class for these phenomena, we introduce an exactly solvable model called Fly-and-Die (FnD) dynamics that generates anomalous diffusion, and we derive analytical expressions for all moments of the displacements, for the position auto-correlation function, and for the velocity auto-correlation functions. The parameters of the model can be mapped to other anomalous transport processes by matching the exponents for the mean square displacement and the prefactor of the corresponding power law. Indeed, this simplification of the SM, generates the same transport regimes as the SM. It is conjectured that the FnD provides the asymptotic behaviour of all the position moments and the auto-correlation functions, for the universality class of systems whose positions statistics are dominated by the ballistic events. The conjecture is motivated by the fact that the sub-dominant terms in the SM and of the FnD contribute like the ballistic fights to the asymptotic behaviour, i.e., they contribute the maximum allowed for a system to belong to such a universality class. Different models in the class may be distinguished considering other variables. This is demonstrated here for the velocity auto-correlation function. Numerical results on the Lévy-Lorentz gas support our conjecture.