Towards a Comprehensive Framework for the Analysis of Anomalous Diffusive Systems
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The modelling of transport processes in biological systems is one of the main theoretical challenges in physics, chemistry and biology. This is motivated by their essential role in the emergence of diseases, like tumour metastases, which originate from the spontaneous migration of cancer cells. Thus, improvements in their understanding could potentially pave the way for an outstanding innovation of present-day techniques in medicine. These processes often exhibit anomalous properties, which are qualitatively described by the power-law scaling of their mean square displacement, compared to the linear one of normal diffusion. Such behaviour has been often successfully explained by the celebrated continuous-time random walk model. However, recent experimental studies revealed the existence of both more complicated mean square displacement behaviour and anomalous features in other characteristic observables, e.g. the position-velocity statistics or the two point correlation functions of either the velocity or the position. Thus, in order to understand the anomalous diffusion recorded in these experiments and assess the microscopic processes underlying the observed macroscopic dynamics, one needs to have a complete tool-kit of techniques and models that can be readily compared with the experimental datasets. In this Thesis, we contribute to the construction of such a complete framework by fully characterising anomalous processes, which are described by means of a continuoustime random walk with general waiting time distributions and/or external forces that are exerted both during the jumps (as in the original model) and the waiting times. In the first case we derive both the joint statistics of these processes and their observables, specifically by obtaining a generalised fractional Feynman-Kac formula, and their multipoint correlation functions and employ them to fit the mean square displacement data of diffusing mitochondria. This result supports the experimental relevance of our formalism, which comprises general formulas for several quantities that can provide readily predictable tests to be checked in experiments. In the second case, we characterise the new anomalous processes by means of Langevin equations driven by a novel type of non Gaussian noise, which reproduces the typical fluctuations of a free diffusive continuous-time random walk. For a constant external force, we also obtain the fractional evolution equations of their position probability density function and show that, contrarily to continuous-time random walks, they are weak Galilean invariant, i.e., their position distribution in different Galilean frames is obtained by shifting the sample variable according to the relative motion of the frames. Thus, these processes provide a suitable frame-invariant framework, that could be employed to investigate the stochastic thermodynamics of anomalous diffusive processes.
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