Non-ergodicity and ageing in anomalous diffusion

Anomalous diffusion, the deviation from classical Brownian motion with its linear-in-time mean squared displacement, is a widespread phenomenon in a large variety of complex systems, ranging from amorphous semiconductors to biological cells. Anomalous diffusion processes are no longer confined by the central limit theorem, that enforces the convergence of Brownian motion to the Gaussian shape of the probability density function. Instead, anomalous processes have a rich variety of physical origins and non-traditional mathematical properties. We here provide a summary of various anomalous diffusion models, paying special attention to their non-ergodic and ageing behaviour. Whether a process is ergodic or not is of vital importance when measurements are evaluated in terms of time averaged observables such as the mean squared displacement. For non-ergodic processes these can no longer be interpreted by comparison to the typically calculated ensemble averaged observables. Breaking of ergodicity occurs in many anomalous diffusion processes. We also consider their ageing properties, the explicit dependence of observables on the time span between the original system initiation and the start of the measurement.