Lecture on the anomalous diffusion in Condensed Matter Physics

Diffusion is a natural or artificial process that governs many phenomena in nature. The most known diffusion is the Brownian or normal motion, where the mean-square-displacement of the tracer (diffusive particle among others) increases as the square-root of time. It is not the case, however, for complex systems, where the diffusion is rather slow, because at small-scales, these media present an heterogenous structure. This kind of slow motion is called subdiffusion, where the associated mean-square-displacement increases in time, with a non trivial exponent, alpha, whose value is between 0 and 1. In this review paper, we report on new trends dealing with the study of the anomalous diffusion in Condensed Matter Physics. The study is achieved using a theoretical approach that is based on a Generalized Langevin Equation. As particular crowded systems, we choose the so-called Pickering emulsions (oil-in-water), and we are interested in how the dispersed droplets (protected by small solid charged nanoparticles) can diffuse in the continuous phase (water). Dynamic study is accomplished through the mean-square-displacement and the velocity-autocorrelation-function. Finally, a comparison with Molecular Dynamics data is made.