Breaking the Brownian motionAn analysis of anomalous diffusion

Resumen

Although today the concept of diffusion is understood as both stochastic (microscopic) and physical, deterministic (macroscopic), before Einstein’s works on Brownian motion both definitions ran in parallel but did not intersect. From that moment on, the fundamental statistical properties of the Brownian motion are valid both for the time evolution of stock price and for the diffusion of small particles submerged in water. However, there are countless diffusive processes in which these statistics break. In these cases we talk about an anomalous diffusion process and, although we need new formalisms to analyse this kind of dynamics, we can use some ideas from the well‐known theories of Brownian motion. The present manuscript is devoted to the study and the analysis of the anomalous dynamics. Strikingly, the causes which lead to anomalous diffusion are quite disparate, many of them are still under study. Anomalous diffusion has been observed a large number of biological environments, porous media and complex fluids. Quantum systems such as ultra‐cold atoms, exciton populations and plasma present anomalous diffusion too. The anomalous diffusion phenomenon can also be induced: in active matter usually appears anomalous diffusion associated to a self‐organised dynamics. The hydrodynamic coupling between displacement can be cause of anomalous diffusion, in this case, also collective. Anomalous diffusion usually appears in disordered or inhomogeneous media, where a diffusive particle (tracer) encounters obstacles which slow down its dynamics. This is translated in a sub‐linear dependence of the mean square displacement with time, usually called subdiffusion and characterised by a subdiffusive exponent. In Chapter 3 we analyse the diffusion of inert tracers in several fixed networks and gel structures, finding an anomalous behaviour of these tracers. We propose a relation between the subdiffusive exponent and the fraction of space that tracers have for moving inside the networks. The proposed ansatz is able to reproduce data coming from previous studies, both experimental and computational, of anomalous diffusion in gels and disordered media. Interestingly, besides the mentioned subdiffusive effect other anomalous behaviour appears as a consequence of the presence of hydrodynamic interactions. At long times, traces seem to perform long jumps when they diffuse through intricate fractal paths, leading to superdiffusive dynamics (the anomalous diffusion exponent is greater than one) compared to what occurs when the hydrodynamic coupling is not considered. In active matter is also common to find anomalous diffusion. Applying electromagnetic fields on inert particles one can control the dynamics of the system. In particular, the techniques of activation through light force fields to enhance the diffusion of small particles submerged in water are very popular. In Chapter 4 we study numerically the activated diffusion of nanoscale dumbbells whose ends are plasmonic particles which move in a optical vortex lattice. Playing with the length of the chain and the intensity of force field we can manipulate the behaviour of a dumbbell: it can move with enhanced diffusion coefficient, get trapped in a vortex, or anomalously diffuse along the diagonal of the optical plane as a consequence of the hydrodynamic coupling between the beads of the chain. This last dynamics generates collective phenomena which can lead to a superdiffusive (collective) dynamics. The influence of the diffusing particles on the environment can modify the dynamics of the particles themselves. Chapter 5 considers the dynamics of excitons (electron‐hole pairs) diffusion on perovskite layers. Experimentally it has been measured that an exciton population presents anomalous diffusion when it moves in two‐dimensional metal-halide perovskites. This anomalous behaviour is due to the presence of defects in the material, where excitons get trapped. Interestingly, the excitons themselves can tune the density of defects: when one exciton gets trapped the defect where the exciton fell disappears. Besides, the exciton is the bound state of an electron and a hole, so when they recombine a photon is emitted. The number of excitons, therefore, is not constant. Both effects, the influence on the medium and the non‐conservation of the mass, are the cause of the anomalous dynamics observed. In this work we develop a model, based on a Smoluchowski equation where the mass is not conserved, which reproduce the full dynamics of an exciton population observed experimentally. In addition, we perform numerical simulations which allow us to estimate important quantities such as the defect density, the diffusion coefficient (which decay exponentially), and the radiative decay rates. The three works already mentioned formed this thesis, in which the main character is the anomalous diffusion and how the breaking of the statistical properties of the Brownian motion leads to it.