Author: Marc Suñé Simon

Publisher:

Published: 2016

Total Pages: 238

ISBN-13:

Brownian motion refers to the random movement that undergo mesosized particles suspended in a simple sol- vent. Einstein's probabilistic approach to the Brownian motion is founded on the principal that it is on account of the molecular motions of heat; it can be summarized in three postulates: particles do not to interact with each other, the motion is memoryless at long times, and the distribution of displacements possesses at least two lower moments. According to the Einstein's theory, the displacements of Brownian particles ought to exhibit a Gaussian distribution whose variance is proportional to time through the diffusion coefficient, that involves the temperature and the friction coefficient. May a constant external force be applied, the mean displacement scales linearly with time. This scenario is referred to as normal transport and diffusion. The thesis aims at exploring the deviations of normal transport and diffusion to exhibit Brownian particles in a disordered medium. The method of choice are numerical simulations of the classical Langevin equation, a generalization of Newtonian equations so as to account for the Brownian trajectories. To grasp the influence of the disorder's attributes on Brownian motion is the main focus of the thesis. Further, the outcome sheds light into the physical foundations of the anomalous transport and diffusion. Complementarily, some refinements are made on the algorithms employed to simulate the stochastic differential equations. First, it is reviewed the Brownian motion in a periodic potential. According to the attained outcome, some hypothesis are conjectured for the subsequent explorations in disordered media: transport anomalies--if any-- would be only of subtransport type when the disorder is static, enhanced diffusion and superdiffusion are likely to be reached, and anomalous transport and diffusion regimes might be transient in dynamic landscapes. For overdamped Brownian particles in a disordered static potential, the anomalous regimes are characterized by the time exponents that exhibit the statistical moments of the ensemble of particle trajectories, as well as by the particle displacement distributions and the clouds of particles. This case of study bears out that the length scale of the roughness of the potential is an essential parameter in the understanding of the effect of disorder. Besides, the shape of the particle density histograms and the particle clouds have been proved to be related to the anomalies. The analogous scenario in the underdamped limit leads to the instantaneous velocity distributions, that disclose appealing properties of the system. This case of study proves that the anomalous transport and diffusion regimes occur no matter the damping, yet they come about at higher forces for high friction conditions. Overdamped Brownian motion of particles in random landscapes of moving deformable obstacles is also studied. It is settled an effective set of quantities to portray the transport and diffusion properties. The characteristic time scale constrains the time span of anomalies, and thus the subsequent steady transport and diffusion coefficients. For a given density of obstacles, both trafficking and diffusion are favored by wider and therefore fewer obstacles. To end, a high density of obstacles hinders both transport and dispersion. Algorithms to carry out the numerical simulations are discussed. A novel method to build Gaussian potential landscapes with arbitrary spatial correlation functions and the only requirement of isotropy is developed. It has the particularity that, although it uses the Fourier space, its constraints are in real space. A refreshing architec- ture for simulating random dynamic obstacles is also covered. Finally, two supplementary physical systems are addressed; the physics of particles undergoing changing viscosi- ties and confinement to quasi 2 d layers, and the transport of the motor KIF1A in a two-dimensional ratchet model that mimics a microtubule.