Normal and anomalously slow diffusion under external fields
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In the present thesis we investigated normal and anomalously slow diffusion under the influence of forces periodic in space and diffusion with anomalously slow relaxation under the influence of forces changing in time, focusing on one-dimensional models. The motivation for such studies comes from condensed matter phyics, materials science, chemical physics, nanotechnology, and molecular biology. Concerning the processes undergoing normal diffusion, the transport of Brownian particles on periodic substrates with one and two minima per period in the presence and in the absence of an external applied bias was studied. We used piecewise linear potentials which allowed us to derive the analytical expressions for the current and diffusion coefficient. We also investigated the diffusion of dimers consisting of two harmonically interacting Brownian monomers in a washboard potential, comparing the results with that of a single monomer. Concerning the anomalously slow diffusion in periodic substrates, the Lifson-Jackson result for the diffusion coefficient in a periodic potential and the Stratonovich solution for the stationary current in a washboard potential were generalized to the case of anomalous transport. Finally, the dynamics of anomalously slow processes in time-varying potential landscapes were discussed within the continuous time random walk and fractional Fokker-Planck equation descriptions. It was demonstrated that the common form of the fractional Fokker-Planck equation is not valid for time-dependent forces. A modified form of the fractional Fokker-Planck equation was derived, valid, however, for dichotomously alternating fields only. As an exactly solvable example, a periodic rectangular force was studied; the behavior of the transport was found to be very different compared to the case of normal transport processes.