Transport properties and diffusion of brownian particles in the deformable potential

Abstract

In this thesis, we analyze the influence of the deformable potential on the dynamics of Brownian particles and the formation of localized modes in nonlinear deformable lattices. Firstly, the directed transport in a one-dimensional overdamped, Brownian motor subjected to a travelling-wave potential with variable shape and exposed to an external bias is studied numerically. In the whole thesis, we focus our attention on the class of Remoissenet-Peyrard parametrized on-site potentials with slight modification, whose shape can be varied as a function of a parameter r, recovering the sine-Gordon shape as the special case. We demonstrate that in the presence of the travelling-wave potential the observed dynamical properties of the Brownian motor, which crucially depends on the travelling-wave speed, the intensity of the noise and the external load, respectively, is significantly influenced also by the geometry of the system. In particular, we notice that systems with broad wells and sharp barriers favour the transport under the influence of an applied load. The efficiency of the transport of Brownian motors in deformable systems remains equal to 1 (in the absence of an applied load) up to a critical value of the travelling wave speed greater than that of the pure sine-Gordon shape. Secondly, using the Langevin-Monte-Carlo method, we show that the average velocity of Brownian particles is an increasing function of the shape parameter in the overdamped case, and a decreasing function of the shape parameter in the underdamped case. In the presence of the deformable travelling-wave potential, for negative as well as positive values of the shape parameter, the underdamped case favors the transport properties in the medium. The average velocity needed to cross the potential barriers is lowest in the underdamped case. Moreover, the effective diffusion coefficient in both cases exhibits peaks, and the diffusion process enhancement is discussed for some values of the shape parameter. The distribution of Brownian particles is also analyzed in the deformed system by using the Smoluchowski equation and the finite-element methods. In the presence of an external load, the deformable potential tilts. Using the matrix continued fraction method, we compute the diffusion coefficient of Brownian particles via the dynamics factor structure at low temperature and intermediate values of friction coefficient. It is numerically found that the transport properties of Brownian particles such as the effective diffusion coefficient, the average velocity and the distribution probability are sensitive to the shape parameter r of the modified Nonsinusoidal Remoissenet-Peyrard deformable potential. The bistable behaviour and the distribution of velocity which also shed light on the diffusion anomalies are discussed for some values of the shape parameter. We show that for negative values of the shape parameter (r < 0), the average velocity versus the external tilting of Brownian particles is optimized, while for positive values (r > 0), the average velocity of Brownian particles collapses due to the geometry of the system combined with the friction. We find a power law for the effective diffusion coefficient in terms of the shape parameter r, and show that, it evolves as Deffmax ∼ r2.