Rectification Through Entropic Barriers

The dynamics of Brownian motion has widespread applications extending from transport in designed micro-channels up to its prominent role for inducing transport in molecular motors and Brownian motors. Here, Brownian transport is studied in micro-sized, two dimensional periodic channels, exhibiting periodically varying cross sections. The particles in addition are subjected to a constant external force acting alongside the direction of the longitudinal channel axis. For a fixed channel geometry, the dynamics of the two dimensional problem is characterized by a single dimensionless parameter which is proportional to the ratio of the applied force and the temperature of the environment. In such structures entropic effects may play a dominant role. Under certain conditions the two dimensional dynamics can be approximated by an effective one dimensional motion of the particle in the longitudinal direction. The Langevin equation describing this reduced, one dimensional process is of the type of the Fick-Jacobs equation. It contains an entropic potential determined by the varying extension of the eliminated transversal channel direction, and a correction to the diffusion constant that introduces a space dependent diffusion. We analyze the influence of broken channel symmetry and the validity of the FickJacobs equation. For the nonlinear mobility we find a temperature dependence which is opposite to that known for particle transport in periodic energetic potentials. The influence of entropic effects is discussed for both, the nonlinear mobility, and the effective diffusion constant. In case of broken reflection symmetry rectification occurs and there is a favored direction for particle transport. The rectification effect could be maximized due to the optimal chosen absolute value of the applied bias.