  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
an 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 particle 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 channel direction, and a correction to the
diffusion constant that introduces a space dependent diffusion. Different forms
of this correction term have been suggested before, which we here compare for a
particular class of models. We analyze the regime of validity of the
Fick-Jacobs equation, both by means of analytical estimates and the comparisons
with numerical results for the full two dimensional stochastic dynamics. For
the nonlinear mobility we find a temperature dependence which is opposite to
that known for particle transport in periodic potentials. The influence of
entropic effects is discussed for both, the nonlinear mobility and the
effective diffusion constant.
