Entropic particle transport in periodic channels
  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. 
P.S. Burada, G. Schmid, P. Talkner, P. H\"anggi, D. Reguera, and J.M.   Rub\'i