Net particle current in an adiabatically driven unbiased inhomogeneous inertial system in a periodic potential

We extend Risken's matrix continued fraction method (MCFM) to solve the Fokker-Planck equation to calculate the particle current in an inertial symmetric (sinusoidal) periodic potential under the action of a constant force. We consider particle motion in a medium with a small friction coefficient varying periodically in space as the potential but with a finite phase difference phi(not equal n pi, n = 0, 1, 2,...). Though the frictional inhomogeneity is considered a less likely candidate to trigger the ratchet effect than the temperature inhomogeneity, for the former leaves the static equilibrium distribution unaffected as opposed to the latter, it results in a finite net algebraic sum of currents with symmetrically time-distributed small applied forces +/-|F|. The MCFM results, with very rich and interesting qualitative characteristics, are supported by Langevin dynamic simulations. The effects of different uniform F, temperature T and average friction coeffcient gamma(0) on the performance characteristics of the inhomogeneous ratchet are presented.