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The rectification of thermal motion can give rise to a steady state
flow of particles. This process is believed to occur in nature and to
be of central importance for intra-cellular transport. Ajdari and
Prost have proposed an "on-off" or "flashing" ratchet and Magnasco
has proposed a similar "tilting" or "rocking" ratchet mechanism. These developments led to new and active fields of research in statistical physics and physical chemistry. Recent work by Gillespie and Eisenberg suggests that the effectiveness of the natural transport process, in biological ion channels, depends strongly on how we model the effect of ion to ion interactions. At high local ion concentrations the effect of the crowding of charge is
significant. It is necessary to include this effect in the models. If
we are interested in average ion currents then we can replace the
complicated many-body problem with a time-average mean-field for the
distribution of charge.
To date, all analyses of artificial, human-made, ratchets require us
to neglect the effect of distributed charge. This means that the
analysis is only strictly valid for dilute solutions. The purpose of
our present paper is to include the effect of distributed charge in
the analysis of artificial Brownian ratchets.
We formulate the Brownian ratchet problem for the case where distributed charge is significant. We investigate methods of solution
and find that the finite difference approach is not adequate because
the governing equations are very "stiff." We propose an alternative
approach based on Fourier series.
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