  Kinetic equations for the stationary state distribution function of ions
moving through narrow pores are solved for a number of one-dimensional models
of single ion transport. Ions move through pores of length $L$, under the
action of a constant external field and of a concentration gradient. The
interaction of single ions with the confining pore surface and with water
molecules inside the pore are modelled by a Fokker-Planck term in the kinetic
equation, or by uncorrelated collisions with thermalizing centres distributed
along the pore. The temporary binding of ions to polar residues lining the pore
is modelled by stopping traps or energy barriers. Analytic expressions for the
stationary ion current through the pore are derived for several versions of the
model, as functions of key physical parameters. In all cases, saturation of the
current at high fields is predicted. Such simple models, for which results are
analytic, may prove useful in the study of the current/voltage relations of ion
channels through membranes.
