  We study the disorder-induced deterministic dispersion of particles uniformly
driven in an array of narrow tracks. For different toy models with quenched
disorder we obtain exact analytical expressions for the steady-state mean
velocity $v$ and the dispersion constant $D$ for any driving force $f$ above a
putative depinning threshold. For short-range correlated pinning forces we find
that at large drives $D\sim 1/v$ for random-field type of disorder while $D
\sim 1/v^3$ for the random-bond type. We show numerically that these results
are robust: the same scaling holds for models of massive damped particles, soft
particles, particles in quasi-one dimensional or two dimensional tracks, and
for a model of a magnetic domain wall with two degrees of freedom driven either
by electrical current or magnetic field. Crossover and finite temperature
effects are discussed. The universal features we identify may be relevant for
describing the fluctuating dynamics of stable localized objects such solitons,
superconducting vortices, magnetic domain walls and skyrmions, and colloids
driven in quasi one-dimensional track arrays. In particular, the drive
dependence of $D$ appears as a sensitive tool for characterizing and assessing
the nature of disorder in the host materials.
