  We discuss the approximate phenomenological description of the motion of a
single second-class particle in a two-species totally asymmetric simple
exclusion process (TASEP) on a 1D lattice. Initially, the second class particle
is located at the origin and to its left, all sites are occupied with first
class particles while to its right, all sites are vacant. Ferrari and Kipnis
proved that in any particular realization, the average velocity of the second
class particle tends to a constant, but this mean value has a wide variation in
different histories. We discuss this phenomena, here called the TASEP Speed
Process, in an approximate effective medium description, in which the second
class particle moves in a random background of the space-time dependent average
density of the first class particles. We do this in three different
approximations of increasing accuracy, treating the motion of the second-class
particle first as a simple biassed random walk in a continuum Langevin
equation, then as a biased Markovian random walk with space and time dependent
jump rates, and finally as a Non-Markovian biassed walk with a non-exponential
distribution of waiting times between jumps. We find that, when the
displacement at time $T$ is $x_0$, the conditional expectation of displacement,
at time $zT$ ($z>1$) is $zx_0$, and the variance of the displacement only
varies as $z(z-1)T$. We extend this approach to describe the trajectories of a
tagged particle in the case of a \emph{finite} lattice, where there are $L$
classes of particles on an $L$-site line, initially placed in the order of
increasing class number. Lastly, we discuss a variant of the problem in which
the exchanges between adjacent particles happened at rates proportional to the
difference in their labels.
