  We consider an exclusion process on a periodic one-dimensional lattice where
all particles perform simple symmetric exclusion at rate $1$ except for a
single tracer particle, which performs partially simple asymmetric exclusion
with rate $p$ to the right and rate $q$ to the left. This model was first
considered by Ferrari, Goldstein and Lebowitz (Progr. Phys., 1985) as a test
for the validity of the Einstein relation in microscopic systems.
  The main thrust of this work is an exact solution for the steady state of
this exclusion process. We show that the stationary probabilities factorize and
give an exact formula for the nonequilibrium partition function. Perhaps
surprisingly, we find that the nonequilibrium free energy in the steady state
is not well-defined for this system in the thermodynamic limit for any values
of $p$ and $q$ if $p \neq q$. We provide formulas for the current and two-point
correlations. When the tracer particle performs asymmetric exclusion ($q=0$),
the results are shown to simplify significantly and we find an unexpected
connection with the combinatorics of set partitions. Finally, we study the
system from the point of view of the tracer particle, the so-called environment
process. In the environment process, we show that the density of particles
decays exponentially with the scaled position in front of the tracer particle
in the thermodynamic limit.
