  When a particle diffuses in a medium with spatially dependent friction
coefficient $\alpha(r)$ at constant temperature $T$, it drifts toward the low
friction end of the system even in the absence of any real physical force $f$.
This phenomenon, which has been previously studied in the context of
non-inertial Brownian dynamics, is termed "spurious drift", although the drift
is real and stems from an inertial effect taking place at the short temporal
scales. Here, we study the diffusion of particles in inhomogeneous media within
the framework of the inertial Langevin equation. We demonstrate that the
quantity which characterizes the dynamics with non-uniform $\alpha(r)$ is not
the displacement of the particle $\Delta r=r-r^0$ (where $r^0$ is the initial
position), but rather $\Delta A(r)=A(r)-A(r^0)$, where $A(r)$ is the primitive
function of $\alpha(r)$. We derive expressions relating the mean and variance
of $\Delta A$ to $f$, $T$, and the duration of the dynamics $\Delta t$. For a
constant friction coefficient $\alpha(r)=\alpha$, these expressions reduce to
the well known forms of the force-drift and fluctuation-dissipation relations.
We introduce a very accurate method for Langevin dynamics simulations in
systems with spatially varying $\alpha(r)$, and use the method to validate the
newly derived expressions.
