  We study analytically the dynamics and the micro-structural changes of a host
medium caused by a driven tracer particle moving in a confined, quiescent
molecular crowding environment. Imitating typical settings of active
micro-rheology experiments, we consider here a minimal model comprising a
geometrically confined lattice system -- a two-dimensional strip-like or a
three-dimensional capillary-like -- populated by two types of hard-core
particles with stochastic dynamics -- a tracer particle driven by a constant
external force and bath particles moving completely at random. Resorting to a
decoupling scheme, which permits us to go beyond the linear-response
approximation (Stokes regime) for arbitrary densities of the lattice gas
particles, we determine the force-velocity relation for the tracer particle and
the stationary density profiles of the host medium particles around it. These
results are validated a posteriori by extensive numerical simulations for a
wide range of parameters. Our theoretical analysis reveals two striking
features: a) We show that, under certain conditions, the terminal velocity of
the driven tracer particle is a nonmonotonic function of the force, so that in
some parameter range the differential mobility becomes negative, and b) the
biased particle drives the whole system into a nonequilibrium steady-state with
a stationary particle density profile past the tracer, which decays
exponentially, in sharp contrast with the behavior observed for unbounded
lattices, where an algebraic decay is known to take place.
