  Aiming to establish a rigorous link between macroscopic random motion
(described e.g. by Langevin-type theories) and microscopic dynamics, we have
undertaken a kinetic-theoretical study of the dynamics of a classical
test-particle weakly coupled to a large heat-bath in thermal equilibrium. Both
subsystems are subject to an external force field. From the (time-non-local)
generalized master equation a Fokker-Planck-type equation follows as a
"quasi-Markovian" approximation. The kinetic operator thus defined is shown to
be ill-defined; in specific, it does not preserve the positivity of the
test-particle distribution function $f(\mathbf{x}, \mathbf{v}; t)$. Adopting an
alternative approach, previously proposed for quantum open systems, is proposed
to lead to a correct kinetic operator, which yields all the expected
properties. A set of explicit expressions for the diffusion and drift
coefficients are obtained, allowing for modelling macroscopic diffusion and
dynamical friction phenomena, in terms of an external field and intrinsic
physical parameters.
