  In this expository article, we present a systematic formal derivation of the
Kubo formula for the linear-response current due to a time-harmonic electric
field applied to non-interacting, spinless charged particles in a finite volume
in the quantum setting. We model dissipation in a transparent way by assuming a
sequence of scattering events occurring at random-time intervals modeled by a
Poisson distribution. By taking the large-volume limit, we derive special cases
of the formula for free electrons, continuum and tight-binding periodic
systems, and the nearest-neighbor tight-binding model of graphene. We present
the analogous formalism with dissipation to derive the Drude conductivity of
classical free particles.
