  We expand on two previous developments in the modeling of discrete-time
Langevin systems. One is the well-documented Gr{\o}nbech-Jensen Farago (GJF)
thermostat, which has been demonstrated to give robust and accurate
configurational sampling of the phase space. Another is the recent discovery
that also kinetics can be accurately sampled for the GJF method. Through a
complete investigation of all possible finite difference approximations to the
velocity, we arrive at two main conclusions:~1) It is not possible to define a
so-called on-site velocity such that kinetic temperature will be correct and
independent of the time step, and~2) there exists a set of infinitely many
possibilities for defining a two-point (leap-frog) velocity that measures
kinetic energy correctly for linear systems in addition to the correct
configurational statistics obtained from the GJF algorithm. We give explicit
expressions for the possible definitions, and we incorporate these into
convenient and practical algorithmic forms of the normal Verlet-type algorithms
along with a set of suggested criteria for selecting a useful definition of
velocity.
