Defining velocities for accurate kinetic statistics in the Grønbech-Jensen Farago thermostat

We expand on two previous developments in the modeling of discrete-time Langevin systems. One is the well-documented Grønbech-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.