  We systematically develop beneficial and practical velocity measures for
accurate and efficient statistical simulations of the Langevin equation with
direct applications to computational statistical mechanics and molecular
dynamics sampling. Recognizing that the existing velocity measures for the most
statistically accurate discrete-time Verlet-type algorithms are inconsistent
with the simulated configurational coordinate, we seek to create and analyze
new velocity companions that both improve existing methods as well as offer
practical options for implementation in existing computer codes. The work is
based on the set of GJ methods that, of all methods, for all time steps within
the stability criteria correctly reproduces the most basic statistical features
of a Langevin system; namely correct Boltzmann distribution for harmonic
potentials and correct transport in the form of drift and diffusion for linear
potentials. Several new accompanying velocities exhibiting correct drift are
identified, and we expand on an earlier conclusion that only half-step
velocities can exhibit correct, time-step independent Maxwell-Boltzmann
distributions. Specific practical and efficient algorithms are given in
familiar forms, and these are used to numerically validate the analytically
derived expectations.
