No on-site finite-difference velocity can give time-step-independent kinetic temperature for the GJF scheme, while an infinite one-parameter family of two-point velocities does.
Verlet, Phys
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Defining velocities for accurate kinetic statistics in the GJF thermostat
No on-site finite-difference velocity can give time-step-independent kinetic temperature for the GJF scheme, while an infinite one-parameter family of two-point velocities does.