For undamped motion in 1D and 2D single-well power-law potentials, random hard velocity reversals do not change the stationary position and velocity distributions, which equal the deterministic time averages.
Sekimoto,Stochastic Energetics(Springer Verlag, Berlin, 2010), Vol
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Deterministic and randomized motions in single-well potentials
For undamped motion in 1D and 2D single-well power-law potentials, random hard velocity reversals do not change the stationary position and velocity distributions, which equal the deterministic time averages.