  Recent molecular dynamics simulations show that a dilute relativistic gas
equilibrates to a Juettner velocity distribution if ensemble velocities are
measured simultaneously in the observer frame. The analysis of relativistic
Brownian motion processes, on the other hand, implies that stationary
one-particle distributions can differ depending on the underlying
time-parameterizations. Using molecular dynamics simulations, we demonstrate
how this relativistic phenomenon can be understood within a deterministic model
system. We show that, depending on the time-parameterization, one can
distinguish different types of soft ergodicity on the level of the one-particle
distributions. Our analysis further reveals a close connection between time
parameters and entropy in special relativity. A combination of different
time-parameterizations can potentially be useful in simulations that combine
molecular dynamics algorithms with randomized particle creation, annihilation,
or decay processes.
