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The following question is about the implementation of molecular dynamics simulations with neighbor lists. In cases where the paths of individual particles are (i) stochastic and (ii) somewhat rough, without well-defined velocities, it seems reasonable (for performance' sake) to maintain neighbor lists in a way that may allow occasional unphysical overlaps between particles, but only with a low probability, and that otherwise prevents the vast majority of particles from overlapping. Particles may be capable of moving distances of order $N\sqrt{\delta t}$ in $N$ steps, but owing to randomness are typically confined to a ball whose radius is of order $\sqrt{N\delta t}$. In this case, would it be possible to reduce overlap error even further with a 'replay' step, where configurations with unphysical overlaps are discarded by restoring the simulation to an earlier snapshot, or could doing so introduce subtle statistical anomalies that would be difficult to address? The main risk seems to be in systematically undersampling orbits that are poorly represented by the neighbor list; that is, a (possibly small but nontrivial) portion of orbits, in which two particles approach each other closely from a moderate distance within a short time scale, would be omitted. Is this actually the case? And if so, are there additional and possibly more subtle fallacies being swept under the rug? (Consider, for example, the opposite limit where the neighbor list is rarely updated at all, unphysical overlaps are relatively common, and almost all orbits are omitted except those where all particles avoid each other.)

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