Simple hard-sphere dynamical systems can exhibit chaotic dynamics. Due to finite-precision arithmetic when implemented on a computer, the presence of chaos implies that for a given set of initial data, simulations of such systems on different machines or run with different arithmetic precisions can yield wildly different dynamics for sufficiently long times.
Nonetheless, for e.g. systems of many hard spheres in a box with reflecting boundaries, can it be mathematically proven that qualitative and/or statistical aspects of the system's finite-precision-simulated behavior will approximate its infinite-precision behavior?
I'm particularly curious about simple event-driven simulations of such systems such as that devised by Alder and Wainwright in 1959¹. I would imagine that the answer to the question is yes, because otherwise what's the use of these methods which, as far as I can tell, are ubiquitous? I can't seem to find a good, reasonably specific discussion of this anywhere. References appreciated.
- Alder, B.J. and Wainwright, T.E., 1959. Studies in molecular dynamics. I. General method. The Journal of Chemical Physics, 31(2), pp.459–466.