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.

  1. 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.

2 Answers 2


Yes, it is possible to show that the statistical behavior of the approximate system will reach that of the "exact" system. (This is true even though hard-sphere dynamics do not accurately describe molecular systems!)

The basic premise underlying molecular dynamics is the ergodic theorem, which states that, in the limit of long times, the time average of a statistical measure of the system is equivalent to its ensemble average: $$\lim_{t \to \infty} \bar{A} = \left< A \right>,$$ where $\bar{A}$ is the time average and $\left<A\right>$ is the ensemble average of the quantity $A$.

Now, in principle, the time average must be measured for an infinite period of time to get exact equality, but in practice, the results will begin to converge within experimental error as the time and the system size increases. (The exact amount of time this will take can be measured, as is the decorrelation time required to achieve statistical independence between time samples, which can itself be used as a proxy to determine how long a simulation is required.) It can be shown using backward error analysis that the error of the numerical integration of a Hamiltonian can be made exponentially small; this applies to the integrators in molecular dynamics.

While numerical errors will take you on eventually exponentially divergent trajectories, in principle, so long you're generating the states in a manner consistent with the ensemble one is trying to observe, this should not ultimately impact the results, so long as a sufficient number of samples are collected. This means, for example, if we use a thermostat to ensure constant temperature, we can only equate the results with the results of the canonical ensemble.

An excellent discussion of the ergodic theorem can be found in Tolman's Principles of Statistical Mechanics.

  • $\begingroup$ Ok well I'm glad to hear that my suspicions were correct, but I feel that this answer begs the question a bit. Haven't you buried potential subtleties in the phrase "so long as the states one visits are consistent with the ensemble one is trying to observe?" I could rephrase the question as: "how can we be certain that the results of a given EDMD simulation will be consistent with that ensemble?" If one knows the result one is looking for, then one can just run it and check, but are there mathematical theorems that guarantee this will be the case even with finite-precision arithmetic? $\endgroup$ Commented May 29, 2018 at 22:16
  • $\begingroup$ I should have clarified—what I meant is that you're trying to do an apples-to-apples comparison. That is, NVE statistics should not be compared with an NVT ensemble average, and so on. $\endgroup$
    – aeismail
    Commented May 29, 2018 at 22:23
  • $\begingroup$ Hmm I'm still unclear as to how what you've pointed out answers the question. I understand, for example, that ergodicity guarantees that if there were no numerical errors then an NVE EDMD simulation would yield the right results in the sense that time averages of observables would match ensemble averages, but I'm not convinced that numerical errors won't mess things up. I believe that they won't for some reason (call it intuition), but how can I be certain? How can I be certain that these errors won't cause massive differences between infinite precision and finite precision time averages? $\endgroup$ Commented May 29, 2018 at 22:47
  • $\begingroup$ I've enclosed some references where they prove that the errors can be made arbitrarily small. I hope that helps. $\endgroup$
    – aeismail
    Commented May 30, 2018 at 0:01
  • $\begingroup$ I'm skeptical that backward error analysis of symplectic integrators for hamiltonian systems is sufficient in this context. Perhaps it's the case that one can view the hard sphere system as a limit of a hamiltonian system consisting of soft spheres where the potential energy gets infinitely large for overlapping spheres, but even so it's also not clear to me that an event-driven algorithm would qualify as a symplectic integrator if one were to attempt a limiting description of this form. $\endgroup$ Commented May 30, 2018 at 0:14

Something that is at least very close to what your are asking for is the Shadowing Lemma, which roughly states that near any numerical solution of a dynamical system, you will find a real solution.

I am no expert on this, so I can only provide this a starting point for further research. In particular I do not know whether this translates to non-smooth dynamical systems (such as hard-sphere molecular dynamics) or whether any statement can be made about statistical properties. Still, it should provide a starting point for a literature search.

  • $\begingroup$ Thanks. As it turns out I had come across this lemma when trying to do some research on this question, but as you point out, it wasn't (and still isn't) immediately clear if such a lemma holds for hard sphere systems. It's also not clear to me if it would answer the question of to what extent a numerical simulation of such a system would allow one to compute certain qualitative and/or statistical aspects of the dynamics. +1 all the same -- this does give me more resources to learn from. $\endgroup$ Commented May 29, 2018 at 21:51

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