Instabilities and Non-Reversibility of Molecular Dynamics Trajectories

The theoretical justification of the Hybrid Monte Carlo algorithm depends upon the molecular dynamics trajectories within it being exactly reversible. If computations were carried out with exact arithmetic then it would be easy to ensure such reversibility, but the use of approximate floating point arithmetic inevitably introduces violations of reversibility. In the absence of evidence to the contrary, we are usually prepared to accept that such rounding errors can be made small enough to be innocuous, but in certain circumstances they are exponentially amplified and lead to blatantly erroneous results. We show that there are two types of instability of the molecular dynamics trajectories which lead to this behavior, instabilities due to insufficiently accurate numerical integration of Hamilton's equations, and intrinsic chaos in the underlying continuous fictitious time equations of motion themselves. We analyze the former for free field theory, and show that it is essentially a finite volume effect. For the latter we propose a hypothesis as to how the Liapunov exponent describing the chaotic behavior of the fictitious time equations of motion for an asymptotically free quantum field theory behaves as the system is taken to its continuum limit, and explain why this means that instabilities in molecular dynamics trajectories are not a significant problem for Hybrid Monte Carlo computations. We present data for pure $SU(3)$ gauge theory and for QCD with dynamical fermions on small lattices to illustrate and confirm some of our results.