# Instabilities and Non-Reversibility of Molecular Dynamics Trajectories

## TL;DR

This paper investigates the causes of instability and irreversibility in molecular dynamics trajectories used in Hybrid Monte Carlo algorithms, analyzing numerical and chaotic factors, and providing evidence that these issues are manageable in quantum field theory simulations.

## Contribution

It identifies two main sources of trajectory instability—numerical inaccuracies and intrinsic chaos—and analyzes their effects, offering insights into their impact on Hybrid Monte Carlo methods.

## Key findings

- Instabilities due to numerical integration are finite volume effects.
- Chaotic behavior's Lyapunov exponent decreases towards the continuum limit.
- Empirical data from lattice QCD supports theoretical analysis.

## Abstract

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.