Numerical verification of the microscopic time reversibility of Newton's equations of motion: Fighting exponential divergence

TL;DR

This paper demonstrates that reducing numerical errors below physical perturbations in chaotic N-body simulations restores microscopic time reversibility, challenging the notion that numerical solutions are inherently irreversible due to exponential error growth.

Contribution

It shows that with sufficiently precise numerical methods, microscopic reversibility can be recovered in chaotic gravitational systems, enabling the computation of converging solutions.

Findings

Numerical errors below physical perturbations restore reversibility.

Time reversible algorithms can find initial conditions leading to converging trajectories.

Definitive solutions can be used to validate statistical results of chaotic systems.

Abstract

Numerical solutions to Newton's equations of motion for chaotic self gravitating systems of more than 2 bodies are often regarded to be irreversible. This is due to the exponential growth of errors introduced by the integration scheme and the numerical round-off in the least significant figure. This secular growth of error is sometimes attributed to the increase in entropy of the system even though Newton's equations of motion are strictly time reversible. We demonstrate that when numerical errors are reduced to below the physical perturbation and its exponential growth during integration the microscopic reversibility is retrieved. Time reversibility itself is not a guarantee for a definitive solution to the chaotic N-body problem. However, time reversible algorithms may be used to find initial conditions for which perturbed trajectories converge rather than diverge. The ability to calculate such a converging pair of solutions is a striking illustration which shows that it is possible to compute a definitive solution to a highly unstable problem. This works as follows: If you (i) use a code which is capable of producing a definitive solution (and which will therefore handle converging pairs of solutions correctly), (ii) use it to study the statistical result of some other problem, and then (iii) find that some other code produces a solution S with statistical properties which are indistinguishable from those of the definitive solution, then solution S may be deemed veracious.