What is liquid? Lyapunov instability reveals symmetry-breaking irreversibilities hidden within Hamilton's many-body equations of motion

TL;DR

This paper investigates how Lyapunov instability and symmetry-breaking in Hamiltonian many-body systems reveal hidden irreversibilities and an arrow of time, despite the equations being fundamentally time-reversible.

Contribution

It demonstrates that asymmetries in Lyapunov instabilities in Hamiltonian systems can be linked to thermodynamic irreversibility and the Second Law, using numerical simulations of colliding crystallites.

Findings

Forward and backward Lyapunov instabilities can differ qualitatively.

Asymmetries in stability relate to dissipation and the Second Law.

Numerical analysis over millions of collisions shows these effects clearly.

Abstract

Typical Hamiltonian liquids display exponential "Lyapunov instability", also called "sensitive dependence on initial conditions". Although Hamilton's equations are thoroughly time-reversible, the forward and backward Lyapunov instabilities can differ, qualitatively. In numerical work, the expected forward/backward pairing of Lyapunov exponents is also occasionally violated. To illustrate, we consider many-body inelastic collisions in two space dimensions. Two mirror-image colliding crystallites can either bounce, or not, giving rise to a single liquid drop, or to several smaller droplets, depending upon the initial kinetic energy and the interparticle forces. The difference between the forward and backward evolutionary instabilities of these problems can be correlated with dissipation and with the Second Law of Thermodynamics. Accordingly, these asymmetric stabilities of Hamilton's equations can provide an "Arrow of Time". We illustrate these facts for two small crystallites colliding so as to make a warm liquid. We use a specially-symmetrized form of Levesque and Verlet's bit-reversible Leapfrog integrator. We analyze trajectories over millions of collisions with several equally-spaced time reversals.