Time's Arrow for Shockwaves ; Bit-Reversible Lyapunov and "Covariant" Vectors ; Symmetry Breaking

TL;DR

This paper explores how time-reversible Newton-Hamilton equations produce irreversible shock entropy, revealing that Lyapunov vectors exhibit time-symmetry breaking, thus providing insight into the microscopic origins of the Second Law of Thermodynamics.

Contribution

It demonstrates that despite time-reversible dynamics, Lyapunov vectors reveal a symmetry breaking consistent with Time's Arrow in shockwave simulations.

Findings

Lyapunov vectors differ significantly forward and backward in time.

Time-symmetry breaking in Lyapunov vectors correlates with irreversibility.

Bit-reversible algorithms enable analysis of past states consistent with present irreversibility.

Abstract

Strong shockwaves generate entropy quickly and locally. The Newton-Hamilton equations of motion, which underly the dynamics, are perfectly time-reversible. How do they generate the irreversible shock entropy? What are the symptoms of this irreversibility? We investigate these questions using Levesque and Verlet's bit-reversible algorithm. In this way we can generate an entirely imaginary past consistent with the irreversibility observed in the present. We use Runge-Kutta integration to analyze the local Lyapunov instability of the forward and backward processes so as to identify those particles most intimately connected with the irreversibility described by the Second Law of Thermodynamics. Despite the perfect time symmetry of the particle trajectories, the fully-converged vectors associated with the largest Lyapunov exponents, forward and backward in time, are qualitatively different. The vectors display a time-symmetry breaking equivalent to Time's Arrow. That is, in autonomous Hamiltonian shockwaves the largest local Lyapunov exponents, forward and backward in time, are quite different.