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.
Highlights
Doug Henderson and John Barker helped to set the stage for our own Nonequilibrium developments through their equilibrium work on Thermodynamic Perturbation Theory [1]
This novel approach solved the problem of calculating accurate liquid-state thermodynamics by approximating the structure of a liquid with hard-sphere or soft-sphere pair distribution functions
In our 2004 contribution, we described Nonequilibrium Molecular Dynamics, the offshoot of classical mechanics designed to treat mechanical and thermal gradients according to generalizations of Gibbs’ statistical mechanics
Summary
Doug Henderson and John Barker helped to set the stage for our own Nonequilibrium developments through their equilibrium work on Thermodynamic Perturbation Theory [1]. This novel approach solved the problem of calculating accurate liquid-state thermodynamics by approximating the structure of a liquid with hard-sphere or soft-sphere pair distribution functions. In our 2004 contribution, we described Nonequilibrium Molecular Dynamics, the offshoot of classical mechanics designed to treat mechanical and thermal gradients according to generalizations of Gibbs’ statistical mechanics. Simulation and Control of Chaotic Nonequilibrium Systems has just been published by World Scientific Publishers in Singapore and released in March 2015
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