Time-Reversible Thermodynamic Irreversibility : One-Dimensional Heat-Conducting Oscillators and Two-Dimensional Newtonian Shockwaves

TL;DR

This paper explores how both dissipative and conservative systems, despite being time-reversible at the fundamental level, exhibit thermodynamic irreversibility through different mechanisms, illustrating key features of Lyapunov instability.

Contribution

It demonstrates that time-reversible mechanics can lead to thermodynamic irreversibility via different pathways in oscillator and shockwave models.

Findings

Oscillator systems approach an attractive dissipative limit cycle.

Shockwave simulations produce reversible near-equilibrium rarefaction fans.

Both systems exhibit notable Lyapunov instability features.

Abstract

We analyze the time-reversible mechanics of two irreversible simulation types. The first is a dissipative one-dimensional heat-conducting oscillator exposed to a temperature gradient in a three-dimensional phase space with coordinate $q$, momentum $p$, and thermostat control variable $\zeta$. The second type simulates a conservative two-dimensional $N$-body fluid with $4N$ phase variables $\{q,p\}$ undergoing shock compression. Despite the time-reversibility of each of the three oscillator equations and all of the $4N$ manybody motion equations both types of simulation are irreversible, obeying the Second Law of Thermodynamics. But for different reasons. The irreversible oscillator seeks out an attractive dissipative limit cycle. The likewise irreversible, but thoroughly conservative, Newtonian shockwave eventually generates a reversible near-equilibrium pair of rarefaction fans. Both problem types illustrate interesting features of Lyapunov instability.