Stochastic Behavior of Resonant Nearly Linear Oscillator Systems in the Limit of Zero Nonlinear Coupling

Abstract

This paper investigates the classical motion of oscillator systems governed by Hamiltonians having the nearly linear form $H=\Sigma \stackrel{N}{k=1\mathrm{}}\frac{{\omega }_k}{2}({P_k}^2+{Q_k}^2)+\gamma (V_3+V_4+\cdots ),$ where $N$ is the number of oscillators, ${\omega }_k$ are the positive frequencies of the harmonic approximation, $\gamma$ is the nonlinear coupling parameter, and $V_3$, $V_4$, etc., are cubic, quartic, etc., polynomials in $Q_k$ and $P_k$. The purpose of this investigation is to demonstrate that macroscopic irreversibility is an inherent property of physical, nearly linear oscillator systems even in the limit as $\gamma$ tends to zero. Irreversibility occurs for these systems because of the appearance of resonance overlap, which causes the system trajectories to wander more or less randomly over part or most of the energy surface. Resonance overlap can occur for arbitrarily small but nonzero $\gamma$ provided that $N\ge 3$ and that the ${\omega }_k$ satisfy commensurability conditions which allow the $V_3$ and/or $V_4$ interaction terms to resonantly couple all internal degrees of freedom. These results are demonstrated through an extensive computer study of the case $N=3\mathrm{}$. This case is especially suitable for study since it possesses much of the complexity of the full many-body problem and yet is sufficiently simple to yield to level-curve analysis which provides an especially lucid pictorial display of the random motion of individual trajectories. In addition, the computer study shows that one consequence of resonance overlap is that system trajectories originally close to each other in phase space can move apart more or less exponentially with time. This exponential stirring of phase space is of the type which Gibbs envisioned as leading to irreversible behavior. In particular, the slightest uncertainty of the initial state leads very quickly to complete uncertainty of the final state. Moreover, resonant oscillator systems share ownership of exponentially divergent trajectories with gaseous systems. Indeed, this property, which is a direct consequence of the equations of motion, is perhaps the ultimate source of irreversibility.