Relaxation and stochasticity in a truncated Toda lattice

This paper summarizes an investigation of the statistical properties of stochastic orbits in one fixed, time-independent potential, namely, the sixth order truncation of the Toda lattice potential, with the aim of identifying a meaningful notion of ‘‘relaxation,’’ and then of correlating this relaxation with the overall stochasticity of the orbits. For variable energies E, localized ensembles of initial conditions were constructed and the subsequent evolution of these initial data then computed numerically. One discovers thereby that, at least above a critical energy ${\mathit{E}}_0$, most ensembles of stochastic orbits exhibit a rapid evolution, exponential in time, towards a time-independent invariant distribution, not microcanonical, the form of which is independent of the choice of the initial ensemble. Moreover, for fixed energy E, the decay rate Λ associated with this exponential approach is independent of the size or location of the phase space region probed by the initial ensemble. This approach towards an invariant measure correlates directly with the sensitive dependence on initial conditions exhibited by the stochastic orbits: A small initial perturbation of the ensemble grows exponentially in time at a rate λ which depends only on the energy E and, within statistical uncertainties, the ratio Λ(E)/λ(E) is independent of E. The fact that an initial ensemble of orbits evolves towards an invariant measure suggests that Lyapunov exponents computed for individual orbits should also have a physical meaning in terms of the shorter time evolution of ensembles of orbits. This intuition is corroborated by calculations that show that, in a well-defined sense, Lyapunov exponents χ(E) characterize the ‘‘average’’ instability of ensembles of orbits that sample the invariant measure.