  Recent work has shown that statistical arguments, seemingly well-justified in
higher dimensions, can also be used to derive reasonable, albeit less accurate,
estimates of the largest Lyapunov exponent ${\chi}$ in lower-dimensional
Hamiltonian systems. This letter explores the detailed assumptions incorporated
into these arguments. The predicted values of ${\chi}$ are insensitive to most
of these details, which can in any event be relaxed straightforwardly, but {\em
can} depend sensitively on the nongeneric form of the auto-correlation function
characterising the time-dependence of an orbit. This dependence on dynamics
implies a fundamental limitation to the application of thermodynamic arguments
to such lower-dimensional systems.

Astrophysics;Chaotic Dynamics