Lyapunov Modes of Two-Dimensional Many-Body Systems; Soft ...

766 Hoover et al.
and macroscopic dissipation results when a Nosé-Hoover thermostat is
used to control a nonequilibrium steady state. In this case the instantaneous
external entropy production rate (due to heat extracted by the
thermostat) is proportional to the sum of the instantaneous Lyapunov
exponents:
S ˙ /k — − C l.
Such microscopic-to-macroscopic connections have focussed attention
on these instantaneous (or ‘‘local’’) Lyapunov exponents, {l}, and their
time averages, {OlP}. A very recent development is the discovery of
‘‘modes’’ (6–13) corresponding to the exponents. Because our current theoretical
understanding of these modes, as well as the exponents themselves, is
still rudimentary, the present work treats only equilibrium systems. These
results can serve as a basis for extending our understanding to the farfrom-equilibrium
systems which are a strong focus of current research in
many-body dynamics.
At equilibrium, and away, the local exponents together with their
associated eigenvectors provide the fundamental microscopic description of
phase-space instability. Recent investigations (6–13) have shown that some of
the Lyapunov exponents, those describing relatively-weak instabilities with
near-zero growth rates, correspond to wavelike eigenvectors, both longitudinal
and transverse. The phase relations in these eigenvectors differ from
those of acoustic waves. The coordinate and momentum displacements in
the Lyapunov ‘‘modes’’ are ‘‘in phase’’ reflecting the exponential time
dependence elt of Lyapunov instability. In acoustic waves the coordinate
, reflecting their
and momentum displacements have a phase difference of p
2
periodic time dependence eiwt .
Computing these instantaneous eigenvectors requires additional
N-body solutions, one for each vector, so that the work for a complete
description varies as the square of the number of particles, N2 . Worse yet,
the additional computational effort involved in keeping of the order of N2 eigenvectors orthonormal, where each eigenvector has of the order of
N components, varies as N3 . Nevertheless, presentday gigahertz serialprocessor
work stations are capable of following the complete spectrum of
eigenvalues and eigenvectors for systems of a thousand particles.
The present work describes detailed computations for soft disks at
equilibrium. Though some qualitative modelike character is present in the
soft-disk results the evidence is much less convincing than corresponding
hard-disk simulations. Likewise, lattice-rotor simulations give no firm
evidence for modes in two dimensions.