

Lyapunov
Instability of ManyBody Systems 


The phasespace trajectory of a manybody system consisting
of atoms or molecules is highly Lyapunov unstable: due to the convex surface
of the particles, a small perturbation of the initial conditions typically
grows, or shrinks, exponentially with time. The rate of this growth/decay
is characterized by a set of Lyapunov exponents, the socalled Lyapunov
spectrum. The Lyapunov instability is ultimately responsible for the statistical
properties of matter.
Two recent discoveries of our group have raised much
interest:
(a) The perturbation associated with the mostunstable directions in phase
space and contributing to the maximum Lyapunov exponent is strongly localized
in space. This localization persists in the thermodynamic limit of infinitelymany
particles.
(b) For harddisk and hardsphere systems the perturbations belonging
to the smallest of the nonvanishing exponents are coherently spread out
in space, reminiscent of the classical modes of fluctuating hydrodynamics.
We refer to these periodic structures as "Lyapunov modes". There
exist nonpropagating "transversal" modes, and propagating "longitudinal"
modes.
We have characterized the Lyapunov modes and have extended
these considerations to large systems close to the thermodynamic limit.
Our results have sparked considerable theoretical activity in various
groups around the world.


Chaos
and Irreversibility 
In 1986 Professor William G. Hoover, Dr. Brad L. Holian and I discovered
that the phasespace probability density of dynamicallythermostated stationary
nonequilibrium systems collapses onto a fractal attractor. The dimension
of such a fractal is smaller than the dimension of the equilibrium phase
space. The dimensionality loss is related to the rate of irreversible
entropy production, and is a consequence of the Second Law of thermodynamics.
It explains, for example, why such systems have an "arrow of time"
and behave irreversibly in spite of their timereversible equations of
motion.
To generate a nonequilibrium flow, an external perturbation is needed.
It drives the system away from equilibrium. However, such a perturbation
performs work on the system and would raise the temperature without a
suitable thermostat or ergostat. It removes the excessive heat and establishes
a stationary state. However, the timereversible thermostats, which we
used in 1986 and which are currently the standard procedure, have led
to an interesting controversy in the literature questioning their physical
significance. Recently, we have been able to show that the dimensionality
loss mentioned above may exceed by far the dimensions contributed by the
thermostatted degrees of freedom for a system far from equilibrium.
We have also demonstrated that it is possible to replace
the timereversible dynamical thermostats by stochastic thermostats and
to construct Lyapunov spectra for such systems, which share the desired
properties with their timereversible counterparts. This shows that dynamical
thermostats are not merely convenient numerical tools but may be a first
stepping stone towards a theory of nonequilibrium stationary states.


Gravitating
Systems 
The asymptotic states of gravitational manybody systems confined to a box
depend on the number of effective constants of the motion. With reflecting
boundaries (a scenario reminiscent of a star cluster caught in the selfconsistent
field of a galaxy), the existence of an additional conserved quantity other
than the energy determines whether three bodies may reach a finite stationary
state or will collapse forever. Our recent work clarifies some of the open
questions and, as usual, provides an interesting new puzzle concerning unexpected
vanishing Lyapunov exponents and, hence, hitherto unknown constraints in
phase space.


Systems
with Negative Specific Heat 
Recent experiments on nuclear fragmentation and atomic clusters indicate
an unusual feature, namely a negative specific heat. This seems to contradict
a general result of Lebowitz and Lieb according to which a Coulomb system
of electrons and nuclei always has a positive specific heat, even in the
microcanonical ensemble. This theorem presupposes ergodicity and the thermodynamic
limit. We have recently demonstrated that the noncompliance of any of these
requirements by an experiment provides a possible pathway to a negative
specific heat.


Structure
and Dynamics of Liquid Mixtures 
In a thorough experimental study of a liquid mixture, benzonitrile  toluene,
we have carried out
i) a number of thermodynamic measurements,
ii) Rayleigh and Raman lightscattering spectroscopy,
iii) NMR 13C spinlattice relaxation measurements, and
iv) measurements of the cross correlation between
dipolar nuclear spin relaxation and chemical shift anisotropy.
Combining these data with computer simulations, we obtain
one of the mostcomprehensive pictures of the molecular
structure and the reorientational dynamics of any liquid mixture available
today.
