More than 100 years after the formulation of statistical mechanics
by Boltzmann, Gibbs, and many others the
dynamical foundations of this theory are still subject of active research.
The ultimate goal is to rigorously derive thermodynamic and statistical
mechanics behaviour from the microscopic equations of motion.
In recent years there has been considerable progress in the theory
of dynamical systems, establishing various links to thermodynamic
behaviour for simple model systems. From a rigorous mathematical
point of view,
the relaxation processes leading to equilibrium states
are well understood for hyperbolic
dynamical systems, and more complicated situations
are subject of current research.
Many interesting results
on the spectrum of the Perron-Frobenius operator
have been derived,
and zeta-function techniques and period orbit theory have reached
an advanced state.
dynamical models exhibiting diffusion and thermodynamic-like behaviour
have been studied in detail and some of their properties are well understood.
of functional central limit theorems have been derived for the iterates
of sufficiently strongly mixing dynamical systems, which can
ultimately help to justify some of the key assumptions
of statistical mechanics.
The proposed Durham symposium will concentrate onto the intersection
between statistical mechanics and dynamical systems. There is variety
of ways how
both theories overlap.
On the one hand,
some results from dynamical systems (proved for simple
model systems) can help to provide
a rigorous foundation for some of the key issues studied
in statistical mechanics (there relevant for more realistic
physical situations). On the other hand,
some formal mathematical methods borrowed from thermodynamics
and statistical mechanics have turned out to be very successfull
for the quantitative analysis and description of
dynamical systems, in particular if these have fractal
attractors or repellers.
These methods, originating from the pioneering
work of Sinai, Ruelle, and Bowen on the thermodynamic formalism
of dynamical systems in the early
seventies, have been further developed in the mean time and
have attracted strong interest not only
among mathematicians but also among physicists.
The quantitative characterization of chaotic motion by thermodynamic
means, the thermodynamic analysis of multifractal sets, the study
of the spectrum of transfer operators and related questions
are now a rapidly evolving branch of nonlinear science,
with applications in many different subject areas.
More recently, generalized statistical mechanics methods
(based on more general entropy measures)
have also been successfully applied
to spatially extended dynamical systems exhibiting complex behaviour,
as well as to nonequilibrium situations.
These techniques generalize the concept of Boltzmann entropy
and have many interesting applications for
systems that interact with
long-range interactions, or that
possess spatial inhomogenities of the temperature field.
The symposium will bring together
people working on both pure and applied aspects of dynamical systems
and statistical mechanics. It will stimulate
discussion, interaction and collaboration between various theoretical
and applied groups
working in dynamical systems, statistical mechanics,
and complexity. It is our intention to have a
is attended by both, pure mathematicians
and applied mathematicians/theoretical physicists.
We hope that both groups will find
a common language that will lead to new collaborations
directions of research.