Imre Péter Tóth (BME MI)
Separation of time scales and averaging in hyperbolic dynamical systems.
<br><br>
Abstract: If we want to understand how the chaotic, hyperbolic behaviour 
of
deterministic Physical models leads to phenomena observed in Statistical
Physics, a useful and interesting approach is to approximate the system 
with
a
stochastic process. If the observables of the system that we wish to study
change "slowly" compared to some "fast evolving" coordinates, and the
behaviour of these "fast variables" is suitably chaotic, then, while the
slow
variables are nearly constant, the fast have enough time to "reach
equilibrium", so the slow variables will be "driven" by some "average
values",
or possibli equilibrium fluctuations of the fast ones. This phenomenon can
be
phrased in a mathematically rigorous way, if the system has a parameter,
which
can be tuned to approach a limiting case when the slow variables become
infinitely slow. In sich a limit, after rescaling time properly, the 
process
of slow variables may converge to some "process living on its own".
Depending
on whether the "average force" driving the slow variables is zero or not,
one
may have to use different scalings to obtain a deterministic or a Markov
limit
process.
Mathematical study of this area is a currently hot topic. In this talk I
discuss one of the fundamental works of the topic, a paper of Dmitry
Dolgopyat
from 2005. In this paper he proves different cases of the above limit
behaviour under technical conditions on "strong mixing" of the fast
variables.
These contitions are phrased using one of the most modern tools in the
science
of hyperbolic dynamical systems, the "standard pairs".