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Abstract:

A fundamental challenge is to understand nonequilibrium statistical mechanics
starting from microscopic chaos in the equations of motion of a many-particle
system. In this thesis we summarize recent theoretical advances along these
lines. We focus on two different approaches to nonequilibrium transport:
One considers Hamiltonian dynamical systems under nonequilibrium boundary
conditions, another one suggests a non-Hamiltonian approach to nonequilibrium
situations created by external electric fields and by temperature or velocity
gradients.
A surprising result related to the former approach is that in simple
low-dimensional periodic models the deterministic transport coefficients
are typically fractal functions of control parameters. These {\em fractal
transport coefficients} yield the first central theme of this thesis. We
exemplify this phenomenon by deterministic diffusion in a simple chaotic
map. We then construct an arsenal of analytical and numerical methods for
computing further transport coefficients such as electrical conductivities
andchemical reaction rates. These methods are applied to hierarchies of
chaotic dynamical systems that are successively getting more complex, starting
from abstract one-dimensional maps generalizing a simple random walk on
the line up to particle billiards that should be directly accessible in
experiments. In all cases, the resulting transport coefficients turn out
to be either strictly fractal, or at least to be profoundly irregular.
The impact of random perturbations on these quantities is also investigated.
We furthermore provide some access roads towards a physical understanding
of these fractalities.

The second central theme is formed by a critical assessment of the non-Hamiltonian
approach to nonequilibrium transport. Here we consider situations where
the nonequilibrium constraints pump energy into a system, hence there must
be some thermal reservoir that prevents the system from heating up. For
this purpose a {\em deterministic and time-reversible modeling} of thermal
reservoirs was proposed in form of Gaussian and Nose-Hoover thermostats.
This approach yielded simple relations between fundamental quantities of
nonequilibrium statistical mechanics and of dynamical systems theory. Our
goal is to critically assesses the universality of these results. As a
vehicle of demonstration we employ the driven periodic Lorentz gas, a toy
model for the classical dynamics of an electron in a metal under application
of an electric field. Applying different types of thermal reservoirs to
this system we compare the resulting nonequilibrium steady states with
each other. Along the same lines we discuss an interacting many-particle
system under shear and heat. Finally, we outline an unexpected relationship
between deterministic thermostats and active Brownian particles modeling
biophysical cell motility.