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 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 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.

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 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 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.

Thermoelektrische Bauelemente ermöglichen die Erzeugung von Elektrizität aus überschüssiger Wärme, wie sie in großen Mengen in Geräten und Prozessen entsteht. Effiziente Thermoelektrika benötigen eine hohe thermoelektrische Gütezahl, die durch elektronische und thermische Transporteigenschaften der Materialien bestimmt wird. Die Dissertation untersucht zunächst die elektronischen Transporteigenschaften zweier hochaktueller thermoelektrischer Materialien, des Schichtsystems SnSe und einer komplexen Klathrat-Legierung. Deren theoretische Beschreibung benötigt unterschiedliche Methoden, die während dieses Dissertationsprojektes implementiert, erweitert oder entwickelt wurden. Die Temperaturabhängigkeit der Leitfähigkeit von SnSe wurde mittels der Boltzmann-Transportmethode in Relaxationszeitnäherung untersucht. Wir zeigen, dass nur bei gleichzeitiger Einbeziehung von thermischer Ausdehnung des Kristallgitters und Elektron-Phonon-Streuprozessen eine gute Übereinstimmung mit Experimenten erreicht wird. Die Eigenschaften des Typ-I-Klathrats Ba8AlxSi46-x sind sowohl von der Stöchiometrie als auch von der Al-Konfiguration, d.h. der Anordnung der Al-Atome im Wirtsgitter, abhängig. Für x=16 wurde der Grundzustand als hableitend bestimmt, während Konfigurationen mit höheren Energien metallisch sind. Wir erhalten eine zuverlässige Beschreibung der elektronischen, strukturellen und Transporteigenschaften von Ba8AlxSi46-x bei endlichen Temperaturen durch Mittlungen über Konfigurationen. Mittels einer neu entwickelten Methode zur Berechnung der temperaturabhängigen effektiven Bandstruktur von Legierungen beobachten wir ein temperaturbedingtes Schließen der Bandlücke bei x=16, was mit einem Phasenübergang von partieller Ordnung zu Unordnung bei 582K einher geht. Basierend auf Gedächtnisfunktions-Modellen präsentieren wir ferner eine neue Ab-initio-Methode zur Berechnung der elektrischen Leitfähigkeit von Festkörpern mit einem Unordungspotential beliebiger Kopplungsstärke. / Thermoelectric devices convert heat into electricity, thus enabling the reuse of waste heat produced by all kinds of engines. To make this conversion process profitable, materials with a high thermoelectric figure of merit, ZT, are demanded. ZT depends on electronic and thermal transport properties. In this thesis, we study the electronic transport properties of two emerging thermoelectric materials, the layered material SnSe and a complex type-I clathrate alloy. Their reliable description requires different methodologies, that has been implemented, extended, or developed during this PhD project. For SnSe, the temperature dependence of the conductivity and the Seebeck coefficient is studied using the Boltzmann transport approach in the relaxation time approximation. We show that only by simultaneously accounting for thermal lattice expansion and electron-phonon coupling, a good agreement with experiment is reached. The properties of the type-I clathrate Ba8AlxSi46-x are determined, on the one hand, by its composition, and, on the other hand, by the configuration, i.e., the arrangement of the Al atoms in the host lattice. At the charge-compensated composition x=16, the ground-state configuration is found to be semiconducting, while configurations higher in energy are metallic. We obtain a realistic description of the electronic, structural, and transport properties of Ba8AlxSi46-x at finite temperature by using configurational thermodynamic averages. From a newly developed method to compute the finite-temperature effective band structure of alloys, we observe a temperature-driven closing of the band gap for x=16, which is concomitant with a partial order-disorder phase transition at 582K. We further present a novel ab initio memory-function approach for solids that enables the calculation of the electrical conductivity of solids in a disorder potential at arbitrary coupling strength. An application of the developed formalism is demonstrated with the example of sodium.