Deterministic diffusion is studied in simple, parameter-dependent dynamical systems. The diffusion coefficient is often a fractal function of the control parameter, exhibiting regions of scaling and self-similarity. Firstly, the concepts of chaos and deterministic diffusion are introduced in the context of dynamical systems. The link between deterministic diffusion and physical diffusion is made via random walk theory. Secondly, parameter-dependent diffusion coefficients are analytically derived by solving the Taylor-Green-Kubo formula. This is done via a recursion relation solution of fractal 'generalised Takagi functions'. This method is applied to simple one-dimensional maps and for the first time worked out fully analytically. The fractal parameter dependence of the diffusion coefficient is explained via Markov partitions. Linear parameter dependence is observed which in some cases is due to ergodicity breaking. However, other cases are due to a previously unobserved phenomenon called the 'dominating-branch' effect. A numerical investigation of the two-dimensional 'sawtooth map' yields evidence for a possible fractal structure. Thirdly, a study of different techniques for approximating the diffusion coefficient of a parameter-dependent dynamical system is then performed. The practicability of these methods, as well as their capability in exposing a fractal structure is compared. Fourthly, an analytical investigation into the dependence of the diffusion coefficient on the size and position of the escape holes is then undertaken. It is shown that varying the position has a strong effect on diffusion, whilst the asymptotic regime of small-hole size is dependent on the limiting behaviour of the escape holes. Finally, an exploration of a method which involves evaluating the zeros of a system's dynamical zeta function via the weighted Milnor-Thurston kneading determinant is performed. It is shown how to relate the diffusion coefficient to a zero of the dynamical zeta function before analytically deriving the diffusion coefficient via the kneading determinant.

The way in which macroscopic transport results from microscopic dynamics is one of the important questions in statistical physics. Dynamical systems theory play a key role in a resent advance in this direction. Offering relatively simple models which are easy to study, dynamical systems theory became a standard branch of modern nonequilibrium statistical physics. In the present work the deterministic diffusion generated by simple dynamical systems is considered. The deterministic nature of these systems is more clearly expressed through the dependencies of the transport quantities as functions of systems parameters. For fully hyperbolic dynamical systems these dependencies were found to be highly irregular and, in fact, fractal. The main focus in this work is on nonhyperbolic and on intermittent dynamical systems. First, the climbing sine map is considered which is a nonhyperbolic system with many physical applications. Then we treat anomalous dynamics generated by a paradigmatic subdiffusive map. In both cases these systems display deterministic transport which, under variation of control parameters, is fractal. For both systems we give an explanation of the observed phenomena. The third part of the thesis is devoted to the relation between chaotic and transport properties of dynamical systems. This question lies at the heart of dynamical systems theory. For closed hyperbolic dynamical systems the Pesin theorem links the sum of positive Lyapunov exponents to the Kolmogorov-Sinai entropy. For open hyperbolic systems the escape rate formula is valid. In this work we have formulated generalizations of these formulas for a class of intermittent dynamical systems where the chaotic properties are weaker.

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.

The way in which macroscopic transport results from microscopic dynamics is one of the important questions in statistical physics. Dynamical systems theory play a key role in a resent advance in this direction. Offering relatively simple models which are easy to study, dynamical systems theory became a standard branch of modern nonequilibrium statistical physics. In the present work the deterministic diffusion generated by simple dynamical systems is considered. The deterministic nature of these systems is more clearly expressed through the dependencies of the transport quantities as functions of systems parameters. For fully hyperbolic dynamical systems these dependencies were found to be highly irregular and, in fact, fractal. The main focus in this work is on nonhyperbolic and on intermittent dynamical systems. First, the climbing sine map is considered which is a nonhyperbolic system with many physical applications. Then we treat anomalous dynamics generated by a paradigmatic subdiffusive map. In both cases these systems display deterministic transport which, under variation of control parameters, is fractal. For both systems we give an explanation of the observed phenomena. The third part of the thesis is devoted to the relation between chaotic and transport properties of dynamical systems. This question lies at the heart of dynamical systems theory. For closed hyperbolic dynamical systems the Pesin theorem links the sum of positive Lyapunov exponents to the Kolmogorov-Sinai entropy. For open hyperbolic systems the escape rate formula is valid. In this work we have formulated generalizations of these formulas for a class of intermittent dynamical systems where the chaotic properties are weaker.

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.