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Additivity is one possible generalisation of the concept of extensivity of thermodynamic quantities for finite physical systems in thermal equilibrium. This thesis concerns itself with the additivity of the Gaussian model of a ferromagnet, originally introduced by Berlin and Kac in 1952. The main focus of the thesis lies on the treatment of the next-neighbour Gaussian model on finite lattice domains in two dimensions.

Understanding and predicting the behaviour of liquids at interfaces poses formidable scientific challenges and is highly relevant for the thriving fields of micro- and nanofluidics. External friction forces, such as in liquids slipping over solids, play a central role.
This work presents a well-founded extension of (generalised) fluctuating hydrodynamics to systems with slip boundaries or more general external friction forces. The theory naturally includes thermal fluctuations, which become important on small length scales. Moreover, they are fundamentally related to dissipative processes and reveal microscopic details of friction. A resulting fluctuating slip boundary condition is applied to derive stochastic thin film equations on slip substrates and to calculate the autocorrelation function of the tangential interaction force at a liquid-solid boundary.
In addition, a complementary approach for arbitrary small (classical) scales offers an alternative description. The formalism transfers notions from macroscopic hydrodynamics to the microscale, and external friction appears as a combination of static external forces and additional viscous dissipation.

From the large-scale structure of the universe to exotic states in nuclear matter: random or disordered spatial structures appear on nearly all length scales in very different physical, chemical, or biological systems. In systems with complex structure, there is often a close interconnection of physics and geometry, and physical insight is often best achieved by a rigorous characterization of the structure. This thesis demonstrates how a family of integral geometric shape descriptors, the so-called Minkowski functionals and tensors, provide an intuitive and versatile morphometric analysis. It sensitively and comprehensively describes the geometry in diverse systems on radically different length scales.
The morphometric analysis is refined and applied to mathematical models and simulations of physical systems as well as experimental data sets. For example, the structures appearing in models from stochastic geometry are examined with a particular emphasis on anisotropy. In one of these models, the Minkowski functionals help to better understand and predict a geometrical phase transition. Moreover, a structural characterization across length scales of a physical model, which consists of hard particles, reveals how systems with similar local configurations can nevertheless exhibit a distinctly different global structure. On extremely small length scales, the Minkowski functionals help to characterize complex shapes of exotic states of nuclear matter. Among a variety of these spontaneously forming so-called pasta shapes, a gyroid network is identified, which was, e.g., already found in the wing scales of a butterfly. In a morphometric data analysis, the Minkowski functionals quantify the shape of noise in sky maps from gamma-ray astronomy. Thus, additional geometric information can be extracted from the data without prior assumptions about potential sources. The latter can then be detected by a significant deviation of the structure of the observed sky map from the shape of the background noise. By an enhanced characterization of this background structure, formerly undetected sources can eventually be detected in the same data.
The Minkowski functionals and tensors allow for a better understanding of quite different mathematical models and physical systems as well as a sensitive analysis of experimental observations. Thereby, this morphometric analysis relates seemingly unrelated fields of research.

The manipulation of liquid crystals by external potentials, such as electric or magnetic fields, is of great importance in many industrial and research applications. To gain insight into the impact of the different mechanisms in such a complex multi-particle system is a scientific challenge. Interactions between particles and external potential, particle-particle interactions, and, in case of colloidal systems, hydrodynamic interactions lead, in their interplay, to fascinating dynamical states and unusual diffusion behavior.
This work presents new insights into the dynamics of colloidal liquid crystals with computer simulations. For this purpose several model systems of increasing complexity are studied. The first model system is the most basic model system possible: a system of hard spherocylinders that only interact via excluded volume. In a next step Brownian motion, the random motion of colloidal particles in a fluid, is included via Langevin Dynamics. Finally, also the impact of hydrodynamic interactions between the particles is studied within a Lattice Boltzmann framework.

Triangulations, which can intuitively be described as a tessellation of space into simplicial building blocks, are structures that arise in various different branches of physics: They can be used for describing complicated and curved objects in a discretized way, e.g., in foams, gels or porous media, or for discretizing curved boundaries for fluid simulations or dissipative systems. Interpreting triangulations as (maximal planar) graphs makes it possible to use them in graph theory or statistical physics, e.g., as small-world networks, as networks of spins or in biological physics as actin networks. Since one can find an analogue of the Einstein-Hilbert action on triangulations, they can even be used for formulating theories of quantum gravity. Triangulations have also important applications in mathematics, especially in discrete topology.
Despite their wide occurrence in different branches of physics and mathematics, there are still some fundamental open questions about triangulations in general. It is a prior unknown how many triangulations there are for a given set of points or a given manifold, or even whether there are exponentially many triangulations or more, a question that relates to a well-defined behavior of certain quantum geometry models. Another major unknown question is whether elementary steps transforming triangulations into each other, which are used in computer simulations, are ergodic. Using triangulations as model for spacetime, it is not clear whether there is a meaningful continuum limit that can be identified with the usual and well-tested theory of general relativity.
Within this thesis some of these fundamental questions about triangulations are answered by the use of Markov chain Monte Carlo simulations, which are a probabilistic method for calculating statistical expectation values, or more generally a tool for calculating high-dimensional integrals. Additionally, some details about the Wang-Landau algorithm, which is the primary used numerical method in this thesis, will be examined in detail.

A fluid of hard spherocylinders serves as a simple model system for
Liquid Crystals. Within classical density functional theory, the free energy
of anisotropic hard bodies can be written in terms of weighted densities solely depending on geometry and position of a single oriented particle.
We improve upon this Fundamental Measure Theory by providing the
exact low-density limit and proposing more suitable expressions
for the dense system. To compare the presented approaches, we
discuss the phase diagram of hard spherocylinders as well as
interfacial and elastic properties. The analytic results for some simple hard-body systems constrain the general form of the functional.
This work provides a sophisticated density functional, which allows
to deduce macroscopic phenomena from the geometry of the fluid‘s particles. It is straightforward to describe mixtures and confined
fluids or to extend the theory to soft interaction potentials.