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The crystallographic texture is a key organization feature of many technical and biological materials. In these materials, especially hierarchically structured ones, the preferential alignment of the nano constituents is heavily influencing the macroscopic behaviour of the material. In order to study local crystallographic texture with both high spatial and angular resolution, we developed Texture tomography (TexTOM). This approach allows to model the diffraction data of polycrystalline materials by using the full reciprocal space of the ensemble of crystals and describe the texture in each voxel via a orientation distribution function. This means, it provides 3D reconstructions of the local texture by measuring the probabilities of all crystal orientations. The TexTOM approach addresses limitations associated with existing models: It correlates the intensities from several Bragg reflections, thus reduces ambiguities resulting from symmetry. Further, it yields quantitative probability distributions of local real space crystal orientations without further assumptions on the sample structure. Finally, its efficient mathematical formulation enables reconstructions faster than the time-scale of the experiment. In this manuscript, we present the mathematical model, the inversion strategy and its current experimental implementation. We show characterizations of simulated data as well as experimental data obtained from a synthetic, inorganic model sample, the silica-witherite biomorph. In conclusion, Tex-TOM provides a versatile framework to reconstruct 3D quantitative texture information for polycrystalline samples. In this way, it opens the door for unprecedented insights into the nanostructural makeup of natural and technical materials.

As phenomena that necessarily emerge from the collective behavior of interacting particles, phase transitions continue to be difficult to predict using statistical thermodynamics. A recent proposal called the topological hypothesis suggests that the existence of a phase transition could perhaps be inferred from changes to the topology of the accessible part of the configuration space. This paper instead suggests that such a topological change is often associated with a dramatic change in the configuration space geometry, and that the geometric change is the actual driver of the phase transition. More precisely, a geometric change that brings about a discontinuity in the mixing time required for an initial probability distribution on the configuration space to reach steady-state is conjectured to be related to the onset of a phase transition in the thermodynamic limit. This conjecture is tested by evaluating the diffusion diameter and $\epsilon$-mixing time of the configuration spaces of hard disk and hard sphere systems of increasing size. Explicit geometries are constructed for the configuration spaces of these systems, and numerical evidence suggests that a discontinuity in the $\epsilon$-mixing time coincides with the solid-fluid phase transition in the thermodynamic limit., Comment: 12 pages, 13 figures

Statistical thermodynamics is valuable as a conceptual structure that shapes our thinking about equilibrium thermodynamic states. A cloud of unresolved questions surrounding the foundations of the theory could lead an impartial observer to conclude that statistical thermodynamics is in a state of crisis though. Indeed, the discussion about the microscopic origins of irreversibility has continued in the scientific community for more than a hundred years. This paper considers these questions while beginning to develop a statistical thermodynamics for finite non-equilibrium systems. Definitions are proposed for all of the extrinsic variables of the fundamental thermodynamic relation that are consistent with existing results in the equilibrium thermodynamic limit. The probability density function on the phase space is interpreted as a subjective uncertainty about the microstate, and the Gibbs entropy formula is modified to allow for entropy creation without introducing additional physics or modifying the phase space dynamics. Resolutions are proposed to the mixing paradox, Gibbs' paradox, Loschmidt's paradox, and Maxwell's demon thought experiment. Finally, the extrinsic variables of the fundamental thermodynamic relation are evaluated as functions of time and space for a diffusing ideal gas, and the initial and final values are shown to coincide with the expected equilibrium values when interpreted in a classical context., Comment: 18 pages, 10 figures

Hard sphere systems are often used to model simple fluids. The configuration spaces of hard spheres in a three-dimensional torus modulo various symmetry groups are comparatively simple, and could provide valuable information about the nature of phase transitions. Specifically, the topological changes in the configuration space as a function of packing fraction have been conjectured to be related to the onset of first-order phase transitions. The critical configurations for one to twelve spheres are sampled using a Morse-theoretic approach, and are available in an online, interactive database. Explicit triangulations are constructed for the configuration spaces of the two sphere system, and their topological and geometric properties are studied. The critical configurations are found to be associated with geometric changes to the configuration space that connect previously distant regions and reduce the configuration space diameter as measured by the commute time and diffusion distances. The number of such critical configurations around the packing fraction of the solid-liquid phase transition increases exponentially with the number of spheres, suggesting that the onset of the first-order phase transition in the thermodynamic limit is associated with a discontinuity in the configuration space diameter., Comment: 15 pages, 16 figures

Hard disks systems are often considered as prototypes for simple fluids. In a statistical mechanics context, the hard disk configuration space is generally quotiented by the action of various symmetry groups. The changes in the topological and geometric properties of the configuration spaces effected by such quotient maps are studied for small numbers of disks on a square and hexagonal torus. A metric is defined on the configuration space and the various quotient spaces that respects the desired symmetries. This is used to construct explicit triangulations of the configuration spaces as $\alpha$-complexes. Critical points in a configuration space are associated with changes in the topology as a function of disk radius, are conjectured to be related to the configurational entropy of glassy systems, and could reveal the origins of phase transitions in other systems. The number and topological and geometric properties of the critical points are found to depend on the symmetries by which the configuration space is quotiented.