Motion by Stopping: Rectifying Brownian Motion of Non-spherical Particles

We show that Brownian motion is spatially not symmetric for mesoscopic particles embedded in a fluid if the particle is not in thermal equilibrium and its shape is not spherical. In view of applications on molecular motors in biological cells, we sustain non-equilibrium by stopping a non-spherical particle at periodic sites along a filament. Molecular dynamics simulations in a Lennard-Jones fluid demonstrate that directed motion is possible without a ratchet potential or temperature gradients if the asymmetric non-equilibrium relaxation process is hindered by external stopping. Analytic calculations in the ideal gas limit show that motion even against a fluid drift is possible and that the direction of motion can be controlled by the shape of the particle, which is completely characterized by tensorial Minkowski functionals.Comment: 11 pages, 5 figure

Unimodular lattice triangulations as small-world and scale-free random graphs

Real-world networks, e.g. the social relations or world-wide-web graphs, exhibit both small-world and scale-free behaviour. We interpret lattice triangulations as planar graphs by identifying triangulation vertices with graph nodes and one-dimensional simplices with edges. Since these triangulations are ergodic with respect to a certain Pachner flip, applying different Monte-Carlo simulations enables us to calculate average properties of random triangulations, as well as canonical ensemble averages using an energy functional that is approximately the variance of the degree distribution. All considered triangulations have clustering coefficients comparable with real world graphs, for the canonical ensemble there are inverse temperatures with small shortest path length independent of system size. Tuning the inverse temperature to a quasi-critical value leads to an indication of scale-free behaviour for degrees $k \geq 5$. Using triangulations as a random graph model can improve the understanding of real-world networks, especially if the actual distance of the embedded nodes becomes important.Comment: 17 pages, 6 figures, will appear in New J. Phy

Thermal noise influences fluid flow in thin films during spinodal dewetting

Experiments on dewetting thin polymer films confirm the theoretical prediction that thermal noise can strongly influence characteristic time-scales of fluid flow and cause coarsening of typical length scales. Comparing the experiments with deterministic simulations, we show that the Navier-Stokes equation has to be extended by a conserved bulk noise term to accomplish the observed spectrum of capillary waves. Due to thermal fluctuations the spectrum changes from an exponential to a power law decay for large wavevectors. Also the time evolution of the typical wavevector of unstable perturbations exhibits noise induced coarsening that is absent in deterministic hydrodynamic flow.Comment: 4 pages, 3 figure

Second order analysis of geometric functionals of Boolean models

This paper presents asymptotic covariance formulae and central limit theorems for geometric functionals, including volume, surface area, and all Minkowski functionals and translation invariant Minkowski tensors as prominent examples, of stationary Boolean models. Special focus is put on the anisotropic case. In the (anisotropic) example of aligned rectangles, we provide explicit analytic formulae and compare them with simulation results. We discuss which information about the grain distribution second moments add to the mean values.Comment: Chapter of the forthcoming book "Tensor Valuations and their Applications in Stochastic Geometry and Imaging" in Lecture Notes in Mathematics edited by Markus Kiderlen and Eva B. Vedel Jensen. (The second version mainly resolves minor LaTeX problems.

Morphological Thermodynamics of Fluids: Shape Dependence of Free Energies

We examine the dependence of a thermodynamic potential of a fluid on the geometry of its container. If motion invariance, continuity, and additivity of the potential are fulfilled, only four morphometric measures are needed to describe fully the influence of an arbitrarily shaped container on the fluid. These three constraints can be understood as a more precise definition for the conventional term "extensive" and have as a consequence that the surface tension and other thermodynamic quantities contain, beside a constant term, only contributions linear in the mean and Gaussian curvature of the container and not an infinite number of curvatures as generally assumed before. We verify this numerically in the entropic system of hard spheres bounded by a curved wall.Comment: 4 pages, 3 figures, accepted for publication in PR

Density functional theory for hard-sphere mixtures: the White-Bear version Mark II

In the spirit of the White-Bear version of fundamental measure theory we derive a new density functional for hard-sphere mixtures which is based on a recent mixture extension of the Carnahan-Starling equation of state. In addition to the capability to predict inhomogeneous density distributions very accurately, like the original White-Bear version, the new functional improves upon consistency with an exact scaled-particle theory relation in the case of the pure fluid. We examine consistency in detail within the context of morphological thermodynamics. Interestingly, for the pure fluid the degree of consistency of the new version is not only higher than for the original White-Bear version but also higher than for Rosenfeld's original fundamental measure theory.Comment: 16 pages, 3 figures; minor changes; J. Phys.: Condens. Matter, accepte

Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences

Appearance of the Single Gyroid Network Phase in Nuclear Pasta Matter

Nuclear matter under the conditions of a supernova explosion unfolds into a rich variety of spatially structured phases, called nuclear pasta. We investigate the role of periodic network-like structures with negatively curved interfaces in nuclear pasta structures, by static and dynamic Hartree-Fock simulations in periodic lattices. As the most prominent result, we identify for the first time the {\it single gyroid} network structure of cubic chiral $I4_123$ symmetry, a well known configuration in nanostructured soft-matter systems, both as a dynamical state and as a cooled static solution. Single gyroid structures form spontaneously in the course of the dynamical simulations. Most of them are isomeric states. The very small energy differences to the ground state indicate its relevance for structures in nuclear pasta.Comment: 7 pages, 4 figure