Polynomial and Rational Approximations and the Link between Schröder’s Processes of the First and Second Kind

We show that Schröder’s processes of the first kind and of the second kind to obtain a simple root of a nonlinear equation are related by polynomial and rational approximations

Jamming, glass transition, and entropy in monodisperse and polydisperse hard-sphere packings

This thesis is dedicated to the investigation of properties of computer-generated monodisperse and polydisperse three-dimensional hard-sphere packings, frictional and frictionless. For frictionless packings, we (i) assess their total (fluid) entropy in a wide range of packing densities (solid volume fractions), (ii) investigate the structure of their phase space, (iii) and estimate several characteristic densities (the J-point, the ideal glass transition density, and the ideal glass density). For frictional packings, we estimate the Edwards entropy in a wide range of densities. We utilize the Lubachevsky–Stillinger, Jodrey–Tory, and force-biased packing generation algorithms. We always generate packings of 10000 particles in cubic boxes with periodic boundary conditions. For estimation of the Edwards entropy, we also use experimentally produced and reconstructed packings of fluidized beds. In polydisperse cases, we use the log-normal, Pareto, and Gaussian particle diameter distributions with polydispersities (relative radii standard deviations) from 0.05 (5%) to 0.3 (30%) in steps of 0.05. This work consists of six chapters, each corresponding to a published paper. In the first chapter, we introduce a method to estimate the probability to insert a particle in a packing (insertion probability) through the so-called pore-size (nearest neighbour) distribution. Under certain assumptions about the structure of the phase space, we link this probability to the (total) entropy of packings. In this chapter, we use only frictionless monodisperse hard-sphere packings. We conclude that the two characteristic particle volume fractions (or densities, φ) often associated with the Random Close Packing limit, φ ≈ 0.64 and φ ≈ 0.65, may refer to two distinct phenomena: the J-point and the Glass Close Packing limit (the ideal glass density), respectively. In the second chapter, we investigate the behaviour of jamming densities of frictionless polydisperse packings produced with different packing generation times. Packings produced quickly are structurally closer to Poisson packings and jam at the J-point (φ ≈ 0.64 for monodisperse packings). Jamming densities (inherent structure densities) of packings with sufficient polydispersity that were produced slowly approach the glass close packing (GCP) limit. Monodisperse packings overcome the GCP limit (φ ≈ 0.65) because they can incorporate crystalline regions. Their jamming densities eventually approach the face-centered cubic (FCC) / hexagonal close packing (HCP) crystal density φ = π/(3 √2) ≈ 0.74. These results support the premise that φ ≈ 0.64 and φ ≈ 0.65 in the monodisperse case may refer to the J-point and the GCP limit, respectively. Frictionless random jammed packings can be produced with any density in-between. In the third chapter, we add one more intermediate step to the procedure from the second chapter. We take the unjammed (initial) packings in a wide range of densities from the second chapter, equilibrate them, and only then jam (search for their inherent structures). Thus, we investigate the structure of their phase space. We determine the J-point, ideal glass transition density, and ideal glass density. We once again recover φ ≈ 0.64 as the J-point and φ ≈ 0.65 as the GCP limit for monodisperse packings. The ideal glass transition density for monodisperse packings is estimated at φ ≈ 0.585. In the fourth chapter, we demonstrate that the excess entropies of the polydisperse hard-sphere fluid at our estimates of the ideal glass transition densities do not significantly depend on the particle size distribution. This suggests a simple procedure to estimate the ideal glass transition density for an arbitrary particle size distribution by solving an equation, which requires that the excess fluid entropy shall equal to some universal value characteristic of the ideal glass transition density. Excess entropies for an arbitrary particle size distribution and density can be computed through equations of state, for example the Boublík–Mansoori–Carnahan–Starling–Leland (BMCSL) equation. In the fifth chapter, we improve the procedure from the first chapter. We retain the insertion probability estimation from the pore-size distribution, but switch from the initial assumptions about the structure of the phase space to a more advanced Widom particle insertion method, which for hard spheres links the insertion probability to the excess chemical potential. With the chemical potential at hand, we can estimate the excess fluid entropy, which complies well with theoretical predictions from the BMCSL equation of state. In the sixth chapter, we extend the Widom particle insertion method from the fifth chapter as well as the insertion probability estimation method from the first chapter to determine the upper bound on the Edwards entropy per particle in monodisperse frictional packings. The Edwards entropy counts the number of mechanically stable configurations at a given density (density interval). We demonstrate that the Edwards entropy estimate is maximum at the Random Loose Packing (RLP) limit (φ ≈ 0.55) and decreases with density increase. In this chapter, we accompany computer-generated packings with experimentally produced and reconstructed ones. Overall, this study extends the understanding of the glass transition, jamming, and the Edwards entropy behavior in the system of hard spheres. The results can help comprehend these phenomena in more complex molecular, colloidal, and granular systems

Conformational Ensembles and Sampled Energy Landscapes: Analysis and Comparison

We present novel algorithms and software addressing four core problemsin computational structural biology, namely analyzing a conformationalensemble, comparing two conformational ensembles, analyzing a sampledenergy landscape, and comparing two sampled energy landscapes. Usingrecent developments in computational topology, graph theory, andcombinatorial optimization, we make two notable contributions. First,we a present a generic algorithm analyzing height fields. We then usethis algorithm to perform density based clustering of conformations,and to analyze a sampled energy landscape in terms of basins andtransitions between them. In both cases, topological persistence isused to manage ruggedness. Second, we introduce two algorithms tocompare transition graphs. The first is the classical earth mover distance metric which depends only on local minimum energyconfigurations along with their statistical weights, while the secondincorporates topological constraints inherent to conformationaltransitions.Illustrations are provided on a simplified protein model (BLN69), whosefrustrated potential energy landscape has been thoroughly studied.The software implementing our tools is also made available, and shouldprove valuable wherever conformational ensembles and energy landscapesare used

Contours in Visualization

This thesis studies the visualization of set collections either via or defines as the relations among contours. In the first part, dynamic Euler diagrams are used to communicate and improve semimanually the result of clustering methods which allow clusters to overlap arbitrarily. The contours of the Euler diagram are rendered as implicit surfaces called blobs in computer graphics. The interaction metaphor is the moving of items into or out of these blobs. The utility of the method is demonstrated on data arising from the analysis of gene expressions. The method works well for small datasets of up to one hundred items and few clusters. In the second part, these limitations are mitigated employing a GPU-based rendering of Euler diagrams and mixing textures and colors to resolve overlapping regions better. The GPU-based approach subdivides the screen into triangles on which it performs a contour interpolation, i.e. a fragment shader determines for each pixel which zones of an Euler diagram it belongs to. The rendering speed is thus increased to allow multiple hundred items. The method is applied to an example comparing different document clustering results. The contour tree compactly describes scalar field topology. From the viewpoint of graph drawing, it is a tree with attributes at vertices and optionally on edges. Standard tree drawing algorithms emphasize structural properties of the tree and neglect the attributes. Adapting popular graph drawing approaches to the problem of contour tree drawing it is found that they are unable to convey this information. Five aesthetic criteria for drawing contour trees are proposed and a novel algorithm for drawing contour trees in the plane that satisfies four of these criteria is presented. The implementation is fast and effective for contour tree sizes usually used in interactive systems and also produces readable pictures for larger trees. Dynamical models that explain the formation of spatial structures of RNA molecules have reached a complexity that requires novel visualization methods to analyze these model\''s validity. The fourth part of the thesis focuses on the visualization of so-called folding landscapes of a growing RNA molecule. Folding landscapes describe the energy of a molecule as a function of its spatial configuration; they are huge and high dimensional. Their most salient features are described by their so-called barrier tree -- a contour tree for discrete observation spaces. The changing folding landscapes of a growing RNA chain are visualized as an animation of the corresponding barrier tree sequence. The animation is created as an adaption of the foresight layout with tolerance algorithm for dynamic graph layout. The adaptation requires changes to the concept of supergraph and it layout. The thesis finishes with some thoughts on how these approaches can be combined and how the task the application should support can help inform the choice of visualization modality

On the Computational Cost and Complexity of Stochastic Inverse Solvers

The goal of this paper is to provide a starting point for investigations into a mainly underdeveloped area of research regarding the computational cost analysis of complex stochastic strategies for solving parametric inverse problems. This area has two main components: solving global optimization problems and solving forward problems (to evaluate the misfit function that we try to minimize). For the first component, we pay particular attention to genetic algorithms with heuristics and to multi-deme algorithms that can be modeled as ergodic Markov chains. We recall a simple method for evaluating the first hitting time for the single-deme algorithm and we extend it to the case of HGS, a multi-deme hierarchic strategy. We focus on the case in which at least the demes in the leaves are well tuned. Finally, we also express the problems of finding local and global optima in terms of a classic complexity theory. We formulate the natural result that finding a local optimum of a function is an NP-complete task, and we argue that finding a global optimum is a much harder, DP-complete, task. Furthermore, we argue that finding all global optima is, possibly, even harder (#P-hard) task. Regarding the second component of solving parametric inverse problems (i.e., regarding the forward problem solvers), we discuss the computational cost of hp-adaptive Finite Element solvers and their rates of convergence with respect to the increasing number of degrees of freedom. The presented results provide a useful taxonomy of problems and methods of studying the computational cost and complexity of various strategies for solving inverse parametric problems. Yet, we stress that our goal was not to deliver detailed evaluations for particular algorithms applied to particular inverse problems, but rather to try to identify possible ways of obtaining such results

Edwards statistical mechanics for jammed granular matter

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