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Ab initio study of phase stability of multicomponent alloys
Fikar, Ondřej ; Brož, Pavel (referee) ; Černý, Miroslav (referee) ; Zelený, Martin (advisor)
Ab initio methods are based on purely theoretical findings of quantum physics that can be used to predict among others physical, chemical and mechanical properties of materials. Due to rapid increase in accessibility of computational resources in the recent decades the theoretical prediction of material properties became an integral part of materials design. This work is focused on theoretical prediction of phase stability and solubility of solid solutions. Ab initio calculations based on Density Functional Theory were performed using Projector-Augmented Waves method and thermal dependencies of thermodynamic quantities were obtained using phonon calculations and Monte Carlo simulations. Attention is paid to alloys mainly based on aluminium, silver and magnesium, which were investigated in order to assess the reliability and precision of theoretical predictions of solubility in the solid state. Phase stability of solid solutions was evaluated multiple times including different energy contributions and using various methods in order to determine the influence of each contribution and method on the prediction accuracy. Calculated solubilities are compared with experimental data provided using the CALPHAD method.
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Contour methods in the mathematical theory of phase transitions
Nagy, Oliver ; Zahradník, Miloš (advisor) ; Netočný, Karel (referee)
Title: Contour methods in the mathematical theory of phase transitions Author: Oliver Nagy Department: Department of Mathematical Analysis Supervisor: doc. RNDr. Miloš Zahradník, CSc., Department of Mathematical Analysis Abstract: This thesis concerns itself with three topics, namely polymer models, Pirogov-Sinai theory and one-dimensional Dyson models. It contains a short introduction into all three topics. The introduction to Pirogov-Sinai theory will serve as a starting point for a future expanded introductory exposition, since such a material is missing in the contemporary literature. Research result of the first chapter is a detailed combinatorial analysis of cluster expansion of hard-core repulsive polymer model based on 'self-avoiding polymer trees', leading to simplification of the structure of summation in the partition function. In the case of Dyson models we suggest an alternative definition of contours for the one-dimensional Dyson model with the exponent of polynomially-decaying interaction p ∈ (1, 2) that is usable for study using Pirogov-Sinai methods. Keywords: Contours, polymers, cluster expansion, Pirogov-Sinai theory, Dyson model;
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Exponential function and Mayer expansion
Nagy, Oliver ; Zahradník, Miloš (advisor) ; Loebl, Martin (referee)
Title: Exponential function and Mayer expansion Author: Oliver Nagy Department: Department of Mathematical Analysis Supervisor: doc. RNDr. Miloš Zahradník, CSc., Department of Mathematical Analysis Abstract: The unifying topic of this thesis is cluster expansion in statistical phy- sics. It is divided into three chapters. In the first one we present the necessary mathematical apparatus - selected topics from combinatorics, graph theory and theory of generating functions. The second one is an introduction to cluster expan- sion and abstract polymer model. Finally, in the third chapter we show a new resummation method for partition function of hard-core repulsive abstract poly- mer model. In this resummation we make use of cancellations of terms in partition function to rewrite the sum of clusters to a sum of quilted clusters, or alternati- vely as a sum of "bunches". The methods we use in this final chapter are original and may lead to some new results. Keywords: binomial and multinomial formula; power series; inclusion-exclusion principle; cluster expansion. iii
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