The present thesis is dedicated to the efficient computation of electrostatic interactions in particle systems, which is of great importance in the field of molecular dynamics simulations. In order to compute the therefor required physical quantities with only O(N log N) arithmetic operations, so called particle-mesh methods make use of the well-known Ewald summation approach and the fast Fourier transform (FFT). Typically, such methods are able to handle systems of point charges subject to periodic boundary conditions in all spatial directions. However, periodicity is not always desired in all three dimensions and, moreover, also interactions to dipoles play an important role in many applications. Within the scope of the present work, we consider the particle-particle NFFT method (P²NFFT), a particle-mesh approach based on the fast Fourier transform for nonequispaced data (NFFT). An extension of this method for mixed periodic as well as open boundary conditions is presented. Furthermore, the method is appropriately modified in order to treat particle systems containing both charges and dipoles. Consequently, an efficient algorithm for mixed charge-dipole systems, that additionally allows a unified handling of various types of periodic boundary conditions, is presented for the first time. Appropriate error estimates as well as parameter tuning strategies are developed and verified by numerical examples. / Die vorliegende Arbeit widmet sich der Berechnung elektrostatischer Wechselwirkungen in Partikelsystemen, was beispielsweise im Bereich der molekulardynamischen Simulationen eine zentrale Rolle spielt. Um die dafür benötigten physikalischen Größen mit lediglich O(N log N) arithmetischen Operationen zu berechnen, nutzen sogenannte Teilchen-Gitter-Methoden die Ewald-Summation sowie die schnelle Fourier-Transformation (FFT). Typischerweise können derartige Verfahren Systeme von Punktladungen unter periodischen Randbedingungen in allen Raumrichtungen handhaben. Periodizität ist jedoch nicht immer bezüglich aller drei Dimensionen erwünscht. Des Weiteren spielen auch Wechselwirkungen zu Dipolen in vielen Anwendungen eine wichtige Rolle. Zentraler Gegenstand dieser Arbeit ist die Partikel-Partikel-NFFT Methode (P²NFFT), ein Teilchen-Gitter-Verfahren, welches auf der schnellen Fouriertransformation für nichtäquidistante Daten (NFFT) basiert. Eine Erweiterung dieses Verfahrens auf gemischt periodische sowie offene Randbedingungen wird vorgestellt. Außerdem wird die Methode für die Behandlung von Partikelsystemen, in denen sowohl Ladungen als auch Dipole vorliegen, angepasst. Somit wird erstmalig ein effizienter Algorithmus für gemischte Ladungs-Dipol-Systeme präsentiert, der zusätzlich die Behandlung sämtlicher Arten von Randbedingungen mit einem einheitlichen Zugang erlaubt. Entsprechende Fehlerabschätzungen sowie Strategien für die Parameterwahl werden entwickelt und anhand numerischer Beispiele verifiziert.

The present thesis provides a modularized view on the structure of fast numerical methods for computing Coulomb interactions between charged particles in three-dimensional space. Thereby, the common structure is given in terms of three self-contained algorithmic frameworks that are built on top of each other, namely fast Fourier transform (FFT), nonequispaced fast Fourier transform (NFFT) and NFFT based particle-mesh methods (P²NFFT). For each of these frameworks algorithmic enhancement and parallel implementations are presented with special emphasis on scalability up to hundreds of thousands of parallel processes. In the context of FFT massively parallel algorithms are composed from hardware adaptive low level modules provided by the FFTW software library. The new algorithmic NFFT concepts include pruned NFFT, interlacing, analytic differentiation, and optimized deconvolution in Fourier space with respect to a mean square aliasing error. Enabled by these generalized concepts it is shown that NFFT provides a unified access to particle-mesh methods. Especially, mixed-periodic boundary conditions are handled in a consistent way and interlacing can be incorporated more efficiently. Heuristic approaches for parameter tuning are presented on the basis of thorough error estimates. / Die vorliegende Dissertation beschreibt einen modularisierten Blick auf die Struktur schneller numerischer Methoden für die Berechnung der Coulomb-Wechselwirkungen zwischen Ladungen im dreidimensionalen Raum. Die gemeinsame Struktur ist geprägt durch drei selbstständige und auf einander aufbauenden Algorithmen, nämlich der schnellen Fourier-Transformation (FFT), der nicht äquidistanten schnellen Fourier-Transformation (NFFT) und der NFFT-basierten Teilchen-Gitter-Methode (P²NFFT). Für jeden dieser Algorithmen werden Verbesserungen und parallele Implementierungen vorgestellt mit besonderem Augenmerk auf massiv paralleler Skalierbarkeit. Im Kontext der FFT werden parallele Algorithmen aus den Hardware adaptiven Modulen der FFTW Softwarebibliothek zusammengesetzt. Die neuen NFFT-Konzepte beinhalten abgeschnittene NFFT, Versatz, analytische Differentiation und optimierte Entfaltung im Fourier-Raum bezüglich des mittleren quadratischen Aliasfehlers. Mit Hilfe dieser Verallgemeinerungen bietet die NFFT einen vereinheitlichten Zugang zu Teilchen-Gitter-Methoden. Insbesondere gemischt periodische Randbedingungen werden einheitlich behandelt und Versatz wird effizienter umgesetzt. Heuristiken für die Parameterwahl werden auf Basis sorgfältiger Fehlerabschätzungen angegeben.

The computation of the Coulomb potentials and forces in charged particle systems under 3d-periodic boundary conditions is possible in an efficient way by utilizing the Ewald summation formulas and applying the fast Fourier transform (FFT). In this paper we consider the particle-particle NFFT (P2NFFT) approach, which is based on the fast Fourier transform for nonequispaced data (NFFT) and compare the error behaviors regarding different window functions, which are used in order to approximate the given continuous charge distribution by a mesh based charge density. Typically B-splines are applied in the scope of particle mesh methods, as for instance within the well-known particle-particle particle-mesh (P3M) algorithm. The publicly available P2NFFT algorithm allows the application of an oversampled FFT as well as the usage of different window functions. We consider for the first time also an approximation by Bessel functions and show how the resulting root mean square errors in the forces can be predicted precisely and efficiently. The results show that, if the parameters are tuned appropriately, the Bessel window function is in many cases even the better choice in terms of computational costs. Moreover, the results indicate that it is often advantageous in terms of efficiency to spend some oversampling within the NFFT while using a window function with a smaller support.

<p>Methods for performing large-scale parallel Molecular Dynamics(MD) simulations are investigated. A perspective on the field of parallel MD simulations is given. Hardware and software aspects are characterized and the interplay between the two is briefly discussed. </p><p>A method for performing <i>ab initio </i>MD is described; the method essentially recomputes the interaction potential at each time-step. It has been tested on a system of liquid water by comparing results with other simulation methods and experimental results. Different strategies for parallelization are explored.</p><p>Furthermore, data-parallel methods for short-range and long-range interactions on massively parallel platforms are described and compared. </p><p>Next, a method for treating electrostatic interactions in MD simulations is developed. It combines the traditional Ewald summation technique with the nonuniform Fast Fourier transform---ENUF for short. The method scales as <i>N log N</i>, where <i>N </i>is the number of charges in the system. ENUF has a behavior very similar to Ewald summation and can be easily and efficiently implemented in existing simulation programs.</p><p>Finally, an outlook is given and some directions for further developments are suggested.</p>

O modelo de Ising dipolar bidimensional inclui, além da interação ferromagnética entre os primeiros vizinhos, interações de longo alcance entre os momentos de dipolo magnético dos spins. A presença da interação dipolar muda completamente o sistema, apresentando um rico diagrama de fase, cujas características têm originado inúmeros estudos na literatura. Além disso, a possibilidade de explicar fenômenos observados em filmes magnéticos ultrafinos, os quais possuem diversas aplicações em àreas tecnológicas, também motiva o estudo deste modelo. O estado fundamental ferromagnético do modelo de Ising puro é alterado para uma série de fases do tipo faixas, as quais consistem em domínios ferromagnéticos de largura $h$ com magnetizações opostas. A largura das faixas depende da razao $\\delta$ das intensidades dos acoplamentos ferromagnético e dipolar. Através de simulações de Monte Carlo e técnicas de repesagem em histogramas múltiplos identificamos as temperaturas críticas de tamanho finito para as transições de fase quando $\\delta=2$, o que corresponde a $h=2$. Calculamos o calor específico e a susceptibilidade do parâmetro de ordem, no intervalo de temperaturas onde as transições são observadas, para diferentes tamanhos de rede. As técnicas de repesagem permitem-nos explorar e identificar máximos distintos nessas funções da temperatura e, desse modo, estimar as temperaturas críticas de tamanho finito com grande precisão. Apresentamos evidências numéricas da existência de uma fase nemática de Ising para tamanhos grandes de rede. Em nossas simulações, observamos esta fase para tamanhos de rede a partir de $L=48$. Para verificar o quanto a interação dipolar de longo alcance afeta as estimativas físicas, nós calculamos o tempo de autocorrelação integrado nas séries temporais da energia. Inferimos daí quão severo é o critical slowing down (decaimento lento crítico) para esse sistema próximo às transições de fase termodinâmicas. Os resultados obtidos utilizando um algoritmo de atualização local foram comparados com os resultados obtidos utilizando o algoritmo multicanônico. / Two-dimensional spin model with nearest-neighbor ferromagnetic interaction and long-range dipolar interactions exhibit a rich phase diagram, whose characteristics have been exploited by several studies in the recent literature. Furthermore, the possibility of explain observed phenomena in ultrathin magnetic films, which have many technological applications, also motivates the study of this model. The presence of dipolar interaction term changes the ferromagnetic ground state expected for the pure Ising model to a series of striped phases, which consist of ferromagnetic domains of width $h$ with opposite magnetization. The width of the stripes depends on the ratio $\\delta$ of the ferromagnetic and dipolar couplings. Monte Carlo simulations and reweighting multiple histograms techniques allow us to identify the finite-size critical temperatures of the phase transitions when $\\delta=2$, which corresponds to $h=2$. We calculate, for different lattice sizes, the specific heat and susceptibility of the order parameter around the transition temperatures by means of reweighting techniques. This allows us to identify in these observables, as functions of temperature, the distinct maxima and thereby to estimate the finite-size critical temperatures with high precision. We present numerical evidence of the existence of a Ising nematic phase for large lattice sizes. Our results show that simulations need to be performed for lattice sizes at least as large as $L=48$ to clearly observe the Ising nematic phase. To access how the long-range dipolar interaction may affect physical estimates we also evaluate the integrated autocorrelation time in energy time series. This allows us to infer how severe is the critical slowing down for this system with long-range interaction and nearby thermodynamic phase transitions. The results obtained using a local update algorithm are compared with results obtained using the multicanonical algorithm.

Methods for performing large-scale parallel Molecular Dynamics(MD) simulations are investigated. A perspective on the field of parallel MD simulations is given. Hardware and software aspects are characterized and the interplay between the two is briefly discussed. A method for performing ab initio MD is described; the method essentially recomputes the interaction potential at each time-step. It has been tested on a system of liquid water by comparing results with other simulation methods and experimental results. Different strategies for parallelization are explored. Furthermore, data-parallel methods for short-range and long-range interactions on massively parallel platforms are described and compared. Next, a method for treating electrostatic interactions in MD simulations is developed. It combines the traditional Ewald summation technique with the nonuniform Fast Fourier transform---ENUF for short. The method scales as N log N, where N is the number of charges in the system. ENUF has a behavior very similar to Ewald summation and can be easily and efficiently implemented in existing simulation programs. Finally, an outlook is given and some directions for further developments are suggested.

O modelo de Ising dipolar bidimensional inclui, além da interação ferromagnética entre os primeiros vizinhos, interações de longo alcance entre os momentos de dipolo magnético dos spins. A presença da interação dipolar muda completamente o sistema, apresentando um rico diagrama de fase, cujas características têm originado inúmeros estudos na literatura. Além disso, a possibilidade de explicar fenômenos observados em filmes magnéticos ultrafinos, os quais possuem diversas aplicações em àreas tecnológicas, também motiva o estudo deste modelo. O estado fundamental ferromagnético do modelo de Ising puro é alterado para uma série de fases do tipo faixas, as quais consistem em domínios ferromagnéticos de largura $h$ com magnetizações opostas. A largura das faixas depende da razao $\\delta$ das intensidades dos acoplamentos ferromagnético e dipolar. Através de simulações de Monte Carlo e técnicas de repesagem em histogramas múltiplos identificamos as temperaturas críticas de tamanho finito para as transições de fase quando $\\delta=2$, o que corresponde a $h=2$. Calculamos o calor específico e a susceptibilidade do parâmetro de ordem, no intervalo de temperaturas onde as transições são observadas, para diferentes tamanhos de rede. As técnicas de repesagem permitem-nos explorar e identificar máximos distintos nessas funções da temperatura e, desse modo, estimar as temperaturas críticas de tamanho finito com grande precisão. Apresentamos evidências numéricas da existência de uma fase nemática de Ising para tamanhos grandes de rede. Em nossas simulações, observamos esta fase para tamanhos de rede a partir de $L=48$. Para verificar o quanto a interação dipolar de longo alcance afeta as estimativas físicas, nós calculamos o tempo de autocorrelação integrado nas séries temporais da energia. Inferimos daí quão severo é o critical slowing down (decaimento lento crítico) para esse sistema próximo às transições de fase termodinâmicas. Os resultados obtidos utilizando um algoritmo de atualização local foram comparados com os resultados obtidos utilizando o algoritmo multicanônico. / Two-dimensional spin model with nearest-neighbor ferromagnetic interaction and long-range dipolar interactions exhibit a rich phase diagram, whose characteristics have been exploited by several studies in the recent literature. Furthermore, the possibility of explain observed phenomena in ultrathin magnetic films, which have many technological applications, also motivates the study of this model. The presence of dipolar interaction term changes the ferromagnetic ground state expected for the pure Ising model to a series of striped phases, which consist of ferromagnetic domains of width $h$ with opposite magnetization. The width of the stripes depends on the ratio $\\delta$ of the ferromagnetic and dipolar couplings. Monte Carlo simulations and reweighting multiple histograms techniques allow us to identify the finite-size critical temperatures of the phase transitions when $\\delta=2$, which corresponds to $h=2$. We calculate, for different lattice sizes, the specific heat and susceptibility of the order parameter around the transition temperatures by means of reweighting techniques. This allows us to identify in these observables, as functions of temperature, the distinct maxima and thereby to estimate the finite-size critical temperatures with high precision. We present numerical evidence of the existence of a Ising nematic phase for large lattice sizes. Our results show that simulations need to be performed for lattice sizes at least as large as $L=48$ to clearly observe the Ising nematic phase. To access how the long-range dipolar interaction may affect physical estimates we also evaluate the integrated autocorrelation time in energy time series. This allows us to infer how severe is the critical slowing down for this system with long-range interaction and nearby thermodynamic phase transitions. The results obtained using a local update algorithm are compared with results obtained using the multicanonical algorithm.

We present an efficient method to compute the electrostatic fields, torques and forces in dipolar systems, which is based on the fast Fourier transform for nonequispaced data (NFFT). We consider 3d-periodic, 2d-periodic, 1d-periodic as well as 0d-periodic (open) boundary conditions. The method is based on the corresponding Ewald formulas, which immediately lead to an efficient algorithm only in the 3d-periodic case. In the other cases we apply the NFFT based fast summation in order to approximate the contributions of the nonperiodic dimensions in Fourier space. This is done by regularizing or periodizing the involved functions, which depend on the distances of the particles regarding the nonperiodic dimensions. The final algorithm enables a unified treatment of all types of periodic boundary conditions, for which only the precomputation step has to be adjusted.

No description available.

In this thesis, we study the phenomenon of thermally induced polarisation using a combination of theory and computer simulation. Molecules of sufficiently low symmetry exhibit thermo-molecular orientation when subjected to a temperature gradient, leading to considerable electrostatic fields in polar liquids. Here, we first use non-equilibrium molecular dynamics simulations to study this interesting effect numerically. To this end, we propose an integration algorithm to impose a constant heat flux in simulations and show that it greatly improves energy conservation compared to a previous algorithm. We next investigate the thermal polarisation of water and find that truncation of electrostatic interactions can lead to severe artefacts, such as the wrong sign of polarisation and an overestimation of the electric field. We further show that the quadrupole-moment contribution to the electric field is significant and responsible for an inversion of its sign. To facilitate the theoretical description of electrostatic interactions, we propose a new dipolar model fluid as a perturbation of a Stockmayer fluid. Using this modified Stockmayer model, we provide numerical evidence for the recently proposed phenomenon of thermally induced monopoles. We show that the electrostatic field generated by a pair of heated/cooled colloidal particles immersed in such a solvent can be trivially described by two Coulomb charges. Finally, we propose a mean-field theory to predict the thermo-polarisation effect exhibited by our model fluid theoretically, and demonstrate near quantitative agreement with simulation results.