Estudamos o comportamento crítico do modelo de Ising com interação dipolar, em redes bidimensionais regulares. Este modelo apresenta um cenário fenomenologicamente rico devido ao efeito de frustração causado pela competição entre as interações de troca do Ising puro e a interação dipolar. A criticalidade do modelo foi estudada a partir das relações de escala de tamanho finito para os zeros da função de partição no plano complexo da temperatura. Esta abordagem nunca foi utilizada no estudo do modelo em questão. Nosso estudo se baseia em simulações de Monte Carlo usando o algoritmo multicanônico. O objetivo deste trabalho é obter a temperatura crítica em função do acoplamento (razão entre as intensidades dos acoplamentos ferromagnético e dipolar) e construir uma parte do diagrama de fase do modelo. Diferentes partes do diagrama de fase ainda não apresentam indicações conclusivas a respeito da ordem das linhas de transição. Em particular, há evidências na literatura de um ponto tricrítico para no intervalo [0.90,1.00], mas sua localização precisa não é conhecida. Nossas simulações indicam que o ponto tricrítico não se localiza no intervalo acima. Nossos resultados mostraram que, para [0.89,1.10], a fase do tipo faixas com h=1 passa para a fase tetragonal através de uma transição de segunda ordem. A análise de FSS para os zeros da função de partição na variável temperatura, apresenta, para =1.20, uma transição de fase de segunda ordem e para =1.30, uma transição de fase de primeira ordem. Dessa forma, o ponto tricrítico ocorre somente entre =1.20 e 1.30. Realizamos um estudo complementar baseado na abordagem microcanônica e observamos duas transições de fase de segunda ordem para =1.20 e duas transições de fase de primeira ordem para =1.30, que indica a presença da fase nemática intermediária. / We study the critical behavior of the dipolar Ising model on two-dimensional regular lattices. This model presents a phenomenologically rich scenario due to the effect of frustration caused by the competition between the pure Ising interaction and the dipolar one. To study the criticality of this model we apply finite size scaling relations for the partition function zeros in the complex temperature plane. The partition function zeros analysis has never been used before to study such model with long-range interactions. Our study relies on Monte Carlo simulations using the multicanonical algorithm. Our goal is to obtain the critical temperature as a function of the coupling (the ratio between the ferromagnetic and dipolar couplings) to construct a part of the phase diagram. Different parts of the phase diagram do not present a conclusive results about the order of the phase transition lines.In particular, there is evidence of a tricritical point for [0.90,1.00], but its precise location is unknown. Our simulations indicate that the tricritical point is not located in the above range. Our FSS analysis show that for =1.20 the striped-tetragonal transition is a second-order phase transition and for =1.30 it is a first-order one. Thus, the tricritical point must occur between =1.2 and =1.3. We have used a microcanonical approach to study the criticality of this model too. This approach indicates two second-order phase transitions for =1.20 and two first-order phase transitions for =1.30. Therefore, it presents evidences for the presence of an intermediate nematic phase.

Estudamos o comportamento crítico do modelo de Ising com interação dipolar, em redes bidimensionais regulares. Este modelo apresenta um cenário fenomenologicamente rico devido ao efeito de frustração causado pela competição entre as interações de troca do Ising puro e a interação dipolar. A criticalidade do modelo foi estudada a partir das relações de escala de tamanho finito para os zeros da função de partição no plano complexo da temperatura. Esta abordagem nunca foi utilizada no estudo do modelo em questão. Nosso estudo se baseia em simulações de Monte Carlo usando o algoritmo multicanônico. O objetivo deste trabalho é obter a temperatura crítica em função do acoplamento (razão entre as intensidades dos acoplamentos ferromagnético e dipolar) e construir uma parte do diagrama de fase do modelo. Diferentes partes do diagrama de fase ainda não apresentam indicações conclusivas a respeito da ordem das linhas de transição. Em particular, há evidências na literatura de um ponto tricrítico para no intervalo [0.90,1.00], mas sua localização precisa não é conhecida. Nossas simulações indicam que o ponto tricrítico não se localiza no intervalo acima. Nossos resultados mostraram que, para [0.89,1.10], a fase do tipo faixas com h=1 passa para a fase tetragonal através de uma transição de segunda ordem. A análise de FSS para os zeros da função de partição na variável temperatura, apresenta, para =1.20, uma transição de fase de segunda ordem e para =1.30, uma transição de fase de primeira ordem. Dessa forma, o ponto tricrítico ocorre somente entre =1.20 e 1.30. Realizamos um estudo complementar baseado na abordagem microcanônica e observamos duas transições de fase de segunda ordem para =1.20 e duas transições de fase de primeira ordem para =1.30, que indica a presença da fase nemática intermediária. / We study the critical behavior of the dipolar Ising model on two-dimensional regular lattices. This model presents a phenomenologically rich scenario due to the effect of frustration caused by the competition between the pure Ising interaction and the dipolar one. To study the criticality of this model we apply finite size scaling relations for the partition function zeros in the complex temperature plane. The partition function zeros analysis has never been used before to study such model with long-range interactions. Our study relies on Monte Carlo simulations using the multicanonical algorithm. Our goal is to obtain the critical temperature as a function of the coupling (the ratio between the ferromagnetic and dipolar couplings) to construct a part of the phase diagram. Different parts of the phase diagram do not present a conclusive results about the order of the phase transition lines.In particular, there is evidence of a tricritical point for [0.90,1.00], but its precise location is unknown. Our simulations indicate that the tricritical point is not located in the above range. Our FSS analysis show that for =1.20 the striped-tetragonal transition is a second-order phase transition and for =1.30 it is a first-order one. Thus, the tricritical point must occur between =1.2 and =1.3. We have used a microcanonical approach to study the criticality of this model too. This approach indicates two second-order phase transitions for =1.20 and two first-order phase transitions for =1.30. Therefore, it presents evidences for the presence of an intermediate nematic phase.

Die vorliegende Arbeit befasst sich mit den Gleichgewichtseigenschaften und Phasenübergängen in verdünnten Teilchen- und Polymersystemen, mit einem Fokus auf Teilchenkondensation und Polymeraggregation. Dazu werden sowohl analytische Argumente als auch hochentwickelte Monte Carlo Simulationen verwendet. Um die in dieser Arbeit erreichten Systemgrößen zu simulieren, wurde eine parallele Version der multikanonischen Methode entwickelt. Die Leistungsfähigkeit dieser Erweiterung wird an mehreren relevanten Beispielen demonstriert. Um Teilchenkondensation und Polymeraggregation in finiten Systemen und in geometrisch beschränkten Strukturen besser zu verstehen, wird der Einfluss von verschiedenen Parametern auf die jeweiligen Übergange untersucht. Dies beinhaltet unter anderem die Systemgröße und Dichte, sowie im Speziellen für semiflexible Polymere deren Steifigkeit. Betrachtet werden sowohl kanonische Observablen (Energie, Tropfen- bzw. Aggregatgröße, etc.) mit der dazugehörigen Übergangstemperatur und -breite, als auch eine mikrokanonische Analyse sowie die Barrieren der Freien Energie. Für semiflexible Polymere wird insbesondere der Einfluss von Steifigkeit auf die resultierende Struktur der Aggregate untersucht, die von amorphen Kugeln für flexible Polymere bis hin zu verdrehten Bündeln für steifere Polymere reichen. Ein weiterer Fokus liegt auf der Untersuchung von Übereinstimmungen zwischen den generischen Mechanismen in Kondensation und Aggregation: dem Übergang zwischen einer homogenen Phase und einer inhomogenen (gemischten) Phase. Auf diesem Niveau kann man Polymeraggregation als Kondensation von ausgedehnten Objekten verstehen. Dies zeigt sich vor allem in dem Skalierungsverhalten von kanonischen und mikrokanonischen Observablen, insbesondere an einem unerwarteten aber konsistenten Bereich für mittelgroße (mesoskopische) Systemgrößen.

In dieser Dissertation wird das Skalenverhalten derÜbergangstemperatur von Systemen an diskontinuierlichen Phasenübergängen aus einem Zwei- Zustands-Modell abgeleitet und erweitert. Es wird erläutert, wie sich das Skalenverhalten für periodische Randbedingungen drastisch verändern kann, sobald der Entartungsgrad der geordneten Phasen von der Teilchenzahl abhängt. Eswerden Modellsysteme in zwei und drei Dimensionen betrachtet, deren Zustandssummen mittels analytischer, kombinatorischer Argumente berechnet werden. Für das kompliziertere, isotrope Plaquettemodell in drei Dimensionen können durch diese Rechnungen Ordnungsparameter definiert werden. Diese werden, zusammen mit dem veränderten Skalenverhalten selbskonsistent durch anspruchsvolle und hochpräzise, sogenannte multikanonische Monte-Carlo Simulationen überprüft und bestätigt.

In this thesis we examine methodologies for determining free energy differences (FEDs) of phases via Monte Carlo simulation. We identify and address three generic issues that arise in FED calculations; the choice of representation, the choice of estimator, and the choice of sampling strategy. In addition we discuss how the classical framework may be extended to take into account quantum effects. Key words: Phase Mapping, Phase Switch, Lattice Switch, Simulated Tempering, Multi-stage, Weighted Histogram Analysis Method, Fast Growth, Jarzynski method, Umbrella, Multicanonical, Path Integral Monte Carlo, Path Sampling, Multihamiltonian, fluctuation theorem.

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.

Polarization mode dispersion (PMD) arises as a result of the birefringence in optical fibers, due to inherent asymmetries and deformities from external stresses. The spectral components of the input optical pulse propagate with different group velocities. Consequently, pulse duration increases leading to intersymbol interference between consequent symbols, leading to performance reduction of the coherent systems. In order to compensate for the PMD, we use adaptive linear PMD equalizers. Due to the dynamic and random nature of PMD, it is crucial for a system designer to efficiently simulate the PMD-induced outage probabilities of 10-5. Because of this stringent requirement, it is computationally costly to use the conventional Monte Carlo methods. To overcome this hurdle, Importance Sampling methods, such as the multicanonical Monte Carlo method have been applied in the past in order to efficiently reduce the simulation time required to estimate the statistics of these rare events. The multicanonical Monte Carlo method does not require any prior knowledge of which rare events contribute significantly to the PMD-induced outages. In essence, multicanonical Monte Carlo simulations adaptively bias the input random variables with a priori unknown weights. The PMD emulation model consists of a concatenation of birefringent sections, simulated based on MMC. The objective of this dissertation is to apply, for the first time, the multicanonical Monte Carlo method to accurately and efficiently evaluate the performance of adaptive, blind, feed-forward PMD equalizers employed in coherent polarization division multiplexed (PDM) quadrature phase-shift keying (QPSK) systems in all order PMD emulation model. In the exclusive presence of PMD, we demonstrated that the half-symbol-period-spaced adaptive electronic equalizers, based on the constant modulus algorithm (CMA) equalizers perform slightly better than the decision directed least mean square (DD-LMS) counterparts at links with larger PMD values, whereas the opposite holds true for the low PMD regime. Due to their distinguishable performance in different regimes of the PMD, they provided an even better performance when running DD-LMS after a first round of CMA-based equalization than using either one of the equalization algorithms stand alone. Finally, the joint presence of PMD and intermediate frequency offset or PMD and random differential phase carrier shifts slightly worsened the performance of the coherent PDM QPSK systems, independently of the equalizer. Although these random differential carrier phase shifts are typically omitted in similar PMD studies in intensity modulated/direct detection (IM/DD) systems, they should be taken into account in due to the phase sensitivity of the PDM QPSK coherent systems. / Οι οπτικές ίνες παρουσιάζουν διπλοθλαστικότητα, η οποία οφείλεται σε κατασκευαστικές ατέλειες των οπτικών ινών και σε εξωτερικούς παράγοντες. Η διπλοθλαστικότητα προκαλεί διασπορά μεταξύ των φασματικών συνιστωσών ενός διαμορφωμένου οπτικού σήματος. Κάθε φασματική συνιστώσα, ανάλογα με την πόλωσή της στην είσοδο της οπτικής ίνας, υφίσταται διαφορετική αλλαγή φάσης κατά τη διέλευσή της μέσα από την οπτική ίνα. Το φαινόμενο αυτό ονομάζεται διασπορά τρόπων πόλωσης. Η διασπορά τρόπων πόλωσης στην οπτική ίνα προκαλεί παραμόρφωση του οπτικού σήματος κι αλληλοπαρεμβολή συμβόλων στον οπτικό δέκτη, με αποτέλεσμα τη μείωση της απόδοσης ενός σύμφωνου οπτικού τηλεπικοινωνιακού συστήματος. Για την αντιμετώπιση του φαινομένου, χρησιμοποιούνται οι προσαρμοστικοί γραμμικοί εξισωτές διασποράς τρόπων πόλωσης. Εξαιτίας της στατιστικής φύσης του φαινομένου, πιθανότητες διακοπής της λειτουργίας της τάξεως του 10-5 ενός σύμφωνου συστήματος, τετραδικής διαμόρφωσης φάσης με πολυπλεξία πόλωσης της τάξεως με εξισωτές διασποράς τρόπων πόλωσης, υπολογίστηκαν βάσει της πολυκανονικής Monte Carlo μεθόδου (MMC). Στην MMC μέθοδο. οι παράμετροι στην είσοδο του συστήματος κατευθύνονται, έτσι ώστε στην έξοδο, η (άγνωστη) συνάρτηση πυκνότητας πιθανότητας της παραμέτρου ελέγχου να υπολογίζεται με ακρίβεια ακόμα και στις ουρές της. Το πλεονέκτημα της ΜΜC, σε σχέση με τις μεθόδους δειγματοληψίας σημαντικότητας, είναι ότι δεν απαιτείται καμία γνώση για το ποιες περιοχές στην είσοδο πρέπει να δειγματοληφθούν, ώστε στην έξοδο να προκύψουν τα σπάνια εκείνα γεγονότα που μας ενδιαφέρουν. Με βάση την ΜΜC μέθοδο υλοποιήθηκε και το μοντέλο της ίνας, ως μια αλληλουχία διπλοθλαστικών πλακιδίων. Σκοπός της διδακτορικής διατριβής, είναι η αξιολόγηση της απόδοσης του ενός σύμφωνου συστήματος με χρήση των εξισωτών, συναρτήσει της πιθανότητας διακοπής της λειτουργίας του συστήματος. Για την περίπτωση της αποκλειστικής παρουσίας της διασποράς τρόπων πόλωσης, ο εξισωτής ελαχίστου μέσου τετραγώνου (DD-LMS) έχει αποδοτικότερη λειτουργία, σε σχέση με τον εξισωτή σταθερής περιβάλλουσας (CMA), για χαμηλές τιμές της διασποράς τρόπων πόλωσης, ενώ ο εξισωτής CMA κυριαρχεί στις περιοχές με μεγαλύτερες τιμές της διασποράς τρόπων πόλωσης. Η βέλτιστη λειτουργία του σύμφωνου συστήματος σε μια ευρύτερη περιοχή τιμών της διασποράς τρόπων πόλωσης, επιτυγχάνεται με την χρήση ενός συνδυασμού των δύο εξισωτών CMA και LMS. Η αλληλεπίδραση της διασποράς τρόπων πόλωσης και της ενδιάμεσης συχνότητας επηρεάζει την απόδοση του σύμφωνου συστήματος, όπου ο εξισωτής CMA λειτουργεί αποδοτικότερα σε σχέση με τον εξισωτή DD-LMS, τόσο στις περιοχές χαμηλής όσο και υψηλής τιμής της διασποράς τρόπων πόλωσης. Επίσης, αν στο μοντέλο της ίνας, προσομοιώσουμε και τις τυχαίες διαφορικές ολισθήσεις της φέρουσας συχνότητας μεταξύ των πλακιδίων, λόγω της διπλοθλαστικότητας, τότε η επίδοση των εξισωτών ελαττώνεται. Επομένως, θα πρέπει να λαμβάνονται υπόψιν για την ορθότερη αξιολόγηση της απόδοσης του σύμφωνου συστήματος.

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.