Evaluating links through spectral decomposition

Spectral decomposition has rarely been used to investigate complex networks. In this work we apply this concept in order to define two kinds of link-directed attacks while quantifying their respective effects on the topology. Several other kinds of more traditional attacks are also adopted and compared. These attacks had substantially diverse effects, depending on each specific network (models and real-world structures). It is also shown that the spectrally based attacks have special effects in affecting the transitivity of the networks.

Spin-glass behaviour on random lattices

The ground-state phase diagram of an Ising spin-glass model on a random graph with an arbitrary fraction w of ferromagnetic interactions is analysed in the presence of an external field. Using the replica method, and performing an analysis of stability of the replica-symmetric solution, it is shown that w = 1/2, corresponding to an unbiased spin glass, is a singular point in the phase diagram, separating a region with a spin-glass phase (w < 1/2) from a region with spin-glass, ferromagnetic, mixed and paramagnetic phases (w > 1/2).

A quantitative approach to evolution of music and philosophy

The development of new statistical and computational methods is increasingly making it possible to bridge the gap between hard sciences and humanities. In this study, we propose an approach based on a quantitative evaluation of attributes of objects in fields of humanities, from which concepts such as dialectics and opposition are formally defined mathematically. As case studies, we analyzed the temporal evolution of classical music and philosophy by obtaining data for 8 features characterizing the corresponding fields for 7 well-known composers and philosophers, which were treated with multivariate statistics and pattern recognition methods. A bootstrap method was applied to avoid statistical bias caused by the small sample data set, with which hundreds of artificial composers and philosophers were generated, influenced by the 7 names originally chosen. Upon defining indices for opposition, skewness and counter-dialectics, we confirmed the intuitive analysis of historians in that classical music evolved according to a master apprentice tradition, while in philosophy changes were driven by opposition. Though these case studies were meant only to show the possibility of treating phenomena in humanities quantitatively, including a quantitative measure of concepts such as dialectics and opposition, the results are encouraging for further application of the approach presented here to many other areas, since it is entirely generic.

Self-consistency in non-extensive thermodynamics of highly excited hadronic states

The self-consistency of a thermodynamical theory for hadronic systems based on the non-extensive statistics is investigated. We show that it is possible to obtain a self-consistent theory according to the asymptotic bootstrap principle if the mass spectrum and the energy density increase q-exponentially. A direct consequence is the existence of a limiting effective temperature for the hadronic system. We show that this result is in agreement with experiments. (C) 2012 Elsevier B.V. All rights reserved.

Nonlocal asymmetric exclusion process on a ring and conformal invariance

We present a one-dimensional nonlocal hopping model with exclusion on a ring. The model is related to the Raise and Peel growth model. A nonnegative parameter u controls the ratio of the local backwards and nonlocal forwards hopping rates. The phase diagram, and consequently the values of the current, depend on u and the density of particles. In the special case of half-lling and u = 1 the system is conformal invariant and an exact value of the current for any size L of the system is conjectured and checked for large lattice sizes in Monte Carlo simulations. For u > 1 the current has a non-analytic dependence on the density when the latter approaches the half-lling value.

An equilibrium approach to modelling social interaction

The aim of this work is to put forward a statistical mechanics theory of social interaction, generalizing econometric discrete choice models. After showing the formal equivalence linking econometric multinomial logit models to equilibrium statical mechanics, a multi- population generalization of the Curie-Weiss model for ferromagnets is considered as a starting point in developing a model capable of describing sudden shifts in aggregate human behaviour. Existence of the thermodynamic limit for the model is shown by an asymptotic sub-additivity method and factorization of correlation functions is proved almost everywhere. The exact solution for the model is provided in the thermodynamical limit by nding converging upper and lower bounds for the system's pressure, and the solution is used to prove an analytic result regarding the number of possible equilibrium states of a two-population system. The work stresses the importance of linking regimes predicted by the model to real phenomena, and to this end it proposes two possible procedures to estimate the model's parameters starting from micro-level data. These are applied to three case studies based on census type data: though these studies are found to be ultimately inconclusive on an empirical level, considerations are drawn that encourage further refinements of the chosen modelling approach, to be considered in future work.

Thermal relaxation for particle systems in interaction with several bosonic heat reservoirs

The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.

Computational modeling of surface interactions

Die Wechselwirkung zwischen Proteinen und anorganischen Oberflächen fasziniert sowohl aus angewandter als auch theoretischer Sicht. Sie ist ein wichtiger Aspekt in vielen Anwendungen, unter anderem in chirugischen Implantaten oder Biosensoren. Sie ist außerdem ein Beispiel für theoretische Fragestellungen betreffend die Grenzfläche zwischen harter und weicher Materie. Fest steht, dass Kenntnis der beteiligten Mechanismen erforderlich ist um die Wechselwirkung zwischen Proteinen und Oberflächen zu verstehen, vorherzusagen und zu optimieren. Aktuelle Fortschritte im experimentellen Forschungsbereich ermöglichen die Untersuchung der direkten Peptid-Metall-Bindung. Dadurch ist die Erforschung der theoretischen Grundlagen weiter ins Blickfeld aktueller Forschung gerückt. Eine Möglichkeit die Wechselwirkung zwischen Proteinen und anorganischen Oberflächen zu erforschen ist durch Computersimulationen. Obwohl Simulationen von Metalloberflächen oder Proteinen als Einzelsysteme schon länger verbreitet sind, bringt die Simulation einer Kombination beider Systeme neue Schwierigkeiten mit sich. Diese zu überwinden erfordert ein Mehrskalen-Verfahren: Während Proteine als biologische Systeme ausreichend mit klassischer Molekulardynamik beschrieben werden können, bedarf die Beschreibung delokalisierter Elektronen metallischer Systeme eine quantenmechanische Formulierung. Die wichtigste Voraussetzung eines Mehrskalen-Verfahrens ist eine Übereinstimmung der Simulationen auf den verschiedenen Skalen. In dieser Arbeit wird dies durch die Verknüpfung von Simulationen alternierender Skalen erreicht. Diese Arbeit beginnt mit der Untersuchung der Thermodynamik der Benzol-Hydratation mittels klassischer Molekulardynamik. Dann wird die Wechselwirkung zwischen Wasser und den [111]-Metalloberflächen von Gold und Nickel mittels eines Multiskalen-Verfahrens modelliert. In einem weiteren Schritt wird die Adsorbtion des Benzols an Metalloberflächen in wässriger Umgebung studiert. Abschließend wird die Modellierung erweitert und auch die Aminosäuren Alanin und Phenylalanin einbezogen. Dies eröffnet die Möglichkeit realistische Protein- Metall-Systeme in Computersimulationen zu betrachten und auf theoretischer Basis die Wechselwirkung zwischen Peptiden und Oberflächen für jede Art Peptide und Oberfläche vorauszusagen.

Thermal fluctuations and boundary conditions in the lattice Boltzmann method

The lattice Boltzmann method is a popular approach for simulating hydrodynamic interactions in soft matter and complex fluids. The solvent is represented on a discrete lattice whose nodes are populated by particle distributions that propagate on the discrete links between the nodes and undergo local collisions. On large length and time scales, the microdynamics leads to a hydrodynamic flow field that satisfies the Navier-Stokes equation. In this thesis, several extensions to the lattice Boltzmann method are developed. In complex fluids, for example suspensions, Brownian motion of the solutes is of paramount importance. However, it can not be simulated with the original lattice Boltzmann method because the dynamics is completely deterministic. It is possible, though, to introduce thermal fluctuations in order to reproduce the equations of fluctuating hydrodynamics. In this work, a generalized lattice gas model is used to systematically derive the fluctuating lattice Boltzmann equation from statistical mechanics principles. The stochastic part of the dynamics is interpreted as a Monte Carlo process, which is then required to satisfy the condition of detailed balance. This leads to an expression for the thermal fluctuations which implies that it is essential to thermalize all degrees of freedom of the system, including the kinetic modes. The new formalism guarantees that the fluctuating lattice Boltzmann equation is simultaneously consistent with both fluctuating hydrodynamics and statistical mechanics. This establishes a foundation for future extensions, such as the treatment of multi-phase and thermal flows. An important range of applications for the lattice Boltzmann method is formed by microfluidics. Fostered by the "lab-on-a-chip" paradigm, there is an increasing need for computer simulations which are able to complement the achievements of theory and experiment. Microfluidic systems are characterized by a large surface-to-volume ratio and, therefore, boundary conditions are of special relevance. On the microscale, the standard no-slip boundary condition used in hydrodynamics has to be replaced by a slip boundary condition. In this work, a boundary condition for lattice Boltzmann is constructed that allows the slip length to be tuned by a single model parameter. Furthermore, a conceptually new approach for constructing boundary conditions is explored, where the reduced symmetry at the boundary is explicitly incorporated into the lattice model. The lattice Boltzmann method is systematically extended to the reduced symmetry model. In the case of a Poiseuille flow in a plane channel, it is shown that a special choice of the collision operator is required to reproduce the correct flow profile. This systematic approach sheds light on the consequences of the reduced symmetry at the boundary and leads to a deeper understanding of boundary conditions in the lattice Boltzmann method. This can help to develop improved boundary conditions that lead to more accurate simulation results.

Crystal phases and glassy dynamics in monodisperse hard ellipsoids

We have performed Monte Carlo and molecular dynamics simulations of suspensions of monodisperse, hard ellipsoids of revolution. Hard-particle models play a key role in statistical mechanics. They are conceptually and computationally simple, and they offer insight into systems in which particle shape is important, including atomic, molecular, colloidal, and granular systems. In the high density phase diagram of prolate hard ellipsoids we have found a new crystal, which is more stable than the stretched FCC structure proposed previously . The new phase, SM2, has a simple monoclinic unit cell containing a basis of two ellipsoids with unequal orientations. The angle of inclination is very soft for length-to-width (aspect) ratio l/w=3, while the other angles are not. A symmetric state of the unit cell exists, related to the densest-known packings of ellipsoids; it is not always the stable one. Our results remove the stretched FCC structure for aspect ratio l/w=3 from the phase diagram of hard, uni-axial ellipsoids. We provide evidence that this holds between aspect ratios 3 and 6, and possibly beyond. Finally, ellipsoids in SM2 at l/w=1.55 exhibit end-over-end flipping, warranting studies of the cross-over to where this dynamics is not possible. Secondly, we studied the dynamics of nearly spherical ellipsoids. In equilibrium, they show a first-order transition from an isotropic phase to a rotator phase, where positions are crystalline but orientations are free. When over-compressing the isotropic phase into the rotator regime, we observed super-Arrhenius slowing down of diffusion and relaxation, and signatures of the cage effect. These features of glassy dynamics are sufficiently strong that asymptotic scaling laws of the Mode-Coupling Theory of the glass transition (MCT) could be tested, and were found to apply. We found strong coupling of positional and orientational degrees of freedom, leading to a common value for the MCT glass-transition volume fraction. Flipping modes were not slowed down significantly. We demonstrated that the results are independent of simulation method, as predicted by MCT. Further, we determined that even intra-cage motion is cooperative. We confirmed the presence of dynamical heterogeneities associated with the cage effect. The transit between cages was seen to occur on short time scales, compared to the time spent in cages; but the transit was shown not to involve displacements distinguishable in character from intra-cage motion. The presence of glassy dynamics was predicted by molecular MCT (MMCT). However, as MMCT disregards crystallization, a test by simulation was required. Glassy dynamics is unusual in monodisperse systems. Crystallization typically intervenes unless polydispersity, network-forming bonds or other asymmetries are introduced. We argue that particle anisometry acts as a sufficient source of disorder to prevent crystallization. This sheds new light on the question of which ingredients are required for glass formation.

Caratterizzazione della misura di entropia di singolo nodo nell'ambito della teoria statistica dei network

In questo lavoro si è affrontata la definizione e la caratterizzazione di una misura di entropia di singolo nodo nell’ambito della teoria statistica dei network, per ottenere informazioni a livello di singolo nodo a fini di analisi e classificazione. Sono state introdotte e studiate alcune proprietà di questi osservabili in quanto la Network Entropy, precedentemente definita e utilizzata nello stesso contesto, fornisce un’informazione globale a livello dell’intero network. I risultati delle analisi svolte con questa definizione sono stati confrontati con una seconda definizione di entropia di singolo nodo proveniente dalla letteratura, applicando entrambe le misure allo stesso problema di caratterizzazione di due classi di nodi all’interno di un network

Implementation of network entropy algorithms on hpc machines, with application to high-dimensional experimental data

Network Theory is a prolific and lively field, especially when it approaches Biology. New concepts from this theory find application in areas where extensive datasets are already available for analysis, without the need to invest money to collect them. The only tools that are necessary to accomplish an analysis are easily accessible: a computing machine and a good algorithm. As these two tools progress, thanks to technology advancement and human efforts, wider and wider datasets can be analysed. The aim of this paper is twofold. Firstly, to provide an overview of one of these concepts, which originates at the meeting point between Network Theory and Statistical Mechanics: the entropy of a network ensemble. This quantity has been described from different angles in the literature. Our approach tries to be a synthesis of the different points of view. The second part of the work is devoted to presenting a parallel algorithm that can evaluate this quantity over an extensive dataset. Eventually, the algorithm will also be used to analyse high-throughput data coming from biology.

Coarse-graining and quantum-classical adaptive coupling in soft matter

Die vorliegende Arbeit untersucht den Zusammenhang zwischen Skalen in Systemen weicher Materie, der für Multiskalen-Simulationen eine wichtige Rolle spielt. Zu diesem Zweck wurde eine Methode entwickelt, die die Approximation der Separierbarkeit von Variablen für die Molekulardynamik und ähnliche Anwendungen bewertet. Der zweite und größere Teil dieser Arbeit beschäftigt sich mit der konzeptionellen und technischen Erweiterung des Adaptive Resolution Scheme'' (AdResS), einer Methode zur gleichzeitigen Simulation von Systemen mit mehreren Auflösungsebenen. Diese Methode wurde auf Systeme erweitert, in denen klassische und quantenmechanische Effekte eine Rolle spielen.rnrnDie oben genannte erste Methode benötigt nur die analytische Form der Potentiale, wie sie die meisten Molekulardynamik-Programme zur Verfügung stellen. Die Anwendung der Methode auf ein spezielles Problem gibt bei erfolgreichem Ausgang einen numerischen Hinweis auf die Gültigkeit der Variablenseparation. Bei nicht erfolgreichem Ausgang garantiert sie, dass keine Separation der Variablen möglich ist. Die Methode wird exemplarisch auf ein zweiatomiges Molekül auf einer Oberfläche und für die zweidimensionale Version des Rotational Isomer State (RIS) Modells einer Polymerkette angewandt.rnrnDer zweite Teil der Arbeit behandelt die Entwicklung eines Algorithmus zur adaptiven Simulation von Systemen, in denen Quanteneffekte berücksichtigt werden. Die Quantennatur von Atomen wird dabei in der Pfadintegral-Methode durch einen klassischen Polymerring repräsentiert. Die adaptive Pfadintegral-Methode wird zunächst für einatomige Flüssigkeiten und tetraedrische Moleküle unter normalen thermodynamischen Bedingungen getestet. Schließlich wird die Stabilität der Methode durch ihre Anwendung auf flüssigen para-Wasserstoff bei niedrigen Temperaturen geprüft.

Rigorous results in Spin Glasses and Monomer-Dimer systems

In this work I reported recent results in the field of Statistical Mechanics of Equilibrium, and in particular in Spin Glass models and Monomer Dimer models . We start giving the mathematical background and the general formalism for Spin (Disordered) Models with some of their applications to physical and mathematical problems. Next we move on general aspects of the theory of spin glasses, in particular to the Sherrington-Kirkpatrick model which is of fundamental interest for the work. In Chapter 3, we introduce the Multi-species Sherrington-Kirkpatrick model (MSK), we prove the existence of the thermodynamical limit and the Guerra's Bound for the quenched pressure together with a detailed analysis of the annealed and the replica symmetric regime. The result is a multidimensional generalization of the Parisi's theory. Finally we brie y illustrate the strategy of the Panchenko's proof of the lower bound. In Chapter 4 we discuss the Aizenmann-Contucci and the Ghirlanda-Guerra identities for a wide class of Spin Glass models. As an example of application, we discuss the role of these identities in the proof of the lower bound. In Chapter 5 we introduce the basic mathematical formalism of Monomer Dimer models. We introduce a Gaussian representation of the partition function that will be fundamental in the rest of the work. In Chapter 6, we introduce an interacting Monomer-Dimer model. Its exact solution is derived and a detailed study of its analytical properties and related physical quantities is performed. In Chapter 7, we introduce a quenched randomness in the Monomer Dimer model and show that, under suitable conditions the pressure is a self averaging quantity. The main result is that, if we consider randomness only in the monomer activity, the model is exactly solvable.

Optische Mikroskopie an elektrostatisch stabilisierten kolloidalen Suspensionen unter dem Einfluss geladener Wände - ausgewählte Gleichgewichts- und Nichtgleichgewichtsphänomene

Das Verhalten kolloidaler Suspensionen unter räumlich beschränkter Geometrie ist von großer Bedeutung für die statistische Physik wie auch für die Technologie. Von speziellem Interesse sind Modellsysteme geladener kolloidaler Sphären aufgrund ihrer langreichweitigen und veränderbaren Wechselwirkungen. In dieser Arbeit wurde ein experimenteller Aufbau für die optische mikroskopische Untersuchung solcher, zwischen ebenen Wänden beschränkter Systeme realisiert. Anhand von Piezo-Aktuatoren kann die Zellgeometrie flexibel und präzise eingestellt werden. Unter Verwendung eines Pumpkreislaufs mit einer Ionentauschersäule können kolloidale Suspensionen unter stark entsalzten Bedingungen effizient präpariert werden. Anhand dieses Aufbaus wurde zunächst das Gleichgewichtsphasendiagramm monodisperser geladener kolloidaler Sphären zwischen parallelen Wänden untersucht. Es wurden quantitative Resultate für den Grenzfall starker Entsalzung erzielt, welche mit theoretischen Grundzustandsvorhersagen übereinstimmen. In Doppellagensystemen konnte die Existenz transienter kolloidaler Moiré-Rotationsmuster demonstriert werden, welche besondere zweidimensionale Kristallstrukturen mit komplexer Basis darstellen. Es wurden ferner Nichtgleichgewichtsphänomene untersucht, welche durch Gradienten von lokal freigesetzten Elektrolyten verursacht werden. Durch hauptsächlich diffusioosmotischen Partikeltransport entlang einer einzelnen geladenen Substratoberfläche konnten die Bildung kristalliner Ordnung sowie komplexe, selbstorganisierte Bewegungszustände in einem verdünnten kolloidalen Monolagenfluid bei kleinen Reynolds-Zahlen induziert werden. Interessante Perspektiven für die zukünftige Verwendung des experimentellen Aufbaus ergeben sich aus Beobachtungen verschiedener weiterer Phänomene.