Geometric and Combinatorial Aspects of NonEquilibrium Statistical Mechanics

Non-Equilibrium Statistical Mechanics is a broad subject. Grossly speaking, it deals with systems which have not yet relaxed to an equilibrium state, or else with systems which are in a steady non-equilibrium state, or with more general situations. They are characterized by external forcing and internal fluxes, resulting in a net production of entropy which quantifies dissipation and the extent by which, by the Second Law of Thermodynamics, time-reversal invariance is broken. In this thesis we discuss some of the mathematical structures involved with generic discrete-state-space non-equilibrium systems, that we depict with networks in all analogous to electrical networks. We define suitable observables and derive their linear regime relationships, we discuss a duality between external and internal observables that reverses the role of the system and of the environment, we show that network observables serve as constraints for a derivation of the minimum entropy production principle. We dwell on deep combinatorial aspects regarding linear response determinants, which are related to spanning tree polynomials in graph theory, and we give a geometrical interpretation of observables in terms of Wilson loops of a connection and gauge degrees of freedom. We specialize the formalism to continuous-time Markov chains, we give a physical interpretation for observables in terms of locally detailed balanced rates, we prove many variants of the fluctuation theorem, and show that a well-known expression for the entropy production due to Schnakenberg descends from considerations of gauge invariance, where the gauge symmetry is related to the freedom in the choice of a prior probability distribution. As an additional topic of geometrical flavor related to continuous-time Markov chains, we discuss the Fisher-Rao geometry of nonequilibrium decay modes, showing that the Fisher matrix contains information about many aspects of non-equilibrium behavior, including non-equilibrium phase transitions and superposition of modes. We establish a sort of statistical equivalence principle and discuss the behavior of the Fisher matrix under time-reversal. To conclude, we propose that geometry and combinatorics might greatly increase our understanding of nonequilibrium phenomena.

Statistical mechanics formalism and methods for the analysis of real networks

This thesis provides a thoroughly theoretical background in network theory and shows novel applications to real problems and data. In the first chapter a general introduction to network ensembles is given, and the relations with “standard” equilibrium statistical mechanics are described. Moreover, an entropy measure is considered to analyze statistical properties of the integrated PPI-signalling-mRNA expression networks in different cases. In the second chapter multilayer networks are introduced to evaluate and quantify the correlations between real interdependent networks. Multiplex networks describing citation-collaboration interactions and patterns in colorectal cancer are presented. The last chapter is completely dedicated to control theory and its relation with network theory. We characterise how the structural controllability of a network is affected by the fraction of low in-degree and low out-degree nodes. Finally, we present a novel approach to the controllability of multiplex networks

New perspectives in statistical mechanics and high-dimensional inference

The main purpose of this thesis is to go beyond two usual assumptions that accompany theoretical analysis in spin-glasses and inference: the i.i.d. (independently and identically distributed) hypothesis on the noise elements and the finite rank regime. The first one appears since the early birth of spin-glasses. The second one instead concerns the inference viewpoint. Disordered systems and Bayesian inference have a well-established relation, evidenced by their continuous cross-fertilization. The thesis makes use of techniques coming both from the rigorous mathematical machinery of spin-glasses, such as the interpolation scheme, and from Statistical Physics, such as the replica method. The first chapter contains an introduction to the Sherrington-Kirkpatrick and spiked Wigner models. The first is a mean field spin-glass where the couplings are i.i.d. Gaussian random variables. The second instead amounts to establish the information theoretical limits in the reconstruction of a fixed low rank matrix, the “spike”, blurred by additive Gaussian noise. In chapters 2 and 3 the i.i.d. hypothesis on the noise is broken by assuming a noise with inhomogeneous variance profile. In spin-glasses this leads to multi-species models. The inferential counterpart is called spatial coupling. All the previous models are usually studied in the Bayes-optimal setting, where everything is known about the generating process of the data. In chapter 4 instead we study the spiked Wigner model where the prior on the signal to reconstruct is ignored. In chapter 5 we analyze the statistical limits of a spiked Wigner model where the noise is no longer Gaussian, but drawn from a random matrix ensemble, which makes its elements dependent. The thesis ends with chapter 6, where the challenging problem of high-rank probabilistic matrix factorization is tackled. Here we introduce a new procedure called "decimation" and we show that it is theoretically to perform matrix factorization through it.

Characterization of geothermal reservoirs’ parameters by inverse problem resolution and geostatistical simulations

BTES (borehole thermal energy storage)systems exchange thermal energy by conduction with the surrounding ground through borehole materials. The spatial variability of the geological properties and the space-time variability of hydrogeological conditions affect the real power rate of heat exchangers and, consequently, the amount of energy extracted from / injected into the ground. For this reason, it is not an easy task to identify the underground thermal properties to use when designing. At the current state of technology, Thermal Response Test (TRT) is the in situ test for the characterization of ground thermal properties with the higher degree of accuracy, but it doesn’t fully solve the problem of characterizing the thermal properties of a shallow geothermal reservoir, simply because it characterizes only the neighborhood of the heat exchanger at hand and only for the test duration. Different analytical and numerical models exist for the characterization of shallow geothermal reservoir, but they are still inadequate and not exhaustive: more sophisticated models must be taken into account and a geostatistical approach is needed to tackle natural variability and estimates uncertainty. The approach adopted for reservoir characterization is the “inverse problem”, typical of oil&gas field analysis. Similarly, we create different realizations of thermal properties by direct sequential simulation and we find the best one fitting real production data (fluid temperature along time). The software used to develop heat production simulation is FEFLOW 5.4 (Finite Element subsurface FLOW system). A geostatistical reservoir model has been set up based on literature thermal properties data and spatial variability hypotheses, and a real TRT has been tested. Then we analyzed and used as well two other codes (SA-Geotherm and FV-Geotherm) which are two implementation of the same numerical model of FEFLOW (Al-Khoury model).

Predictability in Social Science, The statistical mechanics approach.

The subject of this work concerns the study of the immigration phenomenon, with emphasis on the aspects related to the integration of an immigrant population in a hosting one. Aim of this work is to show the forecasting ability of a recent finding where the behavior of integration quantifiers was analyzed and investigated with a mathematical model of statistical physics origins (a generalization of the monomer dimer model). After providing a detailed literature review of the model, we show that not only such a model is able to identify the social mechanism that drives a particular integration process, but it also provides correct forecast. The research reported here proves that the proposed model of integration and its forecast framework are simple and effective tools to reduce uncertainties about how integration phenomena emerge and how they are likely to develop in response to increased migration levels in the future.

Bayesian Analysis of Linear Inverse Problems with Applications in Economics and Finance

In my PhD thesis I propose a Bayesian nonparametric estimation method for structural econometric models where the functional parameter of interest describes the economic agent's behavior. The structural parameter is characterized as the solution of a functional equation, or by using more technical words, as the solution of an inverse problem that can be either ill-posed or well-posed. From a Bayesian point of view, the parameter of interest is a random function and the solution to the inference problem is the posterior distribution of this parameter. A regular version of the posterior distribution in functional spaces is characterized. However, the infinite dimension of the considered spaces causes a problem of non continuity of the solution and then a problem of inconsistency, from a frequentist point of view, of the posterior distribution (i.e. problem of ill-posedness). The contribution of this essay is to propose new methods to deal with this problem of ill-posedness. The first one consists in adopting a Tikhonov regularization scheme in the construction of the posterior distribution so that I end up with a new object that I call regularized posterior distribution and that I guess it is solution of the inverse problem. The second approach consists in specifying a prior distribution on the parameter of interest of the g-prior type. Then, I detect a class of models for which the prior distribution is able to correct for the ill-posedness also in infinite dimensional problems. I study asymptotic properties of these proposed solutions and I prove that, under some regularity condition satisfied by the true value of the parameter of interest, they are consistent in a "frequentist" sense. Once I have set the general theory, I apply my bayesian nonparametric methodology to different estimation problems. First, I apply this estimator to deconvolution and to hazard rate, density and regression estimation. Then, I consider the estimation of an Instrumental Regression that is useful in micro-econometrics when we have to deal with problems of endogeneity. Finally, I develop an application in finance: I get the bayesian estimator for the equilibrium asset pricing functional by using the Euler equation defined in the Lucas'(1978) tree-type models.

Coordinated Control of Robotic Swarms in Unknown Environments

This thesis gathers the work carried out by the author in the last three years of research and it concerns the study and implementation of algorithms to coordinate and control a swarm of mobile robots moving in unknown environments. In particular, the author's attention is focused on two different approaches in order to solve two different problems. The first algorithm considered in this work deals with the possibility of decomposing a main complex task in many simple subtasks by exploiting the decentralized implementation of the so called \emph{Null Space Behavioral} paradigm. This approach to the problem of merging different subtasks with assigned priority is slightly modified in order to handle critical situations that can be detected when robots are moving through an unknown environment. In fact, issues can occur when one or more robots got stuck in local minima: a smart strategy to avoid deadlock situations is provided by the author and the algorithm is validated by simulative analysis. The second problem deals with the use of concepts borrowed from \emph{graph theory} to control a group differential wheel robots by exploiting the Laplacian solution of the consensus problem. Constraints on the swarm communication topology have been introduced by the use of a range and bearing platform developed at the Distributed Intelligent Systems and Algorithms Laboratory (DISAL), EPFL (Lausanne, CH) where part of author's work has been carried out. The control algorithm is validated by demonstration and simulation analysis and, later, is performed by a team of four robots engaged in a formation mission. To conclude, the capabilities of the algorithm based on the local solution of the consensus problem for differential wheel robots are demonstrated with an application scenario, where nine robots are engaged in a hunting task.

Excitonic properties of transition metal oxide perovskites and workflow automatization of GW schemes

The Many-Body-Perturbation Theory approach is among the most successful theoretical frameworks for the study of excited state properties. It allows to describe the excitonic interactions, which play a fundamental role in the optical response of insulators and semiconductors. The first part of the thesis focuses on the study of the quasiparticle, optical and excitonic properties of \textit{bulk} Transition Metal Oxide (TMO) perovskites using a G$_0$W$_0$+Bethe Salpeter Equation (BSE) approach. A representative set of 14 compounds has been selected, including 3d, 4d and 5d perovskites. An approximation of the BSE scheme, based on an analytic diagonal expression for the inverse dielectric function, is used to compute the exciton binding energies and is carefully bench-marked against the standard BSE results. In 2019 an important breakthrough has been achieved with the synthesis of ultrathin SrTiO3 films down to the monolayer limit. This allows us to explore how the quasiparticle and optical properties of SrTiO3 evolve from the bulk to the two-dimensional limit. The electronic structure is computed with G0W0 approach: we prove that the inclusion of the off-diagonal self-energy terms is required to avoid non-physical band dispersions. The excitonic properties are investigated beyond the optical limit at finite momenta. Lastly a study of the under pressure optical response of the topological nodal line semimetal ZrSiS is presented, in conjunction with the experimental results from the group of Prof. Dr. Kuntscher of the Augsburg University. The second part of the thesis discusses the implementation of a workflow to automate G$_0$W$_0$ and BSE calculations with the VASP software. The workflow adopts a convergence scheme based on an explicit basis-extrapolation approach [J. Klimeš \textit{et al.}, Phys. Rev.B 90, 075125 (2014)] which allows to reduce the number of intermediate calculations required to reach convergence and to explicit estimate the error associated to the basis-set truncation.

Multiresolution spherical wavelet analysis in global seismic tomography

Every seismic event produces seismic waves which travel throughout the Earth. Seismology is the science of interpreting measurements to derive information about the structure of the Earth. Seismic tomography is the most powerful tool for determination of 3D structure of deep Earth's interiors. Tomographic models obtained at the global and regional scales are an underlying tool for determination of geodynamical state of the Earth, showing evident correlation with other geophysical and geological characteristics. The global tomographic images of the Earth can be written as a linear combinations of basis functions from a specifically chosen set, defining the model parameterization. A number of different parameterizations are commonly seen in literature: seismic velocities in the Earth have been expressed, for example, as combinations of spherical harmonics or by means of the simpler characteristic functions of discrete cells. With this work we are interested to focus our attention on this aspect, evaluating a new type of parameterization, performed by means of wavelet functions. It is known from the classical Fourier theory that a signal can be expressed as the sum of a, possibly infinite, series of sines and cosines. This sum is often referred as a Fourier expansion. The big disadvantage of a Fourier expansion is that it has only frequency resolution and no time resolution. The Wavelet Analysis (or Wavelet Transform) is probably the most recent solution to overcome the shortcomings of Fourier analysis. The fundamental idea behind this innovative analysis is to study signal according to scale. Wavelets, in fact, are mathematical functions that cut up data into different frequency components, and then study each component with resolution matched to its scale, so they are especially useful in the analysis of non stationary process that contains multi-scale features, discontinuities and sharp strike. Wavelets are essentially used in two ways when they are applied in geophysical process or signals studies: 1) as a basis for representation or characterization of process; 2) as an integration kernel for analysis to extract information about the process. These two types of applications of wavelets in geophysical field, are object of study of this work. At the beginning we use the wavelets as basis to represent and resolve the Tomographic Inverse Problem. After a briefly introduction to seismic tomography theory, we assess the power of wavelet analysis in the representation of two different type of synthetic models; then we apply it to real data, obtaining surface wave phase velocity maps and evaluating its abilities by means of comparison with an other type of parametrization (i.e., block parametrization). For the second type of wavelet application we analyze the ability of Continuous Wavelet Transform in the spectral analysis, starting again with some synthetic tests to evaluate its sensibility and capability and then apply the same analysis to real data to obtain Local Correlation Maps between different model at same depth or between different profiles of the same model.

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.

Coordination and Control of Autonomous Mobile Robots Swarms by using Particle Swarm Optimization Algorithm and Consensus Theory

This thesis presents some different techniques designed to drive a swarm of robots in an a-priori unknown environment in order to move the group from a starting area to a final one avoiding obstacles. The presented techniques are based on two different theories used alone or in combination: Swarm Intelligence (SI) and Graph Theory. Both theories are based on the study of interactions between different entities (also called agents or units) in Multi- Agent Systems (MAS). The first one belongs to the Artificial Intelligence context and the second one to the Distributed Systems context. These theories, each one from its own point of view, exploit the emergent behaviour that comes from the interactive work of the entities, in order to achieve a common goal. The features of flexibility and adaptability of the swarm have been exploited with the aim to overcome and to minimize difficulties and problems that can affect one or more units of the group, having minimal impact to the whole group and to the common main target. Another aim of this work is to show the importance of the information shared between the units of the group, such as the communication topology, because it helps to maintain the environmental information, detected by each single agent, updated among the swarm. Swarm Intelligence has been applied to the presented technique, through the Particle Swarm Optimization algorithm (PSO), taking advantage of its features as a navigation system. The Graph Theory has been applied by exploiting Consensus and the application of the agreement protocol with the aim to maintain the units in a desired and controlled formation. This approach has been followed in order to conserve the power of PSO and to control part of its random behaviour with a distributed control algorithm like Consensus.

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.

User scheduling for low earth orbit multi-user multiple-input-multiple-output non-terrestrial network systems

In rural and isolated areas without cellular coverage, Satellite Communication (SatCom) is the best candidate to complement terrestrial coverage. However, the main challenge for future generations of wireless networks will be to meet the growing demand for new services while dealing with the scarcity of frequency spectrum. As a result, it is critical to investigate more efficient methods of utilizing the limited bandwidth; and resource sharing is likely the only choice. The research community’s focus has recently shifted towards the interference management and exploitation paradigm to meet the increasing data traffic demands. In the Downlink (DL) and Feedspace (FS), LEO satellites with an on-board antenna array can offer service to numerous User Terminals (UTs) (VSAT or Handhelds) on-ground in FFR schemes by using cutting-edge digital beamforming techniques. Considering this setup, the adoption of an effective user scheduling approach is a critical aspect given the unusually high density of User terminals on the ground as compared to the on-board available satellite antennas. In this context, one possibility is that of exploiting clustering algorithms for scheduling in LEO MU-MIMO systems in which several users within the same group are simultaneously served by the satellite via Space Division Multiplexing (SDM), and then these different user groups are served in different time slots via Time Division Multiplexing (TDM). This thesis addresses this problem by defining a user scheduling problem as an optimization problem and discusses several algorithms to solve it. In particular, focusing on the FS and user service link (i.e., DL) of a single MB-LEO satellite operating below 6 GHz, the user scheduling problem in the Frequency Division Duplex (FDD) mode is addressed. The proposed State-of-the-Art scheduling approaches are based on graph theory. The proposed solution offers high performance in terms of per-user capacity, Sum-rate capacity, SINR, and Spectral Efficiency.

Kinematic description of the rupture from strong motion data: strategies for a robust inversion

We present a non linear technique to invert strong motion records with the aim of obtaining the final slip and rupture velocity distributions on the fault plane. In this thesis, the ground motion simulation is obtained evaluating the representation integral in the frequency. The Green’s tractions are computed using the discrete wave-number integration technique that provides the full wave-field in a 1D layered propagation medium. The representation integral is computed through a finite elements technique, based on a Delaunay’s triangulation on the fault plane. The rupture velocity is defined on a coarser regular grid and rupture times are computed by integration of the eikonal equation. For the inversion, the slip distribution is parameterized by 2D overlapping Gaussian functions, which can easily relate the spectrum of the possible solutions with the minimum resolvable wavelength, related to source-station distribution and data processing. The inverse problem is solved by a two-step procedure aimed at separating the computation of the rupture velocity from the evaluation of the slip distribution, the latter being a linear problem, when the rupture velocity is fixed. The non-linear step is solved by optimization of an L2 misfit function between synthetic and real seismograms, and solution is searched by the use of the Neighbourhood Algorithm. The conjugate gradient method is used to solve the linear step instead. The developed methodology has been applied to the M7.2, Iwate Nairiku Miyagi, Japan, earthquake. The estimated magnitude seismic moment is 2.6326 dyne∙cm that corresponds to a moment magnitude MW 6.9 while the mean the rupture velocity is 2.0 km/s. A large slip patch extends from the hypocenter to the southern shallow part of the fault plane. A second relatively large slip patch is found in the northern shallow part. Finally, we gave a quantitative estimation of errors associates with the parameters.