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.

Matrix Designs and Methods for Secure and Efficient Compressed Sensing

The idea of balancing the resources spent in the acquisition and encoding of natural signals strictly to their intrinsic information content has interested nearly a decade of research under the name of compressed sensing. In this doctoral dissertation we develop some extensions and improvements upon this technique's foundations, by modifying the random sensing matrices on which the signals of interest are projected to achieve different objectives. Firstly, we propose two methods for the adaptation of sensing matrix ensembles to the second-order moments of natural signals. These techniques leverage the maximisation of different proxies for the quantity of information acquired by compressed sensing, and are efficiently applied in the encoding of electrocardiographic tracks with minimum-complexity digital hardware. Secondly, we focus on the possibility of using compressed sensing as a method to provide a partial, yet cryptanalysis-resistant form of encryption; in this context, we show how a random matrix generation strategy with a controlled amount of perturbations can be used to distinguish between multiple user classes with different quality of access to the encrypted information content. Finally, we explore the application of compressed sensing in the design of a multispectral imager, by implementing an optical scheme that entails a coded aperture array and Fabry-Pérot spectral filters. The signal recoveries obtained by processing real-world measurements show promising results, that leave room for an improvement of the sensing matrix calibration problem in the devised imager.

The foodomics approach to investigate the impact of the matrix structure on food quality: applications of magnetic resonance spectroscopy, relaxometry and imaging

The aim of this thesis is to explore the possible influence of the food matrix on food quality attributes. Using nuclear magnetic resonance techniques, the matrix-dependent properties of different foods were studied and some useful indices were defined to classify food products based on the matrix behaviour when responding to processing phenomena. Correlations were found between fish freshness indices, assessed by certain geometric parameters linked to the morphology of the animal, i.e. a macroscopic structure, and the degradation of the product structure. The same foodomics approach was also applied to explore the protective effect of modified atmospheres on the stability of fish fillets, which are typically susceptible to oxidation of the polyunsaturated fatty acids incorporated in the meat matrix. Here, freshness is assessed by evaluating the time-dependent change in the fish metabolome, providing an established freshness index, and its relationship to lipid oxidation. In vitro digestion studies, focusing on food products with different matrixes, alone and in combination with other meal components (e.g. seasoning), were conducted to investigate possible interactions between enzymes and food, modulated by matrix structure, which influence digestibility. The interaction between water and the gelatinous matrix of the food, consisting of a network of protein gels incorporating fat droplets, was also studied by means of nuclear magnetic relaxometry, in order to create a prediction tool for the correct classification of authentic and counterfeit food products protected by a quality label. This is one of the first applications of an NMR method focusing on the supramolecular structure of the matrix, rather than the chemical composition, to assess food authenticity. The effect of innovative processing technologies, such as PEF applied to fruit products, has been assessed by magnetic resonance imaging, exploiting information associated with the rehydration kinetics exerted by a modified food structure.