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.

GNSS interference management techniques against malicious attacks

This thesis collects the outcomes of a Ph.D. course in Telecommunications Engineering and it is focused on the study and design of possible techniques able to counteract interference signal in Global Navigation Satellite System (GNSS) systems. The subject is the jamming threat in navigation systems, that has become a very increasingly important topic in recent years, due to the wide diffusion of GNSS-based civil applications. Detection and mitigation techniques are developed in order to fight out jamming signals, tested in different scenarios and including sophisticated signals. The thesis is organized in two main parts, which deal with management of GNSS intentional counterfeit signals. The first part deals with the interference management, focusing on the intentional interfering signal. In particular, a technique for the detection and localization of the interfering signal level in the GNSS bands in frequency domain has been proposed. In addition, an effective mitigation technique which exploits the periodic characteristics of the common jamming signals reducing interfering effects at the receiver side has been introduced. Moreover, this technique has been also tested in a different and more complicated scenario resulting still effective in mitigation and cancellation of the interfering signal, without high complexity. The second part still deals with the problem of interference management, but regarding with more sophisticated signal. The attention is focused on the detection of spoofing signal, which is the most complex among the jamming signal types. Due to this highly difficulty in detect and mitigate this kind of signal, spoofing threat is considered the most dangerous. In this work, a possible techniques able to detect this sophisticated signal has been proposed, observing and exploiting jointly the outputs of several operational block measurements of the GNSS receiver operating chain.