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

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.

Computer simulation of ordering and dynamics in liquid crystals in the bulk and close to the surface

The aim of this PhD thesis is to investigate the orientational and dynamical properties of liquid crystalline systems, at molecular level and using atomistic computer simulations, to reach a better understanding of material behavior from a microscopic point view. In perspective this should allow to clarify the relation between the micro and macroscopic properties with the objective of predicting or confirming experimental results on these systems. In this context, we developed four different lines of work in the thesis. The first one concerns the orientational order and alignment mechanism of rigid solutes of small dimensions dissolved in a nematic phase formed by the 4-pentyl,4 cyanobiphenyl (5CB) nematic liquid crystal. The orientational distribution of solutes have been obtained with Molecular Dynamics Simulation (MD) and have been compared with experimental data reported in literature. we have also verified the agreement between order parameters and dipolar coupling values measured in NMR experiments. The MD determined effective orientational potentials have been compared with the predictions of Maier­Saupe and Surface tensor models. The second line concerns the development of a correct parametrization able to reproduce the phase transition properties of a prototype of the oligothiophene semiconductor family: sexithiophene (T6). T6 forms two crystalline polymorphs largely studied, and possesses liquid crystalline phases still not well characterized, From simulations we detected a phase transition from crystal to liquid crystal at about 580 K, in agreement with available experiments, and in particular we found two LC phases, smectic and nematic. The crystal­smectic transition is associated to a relevant density variation and to strong conformational changes of T6, namely the molecules in the liquid crystal phase easily assume a bent shape, deviating from the planar structure typical of the crystal. The third line explores a new approach for calculating the viscosity in a nematic through a virtual exper- iment resembling the classical falling sphere experiment. The falling sphere is replaced by an hydrogenated silicon nanoparticle of spherical shape suspended in 5CB, and gravity effects are replaced by a constant force applied to the nanoparticle in a selected direction. Once the nanoparticle reaches a constant velocity, the viscosity of the medium can be evaluated using Stokes' law. With this method we successfully reproduced experimental viscosities and viscosity anisotropy for the solvent 5CB. The last line deals with the study of order induction on nematic molecules by an hydrogenated silicon surface. Gaining predicting power for the anchoring behavior of liquid crystals at surfaces will be a very desirable capability, as many properties related to devices depend on molecular organization close to surfaces. Here we studied, by means of atomistic MD simulations, the flat interface between an hydrogenated (001) silicon surface in contact with a sample of 5CB molecules. We found a planar anchoring of the first layers of 5CB where surface interactions are dominating with respect to the mesogen intermolecular interactions. We also analyzed the interface 5CB­vacuum, finding a homeotropic orientation of the nematic at this interface.

Structure-Property Relationships and Charge Transport Modeling of Organic Molecular Materials

From the perspective of a new-generation opto-electronic technology based on organic semiconductors, a major objective is to achieve a deep and detailed knowledge of the structure-property relationships, in order to optimize the electronic, optical, and charge transport properties by tuning the chemical-physical characteristics of the compounds. The purpose of this dissertation is to contribute to such understanding, through suitable theoretical and computational studies. Precisely, the structural, electronic, optical, and charge transport characteristics of several promising organic materials recently synthesized are investigated by means of an integrated approach encompassing quantum-chemical calculations, molecular dynamics and kinetic Monte Carlo simulations. Particular care is addressed to the rationalization of optical and charge transport properties in terms of both intra- and intermolecular features. Moreover, a considerable part of this project involves the development of a home-made set of procedures and parts of software code required to assist the modeling of charge transport properties in the framework of the non-adiabatic hopping mechanism applied to organic crystalline materials. As a first part of my investigations, I mainly discuss the optical, electronic, and structural properties of several core-extended rylene derivatives, which can be regarded to as model compounds for graphene nanoribbons. Two families have been studied, consisting in bay-linked perylene bisimide oligomers and N-annulated rylenes. Beside rylene derivatives, my studies also concerned electronic and spectroscopic properties of tetracene diimides, quinoidal oligothiophenes, and oxygen doped picene. As an example of device application, I studied the structural characteristics governing the efficiency of resistive molecular memories based on a derivative of benzoquinone. Finally, as a second part of my investigations, I concentrate on the charge transport properties of perylene bisimides derivatives. Precisely, a comprehensive study of the structural and thermal effects on the charge transport of several core-twisted chlorinated and fluoro-alkylated perylene bisimide n-type semiconductors is presented.