Master Equation: Biological Applications and Thermodynamic Description

It is well known that many realistic mathematical models of biological systems, such as cell growth, cellular development and differentiation, gene expression, gene regulatory networks, enzyme cascades, synaptic plasticity, aging and population growth need to include stochasticity. These systems are not isolated, but rather subject to intrinsic and extrinsic fluctuations, which leads to a quasi equilibrium state (homeostasis). The natural framework is provided by Markov processes and the Master equation (ME) describes the temporal evolution of the probability of each state, specified by the number of units of each species. The ME is a relevant tool for modeling realistic biological systems and allow also to explore the behavior of open systems. These systems may exhibit not only the classical thermodynamic equilibrium states but also the nonequilibrium steady states (NESS). This thesis deals with biological problems that can be treat with the Master equation and also with its thermodynamic consequences. It is organized into six chapters with four new scientific works, which are grouped in two parts: (1) Biological applications of the Master equation: deals with the stochastic properties of a toggle switch, involving a protein compound and a miRNA cluster, known to control the eukaryotic cell cycle and possibly involved in oncogenesis and with the propose of a one parameter family of master equations for the evolution of a population having the logistic equation as mean field limit. (2) Nonequilibrium thermodynamics in terms of the Master equation: where we study the dynamical role of chemical fluxes that characterize the NESS of a chemical network and we propose a one parameter parametrization of BCM learning, that was originally proposed to describe plasticity processes, to study the differences between systems in DB and NESS.

Diagnostic techniques for EHD and MHD interaction

The impact of plasma technologies is growing both in the academic and in the industrial fields. Nowadays, a great interest is focused in plasma applications in aeronautics and astronautics domains. Plasma actuators based on the Magneto-Hydro-Dynamic (MHD) and Electro- Hydro-Dynamic (EHD) interactions are potentially able to suitably modify the fluid-dynamics characteristics around a flying body without utilizing moving parts. This could lead to the control of an aircraft with negligible response time, more reliability and improvements of the performance. In order to study the aforementioned interactions, a series of experiments and a wide number of diagnostic techniques have been utilized. The EHD interaction, realized by means of a Dielectric Barrier Discharge (DBD) actuator, and its impact on the boundary layer have been evaluated by means of two different experiments. In the first one a three phase multi-electrode flat panel actuator is used. Different external flow velocities (from 1 to 20m/s) and different values of the supplied voltage and frequency have been considered. Moreover a change of the phase sequence has been done to verify the influence of the electric field existing between successive phases. Measurements of the induced speed had shown the effect of the supply voltage and the frequency, and the phase order in the momentum transfer phenomenon. Gains in velocity, inside the boundary layer, of about 5m/s have been obtained. Spectroscopic measurements allowed to determine the rotational and the vibrational temperature of the plasma which lie in the range of 320 ÷ 440°K and of 3000 ÷ 3900°K respectively. A deviation from thermodynamic equilibrium had been found. The second EHD experiment is realized on a single electrode pair DBD actuator driven by nano-pulses superimposed to a DC or an AC bias. This new supply system separates the plasma formation mechanism from the acceleration action on the fluid, leading to an higher degree of the control of the process. Both the voltage and the frequency of the nano-pulses and the amplitude and the waveform of the bias have been varied during the experiment. Plasma jets and vortex behavior had been observed by means of fast Schlieren imaging. This allowed a deeper understanding of the EHD interaction process. A velocity increase in the boundary layer of about 2m/s had been measured. Thrust measurements have been performed by means of a scales and compared with experimental data reported in the literature. For similar voltage amplitudes thrust larger than those of the literature, had been observed. Surface charge measurements led to realize a modified DBD actuator able to obtain similar performances when compared with that of other experiments. However in this case a DC bias replacing the AC bias had been used. MHD interaction experiments had been carried out in a hypersonic wind tunnel in argon with a flow of Mach 6. Before the MHD experiments a thermal, fluid-dynamic and plasma characterization of the hypersonic argon plasma flow have been done. The electron temperature and the electron number density had been determined by means of emission spectroscopy and microwave absorption measurements. A deviation from thermodynamic equilibrium had been observed. The electron number density showed to be frozen at the stagnation region condition in the expansion through the nozzle. MHD experiments have been performed using two axial symmetric test bodies. Similar magnetic configurations were used. Permanent magnets inserted into the test body allowed to generate inside the plasma azimuthal currents around the conical shape of the body. These Faraday currents are responsible of the MHD body force which acts against the flow. The MHD interaction process has been observed by means of fast imaging, pressure and electrical measurements. Images showed bright rings due to the Faraday currents heating and exciting the plasma particles. Pressure measurements showed increases of the pressure in the regions where the MHD interaction is large. The pressure is 10 to 15% larger than when the MHD interaction process is silent. Finally by means of electrostatic probes mounted flush on the test body lateral surface Hall fields of about 500V/m had been measured. These results have been used for the validation of a numerical MHD code.

COSMIC-LAB: Unexpected Results from High-resolution Spectra of AGB Stars in Globular Clusters

We have used high-resolution spectra, acquired with UVES@ESO-VLT, to determine the chemical abundances of different samples of AGB and RGB stars in 4 Galactic globular clusters, namely 47Tuc, NGC3201, M22 and M62. For almost all the analyzed AGB stars we found a clear discrepancy between the iron abundance measured from neutral lines and that obtained from single ionized lines, while this discrepancy is not obtained for the RGB samples observed in the same clusters and analyzed with the same procedure. Such a behavior exactly corresponds to what expected in the case of Non-Local Thermodynamical Equilibrium (NLTE) in the star atmosphere. These results have a huge impact on the proper determination of GC chemistry. In fact, one of the most intriguing consequences is that, at odds with previous claims, no iron spread is found in NGC3201 and M22 if the iron abundance is obtained from ionized lines only.

Stability, viscous dissipation and local thermal non-equilibrium in fluid saturated porous media

In this thesis, the field of study related to the stability analysis of fluid saturated porous media is investigated. In particular the contribution of the viscous heating to the onset of convective instability in the flow through ducts is analysed. In order to evaluate the contribution of the viscous dissipation, different geometries, different models describing the balance equations and different boundary conditions are used. Moreover, the local thermal non-equilibrium model is used to study the evolution of the temperature differences between the fluid and the solid matrix in a thermal boundary layer problem. On studying the onset of instability, different techniques for eigenvalue problems has been used. Analytical solutions, asymptotic analyses and numerical solutions by means of original and commercial codes are carried out.

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.

Quantum Correlations in Field Theory and Integrable Systems

In this thesis we will investigate some properties of one-dimensional quantum systems. From a theoretical point of view quantum models in one dimension are particularly interesting because they are strongly interacting, since particles cannot avoid each other in their motion, and you we can never ignore collisions. Yet, integrable models often generate new and non-trivial solutions, which could not be found perturbatively. In this dissertation we shall focus on two important aspects of integrable one- dimensional models: Their entanglement properties at equilibrium and their dynamical correlators after a quantum quench. The first part of the thesis will be therefore devoted to the study of the entanglement entropy in one- dimensional integrable systems, with a special focus on the XYZ spin-1/2 chain, which, in addition to being integrable, is also an interacting model. We will derive its Renyi entropies in the thermodynamic limit and its behaviour in different phases and for different values of the mass-gap will be analysed. In the second part of the thesis we will instead study the dynamics of correlators after a quantum quench , which represent a powerful tool to measure how perturbations and signals propagate through a quantum chain. The emphasis will be on the Transverse Field Ising Chain and the O(3) non-linear sigma model, which will be both studied by means of a semi-classical approach. Moreover in the last chapter we will demonstrate a general result about the dynamics of correlation functions of local observables after a quantum quench in integrable systems. In particular we will show that if there are not long-range interactions in the final Hamiltonian, then the dynamics of the model (non equal- time correlations) is described by the same statistical ensemble that describes its statical properties (equal-time correlations).

Thermodynamic and mechanical properties of polymer-solvent systems

This work presents the results of theoretical and experimental characterization of thermodynamic, mechanical and transport properties in polymer solvent systems. The polymer solvent pairs considered ranged to those in which the polymer is rubbery, to those in which the initially glassy polymeric matrix is plasticized by the action of the low molecular weight species. Advanced Equation of State models have been adopted for thermodynamic modeling,along with a rigorous procedure that enables to extend their applicability to the non equilibrium, glassy region. Mass sorption kinetics had been modeled with phenomenological models and with advanced kinetic models.