Non-Markovian stochastic processes and their applications: from anomalous diffusion to time series analysis

This work provides a forward step in the study and comprehension of the relationships between stochastic processes and a certain class of integral-partial differential equation, which can be used in order to model anomalous diffusion and transport in statistical physics. In the first part, we brought the reader through the fundamental notions of probability and stochastic processes, stochastic integration and stochastic differential equations as well. In particular, within the study of H-sssi processes, we focused on fractional Brownian motion (fBm) and its discrete-time increment process, the fractional Gaussian noise (fGn), which provide examples of non-Markovian Gaussian processes. The fGn, together with stationary FARIMA processes, is widely used in the modeling and estimation of long-memory, or long-range dependence (LRD). Time series manifesting long-range dependence, are often observed in nature especially in physics, meteorology, climatology, but also in hydrology, geophysics, economy and many others. We deepely studied LRD, giving many real data examples, providing statistical analysis and introducing parametric methods of estimation. Then, we introduced the theory of fractional integrals and derivatives, which indeed turns out to be very appropriate for studying and modeling systems with long-memory properties. After having introduced the basics concepts, we provided many examples and applications. For instance, we investigated the relaxation equation with distributed order time-fractional derivatives, which describes models characterized by a strong memory component and can be used to model relaxation in complex systems, which deviates from the classical exponential Debye pattern. Then, we focused in the study of generalizations of the standard diffusion equation, by passing through the preliminary study of the fractional forward drift equation. Such generalizations have been obtained by using fractional integrals and derivatives of distributed orders. In order to find a connection between the anomalous diffusion described by these equations and the long-range dependence, we introduced and studied the generalized grey Brownian motion (ggBm), which is actually a parametric class of H-sssi processes, which have indeed marginal probability density function evolving in time according to a partial integro-differential equation of fractional type. The ggBm is of course Non-Markovian. All around the work, we have remarked many times that, starting from a master equation of a probability density function f(x,t), it is always possible to define an equivalence class of stochastic processes with the same marginal density function f(x,t). All these processes provide suitable stochastic models for the starting equation. Studying the ggBm, we just focused on a subclass made up of processes with stationary increments. The ggBm has been defined canonically in the so called grey noise space. However, we have been able to provide a characterization notwithstanding the underline probability space. We also pointed out that that the generalized grey Brownian motion is a direct generalization of a Gaussian process and in particular it generalizes Brownain motion and fractional Brownain motion as well. Finally, we introduced and analyzed a more general class of diffusion type equations related to certain non-Markovian stochastic processes. We started from the forward drift equation, which have been made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation has been interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the same memory kernel K(t). We developed several applications and derived the exact solutions. Moreover, we considered different stochastic models for the given equations, providing path simulations.

Crustal heterogeneities and complexities of fault processes

In this work we study the relation between crustal heterogeneities and complexities in fault processes. The first kind of heterogeneity considered involves the concept of asperity. The presence of an asperity in the hypocentral region of the M = 6.5 earthquake of June 17-th, 2000 in the South Iceland Seismic Zone was invoked to explain the change of seismicity pattern before and after the mainshock: in particular, the spatial distribution of foreshock epicentres trends NW while the strike of the main fault is N 7◦ E and aftershocks trend accordingly; the foreshock depths were typically deeper than average aftershock depths. A model is devised which simulates the presence of an asperity in terms of a spherical inclusion, within a softer elastic medium in a transform domain with a deviatoric stress field imposed at remote distances (compressive NE − SW, tensile NW − SE). An isotropic compressive stress component is induced outside the asperity, in the direction of the compressive stress axis, and a tensile component in the direction of the tensile axis; as a consequence, fluid flow is inhibited in the compressive quadrants while it is favoured in tensile quadrants. Within the asperity the isotropic stress vanishes but the deviatoric stress increases substantially, without any significant change in the principal stress directions. Hydrofracture processes in the tensile quadrants and viscoelastic relaxation at depth may contribute to lower the effective rigidity of the medium surrounding the asperity. According to the present model, foreshocks may be interpreted as induced, close to the brittle-ductile transition, by high pressure fluids migrating upwards within the tensile quadrants; this process increases the deviatoric stress within the asperity which eventually fails, becoming the hypocenter of the mainshock, on the optimally oriented fault plane. In the second part of our work we study the complexities induced in fault processes by the layered structure of the crust. In the first model proposed we study the case in which fault bending takes place in a shallow layer. The problem can be addressed in terms of a deep vertical planar crack, interacting with a shallower inclined planar crack. An asymptotic study of the singular behaviour of the dislocation density at the interface reveals that the density distribution has an algebraic singularity at the interface of degree ω between -1 and 0, depending on the dip angle of the upper crack section and on the rigidity contrast between the two media. From the welded boundary condition at the interface between medium 1 and 2, a stress drop discontinuity condition is obtained which can be fulfilled if the stress drop in the upper medium is lower than required for a planar trough-going surface: as a corollary, a vertically dipping strike-slip fault at depth may cross the interface with a sedimentary layer, provided that the shallower section is suitably inclined (fault "refraction"); this results has important implications for our understanding of the complexity of the fault system in the SISZ; in particular, we may understand the observed offset of secondary surface fractures with respect to the strike direction of the seismic fault. The results of this model also suggest that further fractures can develop in the opposite quadrant and so a second model describing fault branching in the upper layer is proposed. As the previous model, this model can be applied only when the stress drop in the shallow layer is lower than the value prescribed for a vertical planar crack surface. Alternative solutions must be considered if the stress drop in the upper layer is higher than in the other layer, which may be the case when anelastic processes relax deviatoric stress in layer 2. In such a case one through-going crack cannot fulfil the welded boundary conditions and unwelding of the interface may take place. We have solved this problem within the theory of fracture mechanics, employing the boundary element method. The fault terminates against the interface in a T-shaped configuration, whose segments interact among each other: the lateral extent of the unwelded surface can be computed in terms of the main fault parameters and the stress field resulting in the shallower layer can be modelled. A wide stripe of high and nearly uniform shear stress develops above the unwelded surface, whose width is controlled by the lateral extension of unwelding. Secondary shear fractures may then open within this stripe, according to the Coulomb failure criterion, and the depth of open fractures opening in mixed mode may be computed and compared with the well studied fault complexities observed in the field. In absence of the T-shaped decollement structure, stress concentration above the seismic fault would be difficult to reconcile with observations, being much higher and narrower.

Trasporto di materia in membrane polimeriche e nanocomposite per la separazione di gas

Membrane-based separation processes are acquiring, in the last years, an increasing importance because of their intrinsic energetic and environmental sustainability: some types of polymeric materials, showing adequate perm-selectivity features, appear rather suitable for these applications, because of their relatively low cost and easy processability. In this work have been studied two different types of polymeric membranes, in view of possible applications to the gas separation processes, i.e. Mixed Matrix Membranes (MMMs) and high free volume glassy polymers. Since the early 90’s, it has been understood that the performances of polymeric materials in the field of gas separations show an upper bound in terms of permeability and selectivity: in particular, an increase of permeability is often accompanied by a decrease of selectivity and vice-versa, while several inorganic materials, like zeolites or silica derivates, can overcome this limitation. As a consequence, it has been developed the idea of dispersing inorganic particles in polymeric matrices, in order to obtain membranes with improved perm-selectivity features. In particular, dispersing fumed silica nanoparticles in high free volume glassy polymers improves in all the cases gases and vapours permeability, while the selectivity may either increase or decrease, depending upon material and gas mixture: that effect is due to the capacity of nanoparticles to disrupt the local chain packing, increasing the dimensions of excess free volume elements trapped in the polymer matrix. In this work different kinds of MMMs were fabricated using amorphous Teflon® AF or PTMSP and fumed silica: in all the cases, a considerable increase of solubility, diffusivity and permeability of gases and vapours (n-alkanes, CO2, methanol) was observed, while the selectivity shows a non-monotonous trend with filler fraction. Moreover, the classical models for composites are not able to capture the increase of transport properties due to the silica addition, so it has been necessary to develop and validate an appropriate thermodynamic model that allows to predict correctly the mass transport features of MMMs. In this work, another material, called poly-trimethylsilyl-norbornene (PTMSN) was examined: it is a new generation high free volume glassy polymer that, like PTMSP, shows unusual high permeability and selectivity levels to the more condensable vapours. These two polymer differ each other because PTMSN shows a more pronounced chemical stability, due to its structure double-bond free. For this polymer, a set of Lattice Fluid parameters was estimated, making possible a comparison between experimental and theoretical solubility isotherms for hydrocarbons and alcoholic vapours: the successfully modelling task, based on application of NELF model, offers a reliable alternative to direct sorption measurement, which is extremely time-consuming due to the relevant relaxation phenomena showed by each sorption step. For this material also dilation experiments were performed, in order to quantify its dimensional stability in presence of large size, swelling vapours.

Fiber-reinforced ceramics for thermostructural applications, produced by polymer impregnation pyrolysis

Several CFCC (Continuous Fiber Composite Ceramics) production processes were tested, concluding that PIP (Polymer Impregnation, or Infiltration, Pyrolysis) and CBC (Chemically Bonded Ceramics) based procedures have interesting potential applications in the construction and transportation fields, thanks to low costs to get potentially useful thermomechanical performances. Among the different processes considered during the Doctorate (from the synthesis of new preceramic polymers, to the PIP production of SiC / SiC composites) the more promising results came from the PIP process with poly-siloxanes on basalt fabrics preforms. Low processing time and costs, together with fairly good thermomechanical properties were demonstrated, even after only one or two PIP steps in nitrogen flow. In alternative, pyrolysis in vacuum was also tested, a procedure still not discussed in literature, but which could originate an interesting reduction of production costs, with only a moderate detrimental effect on the mechanical properties. The resulting CFCC is a basalt / SiCO composite that can be applied for continuous operation up to 600°C, also in oxidant environment, as TG and XRD demonstrated. The failure upon loading is generally pseudo-plastic, being interlaminar delamination the most probable rupture mechanism. . The strength depends on several different factors (microstructure, polymer curing and subsequent ceramic phase evolution, fiber pull-out, fiber strength, fiber percentage) and can only be optimized empirically. In order to be open minded in selecting the best technology, also CBC (Chemically Bonded Ceramics) matrixes were considered during this Doctorate, making some preliminary investigations on fire-resistant phosphate cements. Our results on a commercial product evidenced some interesting thermomechanical capabilities, even after thermal treatments. However the experiments showed also phase change and possible cracking and deformations even on slow drying (at 130°C) and easy rehydration upon exposure to environmental humidity.

Development and Detailed Characterization of Innovative, High-Performance Membrane Materials for CO2 Capture Processes

The scope of this dissertation is to study the transport phenomena of small molecules in polymers and membranes for gas separation applications, with particular attention to energy efficiency and environmental sustainability. This work seeks to contribute to the development of new competitive selective materials through the characterization of novel organic polymers such as CANALs and ROMPs, as well as through the combination of selective materials obtaining mixed matrix membranes (MMMs), to make membrane technologies competitive with the traditional ones. Kinetic and thermodynamic aspects of the transport properties were investigated in ideal and non-ideal scenarios, such as mixed-gas experiments. The information we gathered contributed to the development of the fundamental understanding related to phenomenon like CO2-induced plasticization and physical aging. Among the most significant results, ZIF-8/PPO MMMs provided materials whose permeability and selectivity were higher than those of the pure materials for He/CO2 separation. The CANALs featured norbornyl benzocyclobutene backbone and thereby introduced a third typology of ladder polymers in the gas separation field, expanding the structural diversity of microporous materials. CANALs have a completely hydrocarbon-based and non-polar rigid backbone, which makes them an ideal model system to investigate structure-property correlations. ROMPs were synthesized by means of the ring opening metathesis living polymerization, which allowed the formation of bottlebrush polymers. CF3-ROMP reveled to be ultrapermeable to CO2, with unprecedented plasticization resistance properties. Mixed-gas experiments in glassy polymer showed that solubility-selectivity controls the separation efficiency of materials in multicomponent conditions. Finally, it was determined that plasticization pressure in not an intrinsic property of a material and does not represent a state of the system, but rather comes from the contribution of solubility coefficient and diffusivity coefficient in the framework of the solution-diffusion model.