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.

Comparison of affinity chromatography supports for separation of protein

Chromatography is the most widely used technique for high-resolution separation and analysis of proteins. This technique is very useful for the purification of delicate compounds, e.g. pharmaceuticals, because it is usually performed at milder conditions than separation processes typically used by chemical industry. This thesis focuses on affinity chromatography. Chromatographic processes are traditionally performed using columns packed with porous resin. However, these supports have several limitations, including the dependence on intra-particle diffusion, a slow mass transfer mechanism, for the transport of solute molecules to the binding sites within the pores and high pressure drop through the packed bed. These limitations can be overcome by using chromatographic supports like membranes or monoliths. Dye-ligands are considered important alternatives to natural ligands. Several reactive dyes, particularly Cibacron Blue F3GA, are used as affinity ligand for protein purification. Cibacron Blue F3GA is a triazine dye that interacts specifically and reversibly with albumin. The aim of this study is to prepare dye-affinity membranes and monoliths for efficient removal of albumin and to compare the three different affinity supports: membranes and monoliths and a commercial column HiTrapTM Blue HP, produced by GE Healthcare. A comparison among the three supports was performed in terms of binding capacity at saturation (DBC100%) and dynamic binding capacity at 10% breakthrough (DBC10%) using solutions of pure BSA. The results obtained show that the CB-RC membranes and CB-Epoxy monoliths can be compared to commercial support, column HiTrapTM Blue HP, for the separation of albumin. These results encourage a further characterization of the new supports examined.

Retrieval of trace gases vertical profile in the lower atmosphere combining. Differential Optical Absorption Spectroscopy with radiative transfer models

The motivation for the work presented in this thesis is to retrieve profile information for the atmospheric trace constituents nitrogen dioxide (NO2) and ozone (O3) in the lower troposphere from remote sensing measurements. The remote sensing technique used, referred to as Multiple AXis Differential Optical Absorption Spectroscopy (MAX-DOAS), is a recent technique that represents a significant advance on the well-established DOAS, especially for what it concerns the study of tropospheric trace consituents. NO2 is an important trace gas in the lower troposphere due to the fact that it is involved in the production of tropospheric ozone; ozone and nitrogen dioxide are key factors in determining the quality of air with consequences, for example, on human health and the growth of vegetation. To understand the NO2 and ozone chemistry in more detail not only the concentrations at ground but also the acquisition of the vertical distribution is necessary. In fact, the budget of nitrogen oxides and ozone in the atmosphere is determined both by local emissions and non-local chemical and dynamical processes (i.e. diffusion and transport at various scales) that greatly impact on their vertical and temporal distribution: thus a tool to resolve the vertical profile information is really important. Useful measurement techniques for atmospheric trace species should fulfill at least two main requirements. First, they must be sufficiently sensitive to detect the species under consideration at their ambient concentration levels. Second, they must be specific, which means that the results of the measurement of a particular species must be neither positively nor negatively influenced by any other trace species simultaneously present in the probed volume of air. Air monitoring by spectroscopic techniques has proven to be a very useful tool to fulfill these desirable requirements as well as a number of other important properties. During the last decades, many such instruments have been developed which are based on the absorption properties of the constituents in various regions of the electromagnetic spectrum, ranging from the far infrared to the ultraviolet. Among them, Differential Optical Absorption Spectroscopy (DOAS) has played an important role. DOAS is an established remote sensing technique for atmospheric trace gases probing, which identifies and quantifies the trace gases in the atmosphere taking advantage of their molecular absorption structures in the near UV and visible wavelengths of the electromagnetic spectrum (from 0.25 μm to 0.75 μm). Passive DOAS, in particular, can detect the presence of a trace gas in terms of its integrated concentration over the atmospheric path from the sun to the receiver (the so called slant column density). The receiver can be located at ground, as well as on board an aircraft or a satellite platform. Passive DOAS has, therefore, a flexible measurement configuration that allows multiple applications. The ability to properly interpret passive DOAS measurements of atmospheric constituents depends crucially on how well the optical path of light collected by the system is understood. This is because the final product of DOAS is the concentration of a particular species integrated along the path that radiation covers in the atmosphere. This path is not known a priori and can only be evaluated by Radiative Transfer Models (RTMs). These models are used to calculate the so called vertical column density of a given trace gas, which is obtained by dividing the measured slant column density to the so called air mass factor, which is used to quantify the enhancement of the light path length within the absorber layers. In the case of the standard DOAS set-up, in which radiation is collected along the vertical direction (zenith-sky DOAS), calculations of the air mass factor have been made using “simple” single scattering radiative transfer models. This configuration has its highest sensitivity in the stratosphere, in particular during twilight. This is the result of the large enhancement in stratospheric light path at dawn and dusk combined with a relatively short tropospheric path. In order to increase the sensitivity of the instrument towards tropospheric signals, measurements with the telescope pointing the horizon (offaxis DOAS) have to be performed. In this circumstances, the light path in the lower layers can become very long and necessitate the use of radiative transfer models including multiple scattering, the full treatment of atmospheric sphericity and refraction. In this thesis, a recent development in the well-established DOAS technique is described, referred to as Multiple AXis Differential Optical Absorption Spectroscopy (MAX-DOAS). The MAX-DOAS consists in the simultaneous use of several off-axis directions near the horizon: using this configuration, not only the sensitivity to tropospheric trace gases is greatly improved, but vertical profile information can also be retrieved by combining the simultaneous off-axis measurements with sophisticated RTM calculations and inversion techniques. In particular there is a need for a RTM which is capable of dealing with all the processes intervening along the light path, supporting all DOAS geometries used, and treating multiple scattering events with varying phase functions involved. To achieve these multiple goals a statistical approach based on the Monte Carlo technique should be used. A Monte Carlo RTM generates an ensemble of random photon paths between the light source and the detector, and uses these paths to reconstruct a remote sensing measurement. Within the present study, the Monte Carlo radiative transfer model PROMSAR (PROcessing of Multi-Scattered Atmospheric Radiation) has been developed and used to correctly interpret the slant column densities obtained from MAX-DOAS measurements. In order to derive the vertical concentration profile of a trace gas from its slant column measurement, the AMF is only one part in the quantitative retrieval process. One indispensable requirement is a robust approach to invert the measurements and obtain the unknown concentrations, the air mass factors being known. For this purpose, in the present thesis, we have used the Chahine relaxation method. Ground-based Multiple AXis DOAS, combined with appropriate radiative transfer models and inversion techniques, is a promising tool for atmospheric studies in the lower troposphere and boundary layer, including the retrieval of profile information with a good degree of vertical resolution. This thesis has presented an application of this powerful comprehensive tool for the study of a preserved natural Mediterranean area (the Castel Porziano Estate, located 20 km South-West of Rome) where pollution is transported from remote sources. Application of this tool in densely populated or industrial areas is beginning to look particularly fruitful and represents an important subject for future studies.

Mass transport in polymers

The study of mass transport in polymeric membranes has grown in importance due to its potential application in many processes such as separation of gases and vapors, packaging, controlled drug release. The diffusion of a low molecular weight species in a polymer is often accompanied by other phenomena like swelling, reactions, stresses, that have not been investigated in all their aspects yet. Furthermore, novel materials have been developed that include inorganic fillers, reactive functional groups or ions, that make the scenery even more complicated. The present work focused on the experimental study of systems where the diffusion is accompanied by other processes; suitable models were also developed to describe the particular circumstances in order to understand the underlying concepts and be able to design the performances of the material. The effect of solvent-induced deformation in polymeric films during sorption processes was studied since the dilation, especially in constrained membranes, can cause the development of stresses and therefore early failures of the material. The bending beam technique was used to test the effects of the dilation and the stress induced in the polymer by penetrant diffusion. A model based on the laminate theory was developed that accounts for the swelling and is able to predict the stress that raise in the material. The addition of inorganic fillers affects the transport properties of polymeric films. Mixed matrix membranes based on fluorinated, high free volume matrices show attractive performances for separation purposes but there is a need for deeper investigation of the selectivity properties towards gases and vapors. A new procedure based on the NELF model was tested on the experimental data; it allows to predict solubility of every penetrant on the basis of data for one vapor. The method has proved to be useful also for the determination of the diffusion coefficient and for an estimation of the permeability in the composite materials. Oxygen scavenging systems can overcome lack of barrier properties in common polymers that forbids their application in sensitive applications as food packaging. The final goal of obtaining a membrane almost impermeable to oxygen leads to experimental times out of reach. Hence, a simple model was developed in order to describe the transport of oxygen in a membrane with also reactive groups and analyze the experimental data collected on SBS copolymers that show attractive scavenging capacity. Furthermore, a model for predicting the oxygen barrier behavior of a film formed as a blend of OSP in a common packaging material was built, considering particles capable of reactions with oxygen embedded in a non-reactive matrix. Perfluorosulphonic acid ionomers (PFSI) are capturing attention due to a high thermal and chemical resistance coupled with very peculiar transport properties, that make them appropriate to be used in fuel cells. The possible effect of different formation procedure was studied together with the swelling due to water sorption since both water uptake and dilation can dramatically affect the fuel cells performances. The water diffusion and sorption was studied with a FTIR-ATR spectrometer that can give deeper information on the bonds between water molecules and the sulphonic hydrophilic groups and, therefore, on the microstructure of the hydrated ionomer.