Classical limit of the Nelson model

Since the development of quantum mechanics it has been natural to analyze the connection between classical and quantum mechanical descriptions of physical systems. In particular one should expect that in some sense when quantum mechanical effects becomes negligible the system will behave like it is dictated by classical mechanics. One famous relation between classical and quantum theory is due to Ehrenfest. This result was later developed and put on firm mathematical foundations by Hepp. He proved that matrix elements of bounded functions of quantum observables between suitable coherents states (that depend on Planck's constant h) converge to classical values evolving according to the expected classical equations when h goes to zero. His results were later generalized by Ginibre and Velo to bosonic systems with infinite degrees of freedom and scattering theory. In this thesis we study the classical limit of Nelson model, that describes non relativistic particles, whose evolution is dictated by Schrödinger equation, interacting with a scalar relativistic field, whose evolution is dictated by Klein-Gordon equation, by means of a Yukawa-type potential. The classical limit is a mean field and weak coupling limit. We proved that the transition amplitude of a creation or annihilation operator, between suitable coherent states, converges in the classical limit to the solution of the system of differential equations that describes the classical evolution of the theory. The quantum evolution operator converges to the evolution operator of fluctuations around the classical solution. Transition amplitudes of normal ordered products of creation and annihilation operators between coherent states converge to suitable products of the classical solutions. Transition amplitudes of normal ordered products of creation and annihilation operators between fixed particle states converge to an average of products of classical solutions, corresponding to different initial conditions.

New perspectives in statistical mechanics and high-dimensional inference

The main purpose of this thesis is to go beyond two usual assumptions that accompany theoretical analysis in spin-glasses and inference: the i.i.d. (independently and identically distributed) hypothesis on the noise elements and the finite rank regime. The first one appears since the early birth of spin-glasses. The second one instead concerns the inference viewpoint. Disordered systems and Bayesian inference have a well-established relation, evidenced by their continuous cross-fertilization. The thesis makes use of techniques coming both from the rigorous mathematical machinery of spin-glasses, such as the interpolation scheme, and from Statistical Physics, such as the replica method. The first chapter contains an introduction to the Sherrington-Kirkpatrick and spiked Wigner models. The first is a mean field spin-glass where the couplings are i.i.d. Gaussian random variables. The second instead amounts to establish the information theoretical limits in the reconstruction of a fixed low rank matrix, the “spike”, blurred by additive Gaussian noise. In chapters 2 and 3 the i.i.d. hypothesis on the noise is broken by assuming a noise with inhomogeneous variance profile. In spin-glasses this leads to multi-species models. The inferential counterpart is called spatial coupling. All the previous models are usually studied in the Bayes-optimal setting, where everything is known about the generating process of the data. In chapter 4 instead we study the spiked Wigner model where the prior on the signal to reconstruct is ignored. In chapter 5 we analyze the statistical limits of a spiked Wigner model where the noise is no longer Gaussian, but drawn from a random matrix ensemble, which makes its elements dependent. The thesis ends with chapter 6, where the challenging problem of high-rank probabilistic matrix factorization is tackled. Here we introduce a new procedure called "decimation" and we show that it is theoretically to perform matrix factorization through it.

Neurogeometry of stereo vision

This work aims to develop a neurogeometric model of stereo vision, based on cortical architectures involved in the problem of 3D perception and neural mechanisms generated by retinal disparities. First, we provide a sub-Riemannian geometry for stereo vision, inspired by the work on the stereo problem by Zucker (2006), and using sub-Riemannian tools introduced by Citti-Sarti (2006) for monocular vision. We present a mathematical interpretation of the neural mechanisms underlying the behavior of binocular cells, that integrate monocular inputs. The natural compatibility between stereo geometry and neurophysiological models shows that these binocular cells are sensitive to position and orientation. Therefore, we model their action in the space R3xS2 equipped with a sub-Riemannian metric. Integral curves of the sub-Riemannian structure model neural connectivity and can be related to the 3D analog of the psychophysical association fields for the 3D process of regular contour formation. Then, we identify 3D perceptual units in the visual scene: they emerge as a consequence of the random cortico-cortical connection of binocular cells. Considering an opportune stochastic version of the integral curves, we generate a family of kernels. These kernels represent the probability of interaction between binocular cells, and they are implemented as facilitation patterns to define the evolution in time of neural population activity at a point. This activity is usually modeled through a mean field equation: steady stable solutions lead to consider the associated eigenvalue problem. We show that three-dimensional perceptual units naturally arise from the discrete version of the eigenvalue problem associated to the integro-differential equation of the population activity.

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.

Governing the water conservation in the Urmia Lake Basin: addressing macro systems’ fit and micro users’ behavior

To change unadapted water governing systems, and water users’ traditional conducts in line with climate change, understanding of systems’ structures and users’ behaviors is necessary. To this aim, comprehensive and pragmatic research was designed and implemented in the Urmia Lake Basin where due to the severe droughts, and human-made influences, especially through the agricultural development, the lake has been shrunken drastically. To analyze the water governance and conservation issues in the basin, an innovative framework was developed based on mathematical physics concepts and pro-environmental behavior theories. Accordingly, in system level (macro/meso), the problem of fit of the early-shaped water governing system associating with the function of “political-security” and “political-economic” factors in the basin was identified through mean-field models. Furthermore, the effect of a “political-environmental” factor, the Urmia Lake Restoration Program (ULRP), on reforming the system structure and hence its fit was assessed. The analysis results revealed that by revising the provincial boundaries (horizontal alternation) for the entity of Kurdistan province to permit that interact with the headquarter of West Azerbaijan province for its water demand-supply initiatives, the system fit can increase. Also, the constitution of the ULRP (vertical arrangement) not only could increase the structural fit of the water governing system to the basin, but also significantly could enhance the system fit through its water-saving policy. Besides, in individual level (micro), the governing factors of water conservation behavior of the major users/farmers were identified through rational and moral socio-psychological models. In rational approach, incorporating PMT and TPB, the SEM results demonstrated that “Perceived Vulnerability”, “Self-Efficacy”, “Response Efficacy”, “Response Cost”, “Subjective Norms” and “Institutional Trust” significantly affect the water-saving intention/behavior. Likewise, NAM based analysis as a moral approach, uncovered the significant effects of “Awareness of Consequences”, “Appraisal of Responsibility”, “Personal Norms” as well as “Place Attachment” and “Emotions” on water-saving intention.