The maximum entropy principle, pioneered by Jaynes, provides a method for finding the least biased probability distribution for the description of a system or process, given as prior information the expectation values of a set (in general, a small number) of relevant quantities associated with the system. The maximum entropy method was originally advanced by Jaynes as the basis of an information theory inspired foundation for equilibrium statistical mechanics. It was soon realised that the method is very useful to tackle several problems in physics and other fields. In particular it constitutes a powerful tool for obtaining approximate and sometimes exact solutions to several important partial differential equations of theoretical physics. In Chapter 1 a brief review of Shannon’s information measure and Jaynes’ maximum entropy formalism is provided. As an illustration of the maximum entropy principle a brief explanation of how it can be used to derive the standard grand canonical formalism in statistical mechanics is given. The work leading up to this thesis has resulted in the following publications in peer-review research journals: • J.-H. Schönfeldt and A.R. Plastino, Maximum entropy approach to the collisional Vlasov equation: Exact solutions, Physica A, 369 (2006) 408-416, • J.-H. Schönfeldt, N. Jimenez, A.R. Plastino, A. Plastino and M. Casas, Maximum entropy principle and classical evolution equations with source terms, Physica A, 374 (2007) 573-584, • J.-H. Schönfeldt, G.B. Roston, A.R. Plastino and A. Plastino, Maximum entropy principle, evolution equations, and physics education, Rev. Mex. Fis. E, 52 (2)(2006) 151-159. Chapter 2 is based on Schönfeldt and Plastino (2006). Two different ways for obtaining exact maximum entropy solutions for a reduced collisional Vlasov equation endowed with a Fokker-Planck like collision term are investigated. Chapter 3 is based on Schönfeldt et al. (2007). Most applications of the maximum entropy principle to time dependent scenarios involved evolution equations exhibiting the form of a continuity equations and, consequently, preserving normalization in time. In Chapter 3 the maximum entropy principle is applied to evolution equations with source terms and, consequently, not preserving normalization. We explore in detail the structure and main properties of the dynamical equations connecting the time dependent relevant mean values , the associated Lagrange multipliers, the partition function, and the entropy of the maximum entropy scheme. In particular, we compare the H-theorems verified by the maximum entropy approximate solutions with the Htheorems verified by the exact solutions. Chapter 4 is based on Schönfeldt et al. (2006). In chapter 4 it is discussed how the maximum entropy principle can be incorporated into the teaching of aspects of theoretical physics related to, but not restricted to, statistical mechanics. We focus our attention on the study of maximum entropy solutions to evolution equations that exhibit the form of continuity equations (eg. Liouville equation, the diffusion equation the Fokker-Planck equation, etc.). / Dissertation (MSc (Physics))--University of Pretoria, 2008. / Physics / MSc / unrestricted

We show that the nonadditive open systems can be studied in a consistent manner by using a generalized version of S-theorem. This new generalized S-theorem can further be considered as an indication of self-organization in nonadditive open systems as prescribed by Haken. The nonadditive S-theorem is then illustrated by using the modified Van der Pol oscillator. Finally, Tsallis entropy as an equilibrium entropy is studied by using Boltzmann's method of orthodes. This part of dissertation shows that Tsallis ensemble is on equal footing with the microcanonical, canonical and grand canonical ensembles. However, the associated entropy turns out to be Renyi entropy.

The Brandeis dice problem, originally introduced in 1962 by Jaynes as an illustration of the principle of maximum entropy, was solved using the genetic algorithm, and the resulting solution was compared with that obtained analytically. The effect of varying the genetic algorithm parameters was observed, and the optimum values for population size, mutation rate, and mutation interval were determined for this problem. The optimum genetic algorithm program was then compared to a completely random method of search and optimization. Finally, the genetic algorithm approach was extended to several variations of the original problem for which an analytical approach would be impractical.

Predicting the kinetics of a chemical reaction is a challenging task, particularly for systems in states far from equilibrium. This work discusses the use of a relatively new theory known as intrinsic quantum thermodynamics (IQT) and its mathematical framework steepest-entropy-ascent quantum thermodynamics (SEA-QT) to predict the reaction kinetics at atomistic levels of chemically reactive systems in the non-equilibrium realm. IQT has emerged over the last three decades as the theory that not only unifies two of the three theories of physical reality, namely, quantum mechanics (QM), and thermodynamics but as well provides a physical basis for both the entropy and entropy production. The SEA-QT framework is able to describe the evolution in state of a system undergoing a dissipative process based on the principle of steepest-entropy ascent or locally-maximal-entropy generation. The work presented in this dissertation demonstrates for the first time the use of the SEA-QT framework to model the evolution in state of a chemically reactive system as its state relaxes to stable equilibrium. This framework brings a number of benefits to the field of reaction kinetics. Among these is the ability to predict the unique non-equilibrium (kinetic) thermodynamic path which the state of the system follows in relaxing to stable equilibrium. As a consequence, the reaction rate kinetics at every instant of time is known as are the chemical affinities, the reaction coordinates, the direction of reaction, the activation energies, the entropy, the entropy production, etc. All is accomplished without any limiting assumption of stable or pseudo-stable equilibrium. The objective of this work is to implement the SEA-QT framework to describe the chemical reaction process as a dissipative one governed by the laws of quantum mechanics and thermodynamics and to extract thermodynamic properties for states that are far from equilibrium. The F+H2-->HF+H and H+F2-->HF+F reaction mechanisms are used as model problems to implement this framework. / Ph. D.

Driven by a tiny positive cosmological constant, our observable universe asymptotes into a casual patch in de Sitter space in the distant future. Due to the exponential cosmic expansion, a static observer in a de Sitter space is surrounded by a horizon. A semi-classical gravity analysis by Gibbons and Hawking implies that the de Sitter horizon has a temperature and entropy, obeying laws of thermodynamics. Understanding the statistical origin of these thermodynamic quantities requires a precise microscopic model for the de Sitter horizon. With the vision of narrowing the search of such a model with quantum-corrected macroscopic data, we aim to exactly compute the leading quantum (1-loop) corrections to the Gibbons-Hawking entropy, mathematically defined as the logarithm of the effective field theory path integral expanded around the round sphere saddle, i.e. sphere partition functions. This thesis discusses sphere partition functions and their relations to de Sitter (dS) thermodynamics. It consists of three main parts: The first part addresses the subtleties of 1-loop partition functions for totally symmetric tensor fields on 𝑆^{d⁺¹, and generalizes all known results to arbitrary spin 𝑠 ≥ 0 in arbitrary dimensions 𝑑 ≥ 1. Starting from a manifestly covariant and local path integral on the sphere, we carry out a detailed analysis for any massive, shift-symmetric, massless, and partially massless fields. For any field with spin 𝑠 ≥ 1, we find a finite contribution from longitudinal modes; for any massless and partially massless fields, there is a residual group volume factor due to modes generating constant gauge transformations; for any massless and partially massless fields with spin 𝑠 ≥ 2, we derive the phase factor resulted from Wick-rotating negative conformal modes, generalizing the phase factor first obtained by Polchinski for the case of massless spin 2 to arbitrary spins. The second part presents a novel formalism for studying 1-loop quantum de Sitter thermodynamics. We first argue that the Harish-Chandra character for the de Sitter group 𝑆𝑂(1,𝑑+1) provides a manifestly de Sitter-invariant regularization for normal mode density of states in the static patch, without introducing boundary ambiguities as in the traditional brick wall approach. These characters encode quasinormal mode spectrums in the static patch. With these, we write down a simple integral formula for the thermal (quasi)canonical partition function, which straightforwardly generalizes to arbitrary spin representations. Then, we derive a universal formula for 1-loop sphere partition functions in terms of the 𝑆𝑂(1,𝑑+1)$ characters. We find a precise relation between these and the (quasi)canonical partition function mentioned earlier: they are equal for scalars and spinors; for any fields with spin 𝑠 ≥ 1, they differ by ``edge'' degrees of freedom living on the de Sitter horizon. This formalism allows us to efficiently compute the exact 1-loop corrected de Sitter horizon entropy, which as we argue provides non-trivial constraints on microscopic models for the de Sitter horizon. In three dimensions, higher-spin gravity can be alternatively formulated as an sl(𝑛) Chern-Simons theory, which as we show possesses an exponentially large landscape of de Sitter vacua. For each vacuum, we obtain the all-loop exact sphere partition function, given by the absolute value squared of a topological string partition function. Finally, our formalism elegantly proves the relations between generic dS, AdS, and conformal higher-spin partition functions. The last part extends our studies in the previous part to grand (quasi)canonical partition functions on the dS static patch, where we generalize the (quasi)canonical partition functions by allowing non-zero chemical potentials in some of the angular directions. For these, we derive a generalized character integral formula in terms of the full 𝑆𝑂(1,𝑑+1) characters. In three dimensions, we relate them to path integrals on Lens spaces. Similar to its sphere counterpart, the Lens space path integral exhibits a ``bulk-edge'' structure.

Just twenty years have passed since the first papers were published on the application of chemical thermodynamics to the soil-water system (11, 14). Since then, soil physicists have used thermodynamics in an attempt to characterize and l earn more about this intricate system.

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Autophagy is an essential quality control process in which proteins and organelles are degraded. In this work, we first extended our understanding of autophagic degradation in disease by investigating the use of acidic nanoparticles to restore autophagic flux in a neurotoxic model of Parkinson Disease (PD). Normal autophagic degradation follows two key steps. First, material is engulfed to form a double-membraned autophagosome. Next, autophagosomes fuse with an acidic lysosome to degrade the inner membrane contents. Insufficient lysosomal acidity results in autophagic flux arrest, and in PC-12 cells, we characterized the use of polymeric nanoparticles as a tool to restore lysosomal acidity and rescue autophagic flux in PC-12 cells. Specifically, in an MPP+ model of neurotoxicity, we demonstrated that formulations of poly(lactic-co-glycolic acid) nanoparticles (PLGA) improved lysosomal acidity, autophagic flux, and cell health significantly, but is likely limited in efficacy by polymer degradation rate. To improve upon this, we developed a new acidic nanoparticle formulated with a novel polymer backbone (termed acNPs), engineered to degrade within lysosomes and release tetrafluorosuccinic acid, a highly potent acid (pKa ~1.6). On the benchtop, these engineered nanoparticles demonstrated both colloidal instability and acid release within a weakly acidic environment (pH 6.0) similar to a diseased lysosome but not at a neutral pH of 7.4. In cells, acNPs effectively decreased lysosomal pH within disease lysosomes, thereby restoring autophagic flux and mitochondrial activity in PC-12 cells. Encouragingly, we also were able to show efficacy of acNPs in 2D and 3D models of the human midbrain. acNPs readily trafficked within the lysosomes of cells in 2D midbrain cultures and 3D midbrain organoids. Similar to PC-12 cells, when we challenged these cells in a model of neurotoxicity, we observed restoration of viability in human organoids following acNP treatment. Next, we addressed some current challenges regarding the quantification of autophagy within cells. We repurposed measures of economic income inequality to quantify the spatial dispersion of LC3 signal intensity in a starvation model of autophagy, and then compared these measures to other image-based measurements based on their ability to represent LC3-II levels, a robust protein marker of the autophagosome. Our analysis showed these indices outperformed all other generated measurements, including the current standard of autophagy research, LC3 puncta counting. Additionally, we also explored the linear decomposition properties of the generalized entropy index and found it a facile way to evaluate autophagic flux within 3D imaging datasets of multicellular systems. Specifically, we revealed a differential response to nutrient depravation between neurons and astrocytes. Finally, we translated this paradigm to a high throughput cell assay where we demonstrated EC50 and IC50 curves, produced from datasets acquired through both confocal and automated widefield fluorescence microscopy. Our results agree with standard cell assays.

This study explores the applicability of including groundwater recharge and water table variation as additional objective functions in a multi-objective optimization approach to design optimal groundwater monitoring networks. The study was conducted using the Ontario Provincial Groundwater Monitoring Network wells in the Hamilton, Halton, and Credit Valley regions in southern Ontario. The Dual Entropy-Multiobjective Optimization (DEMO) model which has been demonstrated to be sufficiently robust for designing optimum hydrometric networks was used in these analyses. The importance of determining the applicability in using additional design objectives in DEMO, including groundwater recharge and groundwater table seasonal variation, is rooted in the limitations of groundwater data and the time required setting up the models. While recharge allows for the capturing of spatial variability of climate, geomorphology, and geology of the area, the groundwater table series reflect the temporal/seasonal variability. The two set of information are complementary and should provide additional information to the DEMO for optimal network design. Two sources of groundwater recharge data were examined and compared; the recharge provided by the local conservation authorities, calculated using both the Precipitation-Runoff Modeling System (PRMS) and Hydrological Simulation Program--Fortran (HSP--F), and the recharge calculated in situ using only PRMS. The entropy functions are used to identify optimal trade-offs between the maximum possible information content and the minimum shared information between each of the existing and potential monitoring wells. The additional objective functions are used here to quantify the hydrological characteristics of the vadose zone in the aquifer as well as the potential impacts of agricultural, municipal, and industrial uses of groundwater in the area, and thus provide more information for the optimization algorithm to use. Results show that including additional design objectives significantly increases the number of optimal network solutions and provides additional information for potential monitoring well locations. These results suggest that it is worthwhile to include recharge as a design objective if the data is available, and to include groundwater table variation for the design of monitoring wells for shallow groundwater system. / Thesis / Master of Applied Science (MASc)