This research is focused on formulating a new model of entropy production for two-phase flow, including cavitating turbulent flow. In particular, it focuses on the following aspects of the fluid dynamics and the potential contribution of the model to fluid device design. It includes (i) developing a new turbulent entropy model, (ii) a new formula of entropy production rate for two-phase flow including cavitating turbulent flow based on the second law, (iii) applying the technique to study a NACA hydrofoil, and (iv) conducting associated performance analysis of a propeller using post-processing of the CFD results and demonstrating that entropy production of two-phase cavitating flow around the propeller can be correlated to the loss of power output. The first stage consists of formulating the entropy production for laminar channel flow using Gibb’s free energy. This model is validated through the analytically solved Navier-Stokes equations. Subsequently, the single-phase turbulent flow is formulated in a similar manner, but the validations are carried out by comparing the prediction of the model with DNS results. Then, the model of entropy production for two-phase turbulent flow is derived from Gibb’s equation and a version of the Reynolds averaged Navier-Stokes (RANS) equations. The k- ε model is employed to represent the turbulent properties of single phase and two phase flows. A developed inter-phase slip algorithm mixture model is applied to control over coupling of phases. The Rayleigh-Plesset equation is used to model the rate of mass generation of vapour at the inter phase. The standard k-ε turbulence equations are used to describe turbulence in the cavitation flow. The validations of CFD predictions include exploring the force and cavitation characteristics of the NACA 4412 hydrofoil section. The application of this entropy production model in engineering design is presented via the comparisons between CFD results and the experimental data for the velocity distributions behind propeller P5168.

Monte Carlo simulation methods are becoming increasingly important for solving difficult optimization problems. Monte Carlo methods are often used when it is infeasible to determine an exact result via a deterministic algorithm, such as with NP or #P problems. Several recent Monte Carlo techniques employ the idea of importance sampling; examples include the Cross-Entropy method and sequential importance sampling. The Cross-Entropy method is a relatively new Monte Carlo technique that has been successfully applied to a wide range of optimization and estimation problems since introduced by R. Y. Rubinstein in 1997. However, as the problem size increases, the Cross-Entropy method, like many heuristics, can take an exponentially increasing amount of time before it returns a solution. For large problems this can lead to an impractical amount of running time. A main aim of this thesis is to develop the Cross-Entropy method for large-scale parallel computing, allowing the running time of a Cross-Entropy program to be significantly reduced by the use of additional computing resources. The effectiveness of the parallel approach is demonstrated via a number of numerical studies. A second aim is to apply the Cross-Entropy method and sequential importance sampling to biological problems, in particular the multiple change-point problem for DNA sequences. The multiple change-point problem in a general setting is the problem of identifying, given a particular sequence of numbers/characters, a point along that sequence where some property of interest changes abruptly. An example in a biological setting, is identifying points in a DNA sequence where there is a significant change in the proportion of the nucleotides G and C with respect to the nucleotides A and T. We show that both sequential importance sampling and the Cross-Entropy approach yield significant improvements in time and/or accuracy over existing techniques.

In this thesis we study the time evolution of Rényi and entanglement entropies of thermal states in Conformal Field Theory (CFT). These quantities are usually hard to compute but Ryu-Takayanagi (RT) and Hubeny-Rangamani-Takayanagi (HRT) proposals allow us to find the same quantities using calculations in general relativity. We will introduce main concepts of holography, quantum information and conformal field theory that will be used to derive the results of this thesis. In the first part of the thesis, we explicitly compute entanglement entropy of the rotating BTZ black hole by directly applying HRT proposal and finding lengths of spacelike geodesics. Rényi entropy of thermal state perturbed by a local quantum quench is computed by mapping correlators on two glued cylinders to the plane for field theory containing a single free boson and for 2d CFTs in the large c limit. We consider Thermofield Double State (TFD) which is an entangled state in direct product of two 2D CFTs. It is conjectured to be holographically equivalent to the eternal BTZ black hole. TFD state is perturbed by a local quench in one CFT and mutual information between two intervals in two CFTs is computed. We find when mutual information vanishes and interpret this as scrambling time, i.e. time scale required for the system to thermalize. This field theory result is modelled with a massive free falling particle in the BTZ black hole. We have computed the back-reaction of the particle on the metric of BTZ and used RT proposal to find holographic entanglement entropy. Finally, we generalize this calculation to the case of rotating BTZ with inner and outer horizons. It is dual to the CFT with different temperatures for left and right moving modes. We calculate mutual information and scrambling time and find exact agreement between results in the gravity and those in the CFT.

Ce travail de thèse apporte une contribution à l’utilisation de grandeurs thermodynamiques ainsi que géométriques en mécanique. La première partie de ce manuscrit est consacrée à l’étude de la fatigue oligocyclique, et de l’entropie de rupture en fatigue. Des entropies de rupture en fatigue sont estimées expérimentalement par diverses relations et sont comparées aux modèles empiriques utilisés dans la littérature. Il apparait que ces diverses entropies de rupture sont très proches ce qui permet de conclure qu'il existe une entropie de rupture constante liée uniquement au matériau. Pour les modèles empiriques, une extension du modèle de Ramberg-Osgood cyclique prenant en compte la variation temporelle de la contrainte est proposée et une étude sur l'imprécision du modèle de Park et Nelson est réalisée. Puis, une étude des différentes phases durant le test de fatigue est effectuée à travers l’étude de l’endommagement lié à l’entropie accumulée par le matériau. Une extension par l’utilisation du concept d’exergie permet la mise en évidence d’une nouvelle quantité, une exergie associée au travail de déformation plastique faisant intervenir une notion de qualité de la déformation plastique. Dans une deuxième partie, la diffusion de l’entropie d’échelle est étudiée et permet de créer divers comportements dépendants d’échelle. Elle permet d’étudier la log-périodicité d’un fractal déterministe fini (ou préfractal) ou de vérifier la construction de géométries déterministes finies dépendantes d’échelle. Une application de ces modèles dépendants d’échelle est effectuée dans le cadre de la détermination de propriétés mécaniques, pour l’analyse de faciès de rupture et pour la fragmentation. Finalement un lien possible entre comportement mécanique, géométrie et théorie constructale est présenté. / This Phd thesis is a contribution to the use of thermodynamics and geometry in mechanics. The first part of this manuscript is devoted to the study of low cycle fatigue and the notion of fracture fatigue entropy. Fracture fatigue entropies are experimentally estimated by various equations and compared to empirical models used in the litterature. It appears that these diverse fracture fatigue entropies are very close and allows to conclude that a constant fracture fatigue entropy exists only depending on the material. For the empirical models, an extension of the Ramberg-Osgood model is proposed taking into account the temporal variation of the loading, and, a study on the inaccuracy of the Park and Nelson model is realized. Then, a study on the different phases occurring in a fatigue test is done through the study of a damage parameter based on the entropy accumulated by the material. An extension using the concept of exergy allows the highlight of a new quantity, an exergy associated with plastic strain involving a quality factor. In a second part, the diffusion of scale-entropy is studied and permits to create various scale-dependent behaviors. It allows the study of log-periodicity of a finite deterministic fractal (or prefractal), or the verification of finite deterministic scale-dependent geometries. An application of these scale-dependent models is performed within the framework of the determination of mechanical properties, for the analysis of fractured surfaces and for fragmentation. Finally, a possible link between mechanical behavior, geometry and constructal theory is presented.

Alzheimer’s disease (AD) is a neurodegenerative disorder in which the death of brain cells causes memory loss and cognitive decline. As AD progresses, changes in the electrophysiological brain activity take place. Such changes can be recorded by the electroencephalography (EEG) and magnetoencephalography (MEG) techniques. These are the only two neurophysiologic approaches able to directly measure the activity of the brain cortex. Since EEGs and MEGs are considered as the outputs of a nonlinear system (i.e., brain), there has been an interest in nonlinear methods for the analysis of EEGs and MEGs. One of the most powerful nonlinear metrics used to assess the dynamical characteristics of signals is that of entropy. The aim of this thesis is to develop entropy-based approaches for characterization of EEGs and MEGs paying close attention to AD. Recent developments in the field of entropy for the characterization of physiological signals have tried: 1) to improve the stability and reliability of entropy-based results for short and long signals; and 2) to extend the univariate entropy methods to their multivariate cases to be able to reveal the patterns across channels. To enhance the stability of entropy-based values for short univariate signals, refined composite multiscale fuzzy entropy (MFE - RCMFE) is developed. To decrease the running time and increase the stability of the existing multivariate MFE (mvMFE) while keeping its benefits, the refined composite mvMFE (RCmvMFE) with a new fuzzy membership function is developed here as well. In spite of the interesting results obtained by these improvements, fuzzy entropy (FuzEn), RCMFE, and RCmvMFE may still lead to unreliable results for short signals and are not fast enough for real-time applications. To address these shortcomings, dispersion entropy (DispEn) and frequency-based DispEn (FDispEn), which are based on our introduced dispersion patterns and the Shannon’s definition of entropy, are developed. The computational cost of DispEn and FDispEn is O(N) – where N is the signal length –, compared with the O(N2) for popular sample entropy (SampEn) and FuzEn. DispEn and FDispEn also overcome the problem of equal values for embedded vectors and discarding some information with regard to the signal amplitudes encountered in permutation entropy (PerEn). Moreover, unlike PerEn, DispEn and FDispEn are relatively insensitive to noise. As extensions of our developed DispEn, multiscale DispEn (MDE) and multivariate MDE (mvMDE) are introduced to quantify the complexity of univariate and multivariate signals, respectively. MDE and mvMDE have the following advantages over the existing univariate and multivariate multiscale methods: 1) they are noticeably faster; 2) MDE and mvMDE result in smaller coefficient of variations for synthetic and real signals showing more stable profiles; 3) they better distinguish various states of biomedical signals; 4) MDE and mvMDE do not result in undefined values for short time series; and 5) mvMDE, compared with multivariate multiscale SampEn (mvMSE) and mvMFE, needs to store a considerably smaller number of elements. In this Thesis, two restating-state electrophysiological datasets related to AD are analyzed: 1) 148-channel MEGs recorded from 62 subjects (36 AD patients vs. 26 age-matched controls); and 2) 16-channel EEGs recorded from 22 subjects (11 AD patients vs. 11 age-matched controls). The results obtained by MDE and mvMDE suggest that the controls’ signals are more and less complex at respectively short (scales between 1 to 4) and longer (scales between 5 to 12) scale factors than AD patients’ recordings for both the EEG and MEG datasets. The p-values based on Mann-Whitney U-test for AD patients vs. controls show that the MDE and mvMDE, compared with the existing complexity techniques, significantly discriminate the controls from subjects with AD at a larger number of scale factors for both the EEG and MEG datasets. Moreover, the smallest p-values are achieved by MDE (e.g., 0.0010 and 0.0181 for respectively MDE and MFE using EEG dataset) and mvMDE (e.g., 0.0086 and 0.2372 for respectively mvMDE and mvMFE using EEG dataset) for both the EEG and MEG datasets, illustrating the superiority of these developed entropy-based techniques over the state-of-the-art univariate and multivariate entropy approaches. Overall, the introduced FDispEn, DispEn, MDE, and mvMDE methods are expected to be useful for the analysis of physiological signals due to their ability to distinguish different types of time series with a low computation time.

The dynamics of growth and formation of surfaces and interfaces is becoming very important for the understanding of the origin and the behavior of a wide range of natural and industrial dynamical processes. The first part of the paper is focused on the interesting field of the random growth of surfaces and interfaces, which finds application in physics, geology, biology, economics, and engineering among others. In this part it is studied the random growth of surfaces from within the perspective of a single column, namely, the fluctuation of the column height around the mean value, which is depicted as being subordinated to a standard fluctuation-dissipation process with friction g. It is argued that the main properties of Kardar-Parisi-Zhang theory are derived by identifying the distribution of return times to y(0) = 0, which is a truncated inverse power law, with the distribution of subordination times. The agreement of the theoretical prediction with the numerical treatment of the model of ballistic deposition is remarkably good, in spite of the finite size effects affecting this model. The second part of the paper deals with the efficiency of the diffusion entropy analysis (DEA) when applied to the studies of stromatolites. In this case it is shown that this tool can be confidently used for the detection of complexity. The connection between the two studies is established by the use of the DEA itself. In fact, in both analyses, that is, the random growth of interfaces and the study of stromatolites, the method of diffusion entropy is able to detect the real scaling of the system, namely, the scaling of the process is determined by genuinely random events, also called critical events.

This study explores the use of the information theory entropy equation in representations of videos for children. The calculated rates of information in the videos are calibrated to the corresponding perceived rates of information as elicited from the twelve 7 to 10 year old girls who were shown video documents. Entropy measures are calculated for several video elements: set time, set incidence, verbal time, verbal incidence, set constraint, nonverbal dependence, and character appearance. As hypothesized, mechanically calculated entropy measure (CEM) was found to be sufficiently similar to perceived entropy measure (PEM) made by children so that they can be used as useful and predictive elements of representations of children’s videos. The relationships between the CEM and the PEM show that CEM could stand for PEM in order to enrich representations for video documents for this age group. Speculations on transferring the CEM to PEM calibration to different age groups and different document types are made, as well as further implications for the field of information science.

We consider diffeomorphisms of a compact Riemann Surface. A development of Oseledec's Multiplicative Ergodic Theorem is given, along with a development of measure theoretic entropy and dimension. The main result, due to L.S. Young, is that for certain diffeomorphisms of a surface, there is a beautiful relationship between these three concepts; namely that the entropy equals dimension times expansion.

The short-term regulation of blood pressure is influenced by many influences, some being represented by cardiovascular signals and their changes. Because of the complexity of this system, the linear methods for its analysis are not sufficient. Non-linear methods for time series analysis have been devised quite a lot. In the course of this work, the analysis of the relations of several signals was evaluated as the most suitable conditional entropy and the resulting indexes of affinity and directionality. This method was applied to a set of heart-rate signals and systolic and diastolic pressure signals measured on eight adults and nine children. Relationships were analyzed but unfortunately after the statistical test was held the expected information links between the individual signals were not established. The indices were very small - in the hundredths of bits. Finally, optimization of the algorithm of the whole method has been performed and the newly modified function already shows significantly better results, for example strong information binding from the time-series of systolic pressure to a series of diastolic pressures.

The short-term regulation of blood pressure is influenced by many influences, some being represented by cardiovascular signals and their changes. Because of the complexity of this system, the linear methods for its analysis are not sufficient. Non-linear methods for time series analysis have been devised quite a lot. In the course of this work, the analysis of the relations of several signals was evaluated as the most suitable conditional entropy and the resulting indexes of affinity and directionality. This method was applied to a set of heart-rate signals and systolic and diastolic pressure signals measured on eight adults and sixteen children. Relationships were analyzed and after the statistical test was held some information links between the individual signals were not established.