Estudamos alguns aspectos da Mecânica Estatística de sistemas com número finito de partículas. Com exceção do capítulo 1 este número de partículas é da ordem N = 108 ?, onde ? é a dimensão do sistema. Não calculamos apenas funções termodinâmicas, mas procuramos determinar conjuntos de configurações cuja probabilidade de ocorrência no Ensemble Canônico a dada temperatura seja próxima de 1. A partir daí, algumas funções termodinâmicas como calor específico e pressão são calculadas. Desenvolvemos técnicas para determinar estes conjuntos de configurações de grande probabilidade e tratamos em maior detalhe os seguintes modelos: a) gás de rede com potencial atrativo de primeiros vizinhos em 1 dimensão e baixa densidade (N3 << V, N = número de partículas, V = número de sítios na rede). b) um modelo semelhante ao \"Modelo da Gota\" de Fisher, em que partículas numa caixa podem se unir em aglomerados que se movem livremente na caixa sem graus internos de liberdade. Consideramos o modelo em 1 dimensão. c) O mesmo modelo (b) modificando o critério de distinguibilidade. d) O modelo (b) com graus de liberdade internos aos aglomerados. e) O modelo (b) num campo gravitacional uniforme. f) O modelo ferromagnético de Ising em qualquer dimensão com campo magnético externo e condições periódicas ou livres de contorno e em 1 dimensão sem campo externo e com um spin fixo e um extremo de rede. Estudamos ainda o modelo (b) no Ensemble Grã-Canônico e comparamos os resultados neste ensemble fixando o número médio de partículas com os resultados no Ensemble Canônico. / We study some aspects of the Statistical Machanics of systems with finite number of particles. With exception of chapter 1 this number of particles is of the order N = 108 ?, where ? is the dimensions f the system. We don\'t restrict ourselves to the calculation of thermo dynamical functions, instead we look for sets of configurations whose probability of occurrence in the Canonical Ensemble at given temperature is near to 1. This permits us to calculate some thermo dynamical functions like the specific heat and the pressure. Techniques to determine these sets of configurations of great probability are developed and we treat in great detail of the following models: a) A lattice gas with attractive nearest neighbor potential in 1 dimension and low density (N3 << V, N = number of particles, V = number of sites). b) A model similar to Fisher\'s \"Droplet Model\" in which particles inside a box can form clusters which move freely in the box without internal degrees of freedom. We consider the model in 1 dimension. c) The same model (b) with the distinguibility criterium modified. d) The model (b) with internal to the clusters degrees of freedom. e) The model (b) in a uniform gravitational field. f) The ferromagnetic Ising model in any dimension with external field and periodic or free boundary conditions, and in 1 dimension without external field and with one spin fixed in the value +1 in one extreme of the lattice. We study also the model (b) in the Grand-Canonical Ensemble and compare the results in this ensemble fixed the mean number of particles with the results in the Canonical Ensemble.

Apresenta-se uma teoria microscópica para as fases do hélio líquido baseada na existência de sólitons planares no fluido. O trabalho adota a interpretação de London para o gás de Bose-Einstein. E segue principalmente, o trabalho pioneiro de Ventura, que é também uma extensão, para condensados não uniformes, da Teoria de Bogoliubov para a superfluidez. Esta abordagem tem, no sóliton, sua principal faceta e na nuvem térmica e no segundo campo condensado suas principais novidades. A nuvem térmica é constituída por excitações térmicas ligadas ao sóliton. Assumimos, num primeiro momento, e como hipótese central, que a densidade da nuvem térmica é proporcional ao buraco do sóliton. Duas dinâmicas, então, são desenvolvidas: o campo médio, que dá origem ao fluído normal e a dinâmica da nuvem térmica. Essas duas dinâmicas são compatibilizadas e, por autoconsistência, desenvolve-se a Mecânica Estatística dos sólitons, no líquido. Os resultados obtidos estão em bom acordo com os resultados experimentais conhecidos. Há um calor específico com uma divergência tipo em T = 2.1 K e um gap térmico efetivo na região T = 6 - 9 K, que concorda com o gap obtido experimentalmente por espalhamento de nêutrons. A teoria é, rigorosamente, microscópica e foi aperfeiçoada, num segundo momento, com a descoberta do segundo campo condensado. Usando uma lagrangiana efetiva, construída a partir da primeira etapa dos cálculos, aperfeiçoou-se a teoria e a dinâmica sóliton/nuvem térmica pôde ser reproduzida de maneira mais apurada. O fenômeno central dessa abordagem é a condensação de um segundo campo clássico no menor estado de energia. As duas abordagens são equivalentes, mas a segunda é a mais correta e conduz, essencialmente, aos mesmos resultados. / A microscopic theory for the liquid phases of helium-4 is presented, based in the existence of planar solitons in the fluid. The work follows the Londons interpretation of the Bose-Einstein gas. Mainly, it follows the pioneer work of Ventura on superfluidity, which is also an extension for the non-uniform condensates, of the Bogoliubov theory in the subject. This approach has in the soliton its main feature and in the thermal clouds its main novelty. The thermal cloud is constituted by thermal excitations bounded to the soliton. We assume, at the first moment, and as central hypothesis, that the density of the thermal cloud is proportional to the soliton hole (which 1S related to the matter vacancy) in the soliton frame. Two dynamics are developed: the mean-field, that gives origin to the normal fluid and the dynamics of the thermal cloud excitations. These two dynamics are compatibilized and by self-consistency we build the Statistical Mechanics of the solitons in the liquid. The results so obtained are in good agreement with the know results of experiments. There 1S a specific heat with a -divergence at T=2.1 K, and a thermal gap in the range T=6-9 K, which agrees with the neutron sacattering gap. The theory is microscopic in all respects and is improved with the introduction of the second condensed field. Using an effective Lagrangian, we have perfected the theory and have reproduced the soliton/thermal cloud dynamics in a more accurate fashion. The central phenomena in this approach is the condensation of the second classical field in the lowest energy state. The two approaches are equivalent, but the second one is the more correct and gives, essentially, the same results.

Com base em exemplos, nos fundamentos da Mecânica estatística e na teoria ergódiga, é dada uma definição de atrator como uma medida invariante. Vários resultados que corroboram esta definição são demostrados. Caos é relacionado à presença de um atrator com entropia métrica maior que zero. O papel dos expoentes de Lyapunov é analisado e é provado que um atrator caótica possui expoentes de Lyapunov positivos em quase todo ponto, e também que, se um atrator possui todos expoentes de Lyapunov estritamente negativos num conjunto de medida atratora maior que zero, então seu suporte é uma órbita periódica assintoticamente estável. / Here, a definition of an attractor as an invariant measure is given based on Ergodic Theory, foundations of Statistical Mechanics and some examples. Chaos is related to the presence of an attractor with metric entropy grater zero. It is proved that a chaotic attractor has positive Lyapunov exponents almost everywhere, and that, if an attractor has every Lyapunov exponents less than zero in a set of nonzero measure then the support set of the attractor is an asymptotic stable periodic orbit.

D.Phil. (Statistics) / We often wish to determine whether observations occurring in a natural time sequence are from the same distribution or whether changes in distribution have taken place at certain points in time. These time points are called change points. We study tests of the null hypothesis of no change versus the alternative hypothesis of changes in parameter at unknown change points, as well as point- and interval estimation of the change points. For univariate observations we distinguish between two cases. In the one case we consider observations having known, but unequal, variances. In the second case each observation has a variance which is a function of the unknown mean. In the first case we develop graphical procedures which can be used for the detection, as well as for point- and interval estimation, of the change points. The method which we develop in the second case can be used for observations from any distribution, provided a suitable variance stabilizing transformation exists. Binomially distributed observations can be accommodated in both of these settings...

This work concerns itself with the exact study of the dynamical properties of two model systems. After a brief summary of theory and concepts previous work is discussed, and this provides the motivation for the formulation of the first model. This quantum mechanical lattice model and some of its equilibrium properties are described in Chapter II. The dynamical problem to be studied is formulated in Chapter III; this is essentially the study of the time evolution (generated by the Hamiltonian) of a finite system at temperature T₁ coupled to an infinite copy of itself at a temperature T₂ and acting as a heat bath for the system. The problem is solved for the special case, when the coupling as scaled by the parameter &gamma;, takes the value &gamma; = 1. The general case for arbitrary &gamma; values is treated in Chapter IV. It is shown that the system approaches the equilibrium state of the heat bath in a non-exponential manner, provided the spectrum of the Hamiltonian is continuous and does not have a discrete part. This result is in complete accord with the findings of other work summarised in Chapter I. The mixing properties of the model and behaviour of the relaxation rate in the weak coupling limit are studied in Chapter V. The model is shown to fail to behave as a calorimeter and in view of this result the relevance of the concept of mixing to irreversible behaviour is discussed. The main conclusions and results for the model are summarised at the end of Chapter V. The second model discussed, was first introduced by R.J.&nbsp;Glauber to study the dynamics of the Ising chain. The main feature here is that the time evolution is defined through a Master equation, and the associated stochastic operator. It is shown in Chapter VI that exploiting fully the free fermion character of the stochastic operator for the Glauber model, it is possible to provide a simple method to the study of the dynamics of the Ising Chain.

The recent advancement of whole-transcriptome gene expression quantification technology - particularly at the single-cell level - has created a wealth of biological data. An increasingly popular unsupervised analysis is to find one dimensional manifolds or trajectories through such data that track the development of some biological process. Such methods may be necessary due to the lack of explicit time series measurements or due to asynchronicity of the biological process at a given time. This thesis aims to recast trajectory inference from high-dimensional "omics" data as a statistical latent variable problem. We begin by examining sources of uncertainty in current approaches and examine the consequences of propagating such uncertainty to downstream analyses. We also introduce a model of switch-like differentiation along trajectories. Next, we consider inferring such trajectories through parametric nonlinear factor analysis models and demonstrate that incorporating information about gene behaviour as informative Bayesian priors improves inference. We then consider the case of bifurcations in data and demonstrate the extent to which they may be modelled using a hierarchical mixture of factor analysers. Finally, we propose a novel type of latent variable model that performs inference of such trajectories in the presence of heterogeneous genetic and environmental backgrounds. We apply this to both single-cell and population-level cancer datasets and propose a nonparametric extension similar to Gaussian Process Latent Variable Models.

The infinite order perturbation theory of Prigogine and coworkers is used, with some modifications, to discuss the theories of classical irreversible processes due to Bogoliubov, Sandri & Frieman, and Mazur & Biel. The latter authors use the BBKGY hierarchy of equations as a starting point. Accordingly, to discuss these theories the infinite order perturbation theory is written out in such a way that it relates easily to the BBKGY hierarchy. The nature of the assumptions involved in the theories of Bogoliubov and Sandri & Frieman become particularly clear when compared with the infinite order perturbation expansion. The relation of the theory of Mazur & Biel with the cluster expansion of Green is also elucidated. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

The stochastic-continuum model developed by Schwartz and Smith [1988] is a new approach to the traditional continuum methods for solute transport in fractured media. Instead of trying to determine dispersion coefficients and an effective porosity for the hydraulic system, statistics on particle motion (direction, velocity and fracture length) collected from a discretely modeled sub-domain network are used to recreate particle motion in a full-domain continuum model. The discrete sub-domain must be large enough that representative statistics can be collected, yet small enough to be modeled with available resources. Statistics are collected in the discrete sub-domain model as the solute, represented by discrete particles, is moved through the network of fractures. The domain of interest, which is typically too large to be modeled discretely is represented by a continuum distribution of the hydraulic head. A particle tracking method is used to move the solute through the continuum model, sampling from the distributions for direction, velocity and fracture length. This thesis documents extensions and further testing of the stochastic-continuum two-dimensional model and initial work on a three-dimensional stochastic-continuum model. Testing of the model was done by comparing the mass distribution from the stochastic-continuum model to the mass distribution from the same domain modeled discretely. Analysis of the velocity statistics collected in the two-dimensional model suggested changes in the form of the fitted velocity distribution from a gaussian distribution to a gamma distribution, and the addition of a velocity correlation function. By adding these changes to the statistics collected, an improvement in the match of the spatial mass distribution moments between the stochastic-continuum and discrete models was effected. This extended two-dimensional model is then tested under a wide range of network conditions. The differences in the first spatial moments of the discrete and stochastic-continuum models were less than 10%, while the differences in the second spatial moments ranged from 6% to 30%. Initial results from the three-dimensional stochastic-continuum model showed that similar statistics to those used in the two-dimensional stochastic-continuum model can be used to recreate the nature of three-dimensional discrete particle motion. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

Two different aspects of the problem of selecting measurements for statistical pattern recognition are investigated. First, the evaluation of features for multiclass recognition problems by using measures of probabilistic dependence is examined. Secondly, the problem of evaluation and selection of features for a general tree type classifier is investigated. Measures of probabilistic dependence are derived from pairwise distance measures such as Bhattacharyya distance, divergence, Matusita's distance, and discrimination information. The properties for the dependence measures are developed in the context of feature class dependency. Inequalities relating the measures are derived. Also upper and lower bounds on error probability are derived for the different measures. Comparisons of the bounds are made. Feature ordering experiments are performed to compare the measures to error probability and to each other. A fairly general tree type sequential classifier is examined. An algorithm which uses distance measures for clustering probability distributions and which uses dependence and distance measures for ordering features is derived for constructing the decision tree. The concept of confidence in a decision in conjunction with backtracking is introduced in order to make decisions at any node of the tree tentative and reversible. Also, the idea of re-introducing classes at any stage is discussed. Experiments are performed to determine the storage and processing requirements of the classifier, to determine effects of various parameters on performance, and to determine the usefulness of procedures for backtracking and reintroducing of classes. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

This project was designed to study the patterns of variability in fish populations. My hypothesis is that specific population patterns should be related to evolutionary concepts (phylogenetic patterns} , zoogeographic considerations (faunal patterns), and their vertical distributions. These patterns should be detected by comparing certain population parameters [growth parameters (K, LINF), the natural mortality coefficient (M) size at first maturity (LM), age at first maturity (TM), size at age 1 (L1) , the weight-length exponential coefficient (b) , and life span (T95)] which are intrinsic biological features of the population. Comparative methods were used to analyze data from published fish population studies by comparing fish population parameters, individually, in pairs (ratio or linear regression), or grouped together (discriminant analysis or Cooley and Lohnes' classification method), in order to find the similarities or differences among different categories, and then to group these into patterns. Published data provided 682 parameter records from 43 families (171 species) of fishes. My findings suggested that more satisfactory results would be obtained from a greater volume of data. Therefore, all the analyses were based mainly on 15 families with large sample sizes (Bothidae, Clupeidae, Cyprinidae, Engraulidae, Gadidae, Hiodontidae, Osmeridae, Percidae, Pleuronectidae, Salmonidae, Sciaenidae, Scombridae, Scorpaenidae, Sparidae, and Sgualidae). Sample sizes, mean values, standard errors, and coefficients of variation for population parameters and relative characters of the 15 families of fishes are listed in the summary table. These data would enable the extrapolation of results based on many areas for management of other fish stocks where data are lacking. In the majority of families significant linear regression relationships were found between 1/K--LINF, between LM--LINF, and between M--K. This means that fish having a greater asymptotic length (LINF) also have a larger size at first maturity (LM), a lower natural mortality coefficient (M), and a lower rate (K) at which the asymptotic length is reached. Using the F-test and the appropriate t-test as a basis for comparison of variances and means of individual parameters, it is evident that in most cases there are significant differences between families. This confirms one of my hypothesis; namely that differences between families, as shown by population parameters, exist from phylogenetic considerations. By comparing the four characters (K, LINF, LM, and LH/LINF) the fish families can be divided into the following groups: A) Shoaling pelagic fishes - Engraulidae, Clupeidae, and Osmeridae. These families have the highest K values (1.6 for Engraulidae, over 0.4 for the others), the smallest LINF, LM, and a very high LM/LINF ratio (over 0.7). B) Large pelagic fishes - Scombridae. This family has a moderately high K value (around 0.35) and the largest LINF. C) Demersal fishes - Gadidae, Pleuronectidae, Scorpaenidae, Sparidae etc. These families have low K values (less than 0.25), intermediate LINF size, and lower LM/LINF ratios (less than 0.6). D) Freshwater fish - Cyprinidae. This family has K and LINF values which are similar to those of the demersal fishes, but has a smaller LM length and, especially, the lowest LK/LINF (0.4) and TH/T95 (0.2) ratios. Stepwise discriminant analysis based on 7 variables in the 15 families showed that over 90% of the 620 cases considered independently could be correctly classified into the right families. Cooley and Lohnes' classification method was also utilized among species within 5 major families (Clupeidae, Cyprinidae, Gadidae, Pleuronectidae, and Scombridae). Correct classification ranged from 5 8.6% (Pleuronectidae) to 87.6% (Cyprinidae). These results further confirmed the existence of population patterns by examination of population parameters. Cluster analysis based on 7 population parameters displayed the closeness among the 15 families. Dendrograph relationships brought out the ecological, rather than the systematic, affinities between families. / Science, Faculty of / Zoology, Department of / Graduate