The methods of statistical physics are increasingly being employed in a range of interdisciplinary areas. In particular, aspects of complex biological processes have been elucidated by bringing the problems down to the level of simple interactions studied in a statistical sense. In nonequilibrium statistical physics, a one dimensional lattice model known as the totally asymmetric simple exclusion processes (TASEP) has become prominent as a tool for modelling various cellular transport processes. Indeed the context in which the TASEP was first introduced (MacDonald et. al., 1968) was to model ribosome motion along mRNA during protein synthesis. In this work I study a variation of the TASEP in which particles hop along a one dimensional lattice which extends as they reach the end. We introduce this model to describe the unique growth dynamics of filamentous fungi, whereby a narrow fungal filament extends purely from its tip region while being supplied with growth materials from behind the tip. We find that the steady state behaviour of our model reflects that of the TASEP, however there is an additional phase where a dynamic shock is present in the system. I show through Monte Carlo simulation and theoretical analysis that the qualitative behaviour of this model can be predicted with a simple mean-field approximation, while the details of the phase behaviour are accurate only in a refined approximation which takes into account some correlations. I also discuss a further refined mean-field approximation and give a heuristic argument for our results. Next I present an extension of the model which allows the particles to interact with a second lattice, on which they diffuse in either direction. A first order meanfield continuum approximation suggests that the steady states of this system will exhibit some novel behaviour. Through Monte Carlo simulation I discuss the qualitative changes that arise due to the on-off dynamics. Finally I study a model for a second biological phenomenon: the length fluctuations of microtubules. The model describes stochastic polymerisation events at the tip of a microtubule. Using a mean-field theory, we find a transition between regimes where the microtubule grows on average, and where the length remains finite. For low rates of polymerisation and depolymerisation, the transition is in good agreement with Monte Carlo simulation.

No description available.

One of the most central concepts in condensed matter physics is the electronic band structure. Although band theory was established more than 80 years ago, recent developments have led to new insights that are formulated in the framework of topological band theory. In this thesis a subset of topological band theory is presented, with particular focus on topological supercon- ductors and accompanying Majorana fermions. While simple models are used to introduce basic concepts, a physically more realistic model is also studied intensely in the papers. Through self- consistent tight-binding calculations it is confirmed that Majorana fermions appear in vortex cores and at wire end points when the superconductor is in the topologically non-trivial phase. Many other properties such as the topological invariant, experimental signatures in the local density of states and spectral function, unconventional and odd-frequency pairing, the precense of spin-polarized currents and spin-polarization of the Majorana fermions, and a local π-phase shift in the order parameter at magnetic impurities are also investigated.

PhD / When modeling a system as complex as the brain, considerable simplifications are inevitable. The nature of these simplifications depends on the available experimental evidence, and the desired form of model predictions. A focus on the former often inspires models of networks of individual neurons, since properties of single cells are more easily measured than those of entire populations. However, if the goal is to describe the processes responsible for the electroencephalogram (EEG), such models can become unmanageable due to the large numbers of neurons involved. Mean-field models in which assemblies of neurons are represented by their average properties allow activity underlying the EEG to be captured in a tractable manner. The starting point of the results presented here is a recent physiologically-based mean-field model of the corticothalamic system, which includes populations of excitatory and inhibitory cortical neurons, and an excitatory population representing the thalamic relay nuclei, reciprocally connected with the cortex and the inhibitory thalamic reticular nucleus. The average firing rates of these populations depend nonlinearly on their membrane potentials, which are determined by afferent inputs after axonal propagation and dendritic and synaptic delays. It has been found that neuronal activity spreads in an approximately wavelike fashion across the cortex, which is modeled as a two-dimensional surface. On the basis of the literature, the EEG signal is assumed to be roughly proportional to the activity of cortical excitatory neurons, allowing physiological parameters to be extracted by inverse modeling of empirical EEG spectra. One objective of the present work is to characterize the statistical distributions of fitted model parameters in the healthy population. Variability of model parameters within and between individuals is assessed over time scales of minutes to more than a year, and compared with the variability of classical quantitative EEG (qEEG) parameters. These parameters are generally not normally distributed, and transformations toward the normal distribution are often used to facilitate statistical analysis. However, no single optimal transformation exists to render data distributions approximately normal. A uniformly applicable solution that not only yields data following the normal distribution as closely as possible, but also increases test-retest reliability, is described in Chapter 2. Specialized versions of this transformation have been known for some time in the statistical literature, but it has not previously found its way to the empirical sciences. Chapter 3 contains the study of intra-individual and inter-individual variability in model parameters, also providing a comparison of test-retest reliability with that of commonly used EEG spectral measures such as band powers and the frequency of the alpha peak. It is found that the combined model parameters provide a reliable characterization of an individual's EEG spectrum, where some parameters are more informative than others. Classical quantitative EEG measures are found to be somewhat more reproducible than model parameters. However, the latter have the advantage of providing direct connections with the underlying physiology. In addition, model parameters are complementary to classical measures in that they capture more information about spectral structure. Another conclusion from this work was that a few minutes of alert eyes-closed EEG already contain most of the individual variability likely to occur in this state on the scale of years. In Chapter 4, age trends in model parameters are investigated for a large sample of healthy subjects aged 6-86 years. Sex differences in parameter distributions and trends are considered in three age ranges, and related to the relevant literature. We also look at changes in inter-individual variance across age, and find that subjects are in many respects maximally different around adolescence. This study forms the basis for prospective comparisons with age trends in evoked response potentials (ERPs) and alpha peak morphology, besides providing a standard for the assessment of clinical data. It is the first study to report physiologically-based parameters for such a large sample of EEG data. The second main thrust of this work is toward incorporating the thalamocortical system and the basal ganglia in a unified framework. The basal ganglia are a group of gray matter structures reciprocally connected with the thalamus and cortex, both significantly influencing, and influenced by, their activity. Abnormalities in the basal ganglia are associated with various disorders, including schizophrenia, Huntington's disease, and Parkinson's disease. A model of the basal ganglia-thalamocortical system is presented in Chapter 5, and used to investigate changes in average firing rates often measured in parkinsonian patients and animal models of Parkinson's disease. Modeling results support the hypothesis that two pathways through the basal ganglia (the so-called direct and indirect pathways) are differentially affected by the dopamine depletion that is the hallmark of Parkinson's disease. However, alterations in other components of the system are also suggested by matching model predictions to experimental data. The dynamics of the model are explored in detail in Chapter 6. Electrophysiological aspects of Parkinson's disease include frequency reduction of the alpha peak, increased relative power at lower frequencies, and abnormal synchronized fluctuations in firing rates. It is shown that the same parameter variations that reproduce realistic changes in mean firing rates can also account for EEG frequency reduction by increasing the strength of the indirect pathway, which exerts an inhibitory effect on the cortex. Furthermore, even more strongly connected subcircuits in the indirect pathway can sustain limit cycle oscillations around 5 Hz, in accord with oscillations at this frequency often observed in tremulous patients. Additionally, oscillations around 20 Hz that are normally present in corticothalamic circuits can spread to the basal ganglia when both corticothalamic and indirect circuits have large gains. The model also accounts for changes in the responsiveness of the components of the basal ganglia-thalamocortical system, and increased synchronization upon dopamine depletion, which plausibly reflect the loss of specificity of neuronal signaling pathways in the parkinsonian basal ganglia. Thus, a parsimonious explanation is provided for many electrophysiological correlates of Parkinson's disease using a single set of parameter changes with respect to the healthy state. Overall, we conclude that mean-field models of brain electrophysiology possess a versatility that allows them to be usefully applied in a variety of scenarios. Such models allow information about underlying physiology to be extracted from the experimental EEG, complementing traditional measures that may be more statistically robust but do not provide a direct link with physiology. Furthermore, there is ample opportunity for future developments, extending the basic model to encompass different neuronal systems, connections, and mechanisms. The basal ganglia are an important addition, not only leading to unified explanations for many hitherto disparate phenomena, but also contributing to the validation of this form of modeling.

Thesis (Ph. D.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

In recent years many researchers have investigated the use of Markov random fields (MRFs) for computer vision. The computational complexity of the implementation has been a drawback of MRFs. In this paper we derive deterministic approximations to MRFs models. All the theoretical results are obtained in the framework of the mean field theory from statistical mechanics. Because we use MRFs models the mean field equations lead to parallel and iterative algorithms. One of the considered models for image reconstruction is shown to give in a natural way the graduate non-convexity algorithm proposed by Blake and Zisserman.

In this dissertation, we consider a complex biological system known as cortical microtubule (CMT) system, where stochastic dynamics of the components (i.e., the CMTs) are defined in both space and time. CMTs have an inherent spatial dimension of their own, as their length changes over time in addition to their location. As a result of their dynamics in a confined space, they run into and interact with each other according to simple stochastic rules. Over time, CMTs acquire an ordered structure that is achieved without any centralized control beginning with a completely disorganized system. It is also observed that this organization might be distorted, when parameters of dynamicity and interactions change due to genetic mutation or environmental conditions. The main question of interest is to explore the characteristics of this system and the drivers of its self-organization, which is not feasible relying solely on biological experiments. For this, we replicate the system dynamics and interactions using computer simulations. As the simulations successfully mimic the organization seen in plant cells, we conduct an extensive analysis to discover the effects of dynamics and interactions on system characteristics by experimenting with different input parameters. To compare simulation results, we characterize system properties and quantify organization level using metrics based on entropy, average length and number of CMTs in the system. Based on our findings and conjectures from simulations, we develop analytical models for more generalized conclusions and efficient computation of system metrics. As a fist step, we formulate a mean-field model, which we use to derive sufficient conditions for organization to occur in terms of input parameters. Next, considering the parameter ranges that satisfy these conditions, we develop predictive methodologies for estimation of expected average length and number of CMTs over time, using a fluid model, transient analysis, and approximation algorithms tailored to our problem. Overall, we build a comprehensive framework for analysis and control of microtubule organization in plant cells using a wide range of models and methodologies in conjunction. This research also has broader impacts related to the fields of bio-energy, healthcare, and nanotechnology; in addition to its methodological contribution to stochastic modeling of systems with high-level spatial and temporal complexity.

No description available.

Several magnetic materials, in the first approximation, can be described by idealised theoretical models, such as classical Ising or Heisenberg spin systems, and, to some extent, such models are able to qualitatively expose many experimentally observed phenomena. But often, to account for complex behavior of magnetic matter, such models have to be refined by including more terms in Hamiltonian. The compound LiHo_xY_{1-x}F_4, by increasing concentration of nonmagnetic yttrium can be tuned from a diluted ferromagnet to a spin glass. LiHoF_4 is a good realisation of the transverse field Ising model, the simplest model exhibiting a quantum phase transition. In the pure case the magnetic behaviour of this material is well described by mean-field theory. It was believed that when diluted, LiHo_xY_{1-x}F_4 would also manifest itself as a diluted transverse field Ising model which continue to be well described by mean-field theory, and, at sufficient dilution, at zero temperature, exhibit a quantum spin-glass transition. The experimental data did not support such a scenario, and it was pointed out that, to explain physics of LiHo_xY_{1-x}F_4 in transverse magnetic field, the effect of a transverse-field-generated longitudinal random field has to be considered. We explore this idea further in local mean-field studies in which all three parameters: temperature, transverse field and concentration can be consistently surveyed, and where the transverse-field-generated longitudinal random field is explicitly present in the effective spin-1/2 Hamiltonian. We suggest other materials that are possible candidates for studying quantum criticality in the transverse field Ising model, and in the diluted case, for studying the effects of transverse and longitudinal random fields. The compounds we consider are RE(OH)_3, where RE are the rare earth ions Tb^{3+}, Dy^{3+} and Ho^{3+}. Using mean-field theory, we estimate the values of the transverse magnetic field that, at zero temperature, destroy ferromagnetic order to be B_x^c=4.35 T, B_x^c=5.03 T and B_x^c=54.81 T for Ho(OH)_3, Dy(OH)_3 and Tb(OH)_3, respectively. We confirm that Ho(OH)_3 and Tb(OH)_3, similarly to LiHoF_4, can be described by an effective spin-1/2 Hamiltonian. In the case of Dy(OH)_3 there is a possibility of a first order phase transition at transverse field close to B_x^c, and Dy(OH)_3 cannot be described by a spin-1/2 effective Hamiltonian. While diluted dipolar Ising spin glass has been studied experimentally in LiHo_xY_{1-x}F_4 and in numerical simulations, there are no studies of the Heisenberg case. Example materials that are likely candidates to be realisations of the diluted dipolar Heisenberg spin glass are (Gd_xY_{1-x})_2Ti_2O_7, (Gd_xY_{1-x})_2Sn_2O_7 and (Gd_xY_{1-x})_3Ga_5O_{12}. To stimulate interest in experimental studies of these systems we present results of Monte of Carlo simulations of the diluted dipolar Heisenberg spin glass. By performing finite-size scaling analysis of the spin-glass correlation length and the spin-glass susceptibility, we provide a compelling evidence of a thermodynamical spin-glass transition in the model. Frustrated pyrochlore magnets, depending on the character of single ion anisotropy and interplay of different types of interaction over a broad range of energy scales, exhibit a large spectrum of exotic phases and novel phenomena. The pyrochlore antiferromagnet Er_2Ti_2O_7 is characterised by a strong planar anisotropy. Experimental studies reveal that Er_2Ti_2O_7 undergoes a continuous phase transition to a long-range ordered phase with a spin configuration that, in this thesis, is referred to as the Champion-Holdsworth state. Such results are not in agreement with the theoretical prediction that the ground state of the pyrochlore easy-plane antiferromagnet with dipolar interactions complementing the nearest neighbour exchange interactions, is not the Champion-Holdsworth state but the so-called Palmer-Chalker state. On the other hand, Monte Carlo simulations of the easy-plane pyrochlore antiferromagnet indicate a thermal order-by-disorder selection of the Champion-Holdsworth state. To answer the question of whether order-by-disorder selection can be the mechanism at play in Er_2Ti_2O_7, we performed Monte Carlo simulations of the easy-plane pyrochlore antiferromagnet with weak dipolar interactions. We estimate the range strengths of the dipolar interaction such that order-by-disorder selection of the Champion-Holdsworth state is not suppressed. The estimated value of the allowed strength of the dipolar interactions indicates that the model studied is likely insufficient to explain the physics of Er_2Ti_2O_7 and other types of interactions or quantum effects should be considered.

Several magnetic materials, in the first approximation, can be described by idealised theoretical models, such as classical Ising or Heisenberg spin systems, and, to some extent, such models are able to qualitatively expose many experimentally observed phenomena. But often, to account for complex behavior of magnetic matter, such models have to be refined by including more terms in Hamiltonian. The compound LiHo_xY_{1-x}F_4, by increasing concentration of nonmagnetic yttrium can be tuned from a diluted ferromagnet to a spin glass. LiHoF_4 is a good realisation of the transverse field Ising model, the simplest model exhibiting a quantum phase transition. In the pure case the magnetic behaviour of this material is well described by mean-field theory. It was believed that when diluted, LiHo_xY_{1-x}F_4 would also manifest itself as a diluted transverse field Ising model which continue to be well described by mean-field theory, and, at sufficient dilution, at zero temperature, exhibit a quantum spin-glass transition. The experimental data did not support such a scenario, and it was pointed out that, to explain physics of LiHo_xY_{1-x}F_4 in transverse magnetic field, the effect of a transverse-field-generated longitudinal random field has to be considered. We explore this idea further in local mean-field studies in which all three parameters: temperature, transverse field and concentration can be consistently surveyed, and where the transverse-field-generated longitudinal random field is explicitly present in the effective spin-1/2 Hamiltonian. We suggest other materials that are possible candidates for studying quantum criticality in the transverse field Ising model, and in the diluted case, for studying the effects of transverse and longitudinal random fields. The compounds we consider are RE(OH)_3, where RE are the rare earth ions Tb^{3+}, Dy^{3+} and Ho^{3+}. Using mean-field theory, we estimate the values of the transverse magnetic field that, at zero temperature, destroy ferromagnetic order to be B_x^c=4.35 T, B_x^c=5.03 T and B_x^c=54.81 T for Ho(OH)_3, Dy(OH)_3 and Tb(OH)_3, respectively. We confirm that Ho(OH)_3 and Tb(OH)_3, similarly to LiHoF_4, can be described by an effective spin-1/2 Hamiltonian. In the case of Dy(OH)_3 there is a possibility of a first order phase transition at transverse field close to B_x^c, and Dy(OH)_3 cannot be described by a spin-1/2 effective Hamiltonian. While diluted dipolar Ising spin glass has been studied experimentally in LiHo_xY_{1-x}F_4 and in numerical simulations, there are no studies of the Heisenberg case. Example materials that are likely candidates to be realisations of the diluted dipolar Heisenberg spin glass are (Gd_xY_{1-x})_2Ti_2O_7, (Gd_xY_{1-x})_2Sn_2O_7 and (Gd_xY_{1-x})_3Ga_5O_{12}. To stimulate interest in experimental studies of these systems we present results of Monte of Carlo simulations of the diluted dipolar Heisenberg spin glass. By performing finite-size scaling analysis of the spin-glass correlation length and the spin-glass susceptibility, we provide a compelling evidence of a thermodynamical spin-glass transition in the model. Frustrated pyrochlore magnets, depending on the character of single ion anisotropy and interplay of different types of interaction over a broad range of energy scales, exhibit a large spectrum of exotic phases and novel phenomena. The pyrochlore antiferromagnet Er_2Ti_2O_7 is characterised by a strong planar anisotropy. Experimental studies reveal that Er_2Ti_2O_7 undergoes a continuous phase transition to a long-range ordered phase with a spin configuration that, in this thesis, is referred to as the Champion-Holdsworth state. Such results are not in agreement with the theoretical prediction that the ground state of the pyrochlore easy-plane antiferromagnet with dipolar interactions complementing the nearest neighbour exchange interactions, is not the Champion-Holdsworth state but the so-called Palmer-Chalker state. On the other hand, Monte Carlo simulations of the easy-plane pyrochlore antiferromagnet indicate a thermal order-by-disorder selection of the Champion-Holdsworth state. To answer the question of whether order-by-disorder selection can be the mechanism at play in Er_2Ti_2O_7, we performed Monte Carlo simulations of the easy-plane pyrochlore antiferromagnet with weak dipolar interactions. We estimate the range strengths of the dipolar interaction such that order-by-disorder selection of the Champion-Holdsworth state is not suppressed. The estimated value of the allowed strength of the dipolar interactions indicates that the model studied is likely insufficient to explain the physics of Er_2Ti_2O_7 and other types of interactions or quantum effects should be considered.