Anomalous beam diffusion near beam-beam synchro-betatron resonances

The diffusion process near low order synchro-betatron resonances driven by beam-beam interactions at a crossing angle is investigated. Macroscopic observables such as beam emittance, lifetime and beam profiles are calculated. These are followed with detailed studies of microscopic quantities such as the evolution of the variance at several initial transverse amplitudes and single particle probability distribution functions. We present evidence to show that the observed diffusion is anomalous and the dynamics follows a non-Markovian continuous time random walk process. We derive a modified master equation to replace the Chapman-Kolmogorov equation in action-angle space and a fractional diffusion equation to describe the density evolution for this class of processes.Comment: 23 pages, 12 figure

The Monte Carlo and fractional kinetics approaches to the underground anomalous subdiffusion of contaminants

This paper deals with a comparison of Fractional Derivative and Monte carlo approaches to the modelling of anomalous diffusion in the field of particle transport. The goal of this research is to provide a better insight on the behavior of (radioactive) contaminant tracers when flowing through heterogeneous media.Comment: 17 pages, 18 figure

Mathematical analysis of k-path Laplacian operators on simple graphs

A set of links and nodes are the fundamental units or components used to represent complex networks. Over the last few decades, network studies have expanded and matured, increasingly making use of complex mathematical tools. Complex networks play a significant role in the propagation of processes, which include for example the case of epidemic spreading, the diffusion process, synchronisation or the consensus process. Such dynamic processes are critically important in achieving understanding of the behaviour of complex systems at different levels of complexity - examples might be the brain and modern man-made infrastructures. Although part of the study of the diffusion of information in the dynamic processes, it is generally supposed that interactions in networks originate only from a node, spreading to its nearest neighbours, there also exist long-range interactions (LRI), which can be transmitted from a node to others not directly connected. The focus of this study is on dynamic processes on networks where nodes interact with not only their nearest neighbours but also through certain LRIs. The generalised k-path Laplacian operators (LOs) Lk, which account for the hop of a diffusive particle to its non-nearest neighbours in a graph, control this diffusive process, describing hops of nodes vi at distance k; here the distance is measured as the length of the shortest path between two nodes. In this way the introduction of the k-path LOs can facilitate conducting more precise studies of network dynamics in different applications. This thesis aims to study a generalised diffusion equation employing the transformed generalised k-path LOs for a locally finite infinite graph. This generalised diffusion equation promotes both normal and super diffusive processes on infinite graphs. Furthermore, this thesis develops a new theoretical mathematical framework for describing superdiffusion processes that use a transform of the k-path LOs defined on infinite graphs. The choice of the transform appeared to be vitally important as the probability of a long jump should be great enough. As described by other researchers the fractional diffusion equation (FDE) formed the mathematical framework employed to describe this anomalous diffusion. In this regard,it is taken that the diffusive particle is not just hopping to its nearest node but also to any other node of the network with a probability that scales according to the distance between the two places. Initially, we extend the k-path LOs above to consider a connected and locally finite infinite network with a bounded degree and investigate a number of the properties of these operators, such as their self-adjointness and boundedness. Then, three different transformations of the k-path LOs, i.e. the Laplace, Factorial and Mellin transformations as well as their properties, are studied.In addition, in order to show a number of applications of these operators and the transformed ones, the transformed k-path LOs are used to obtain a generalised diffusion process for one-dimensional and two-dimensional infinite graphs.First, the infinite path graph is studied, where it is possible to prove that when the Laplacian- and factorial-transformed operators are used in the generalised diffusion equation, the diffusive processes observed are always normal, independent of the transform parameters. It is then proven analytically that when the k-path LOs are transformed via a Mellin transform and plugged into the diffusion equation, the result is a super diffusive process for certain values of the exponent in the transform. Secondly, we generalise the results on the superdiffusive behaviour generated by transforming k-path LOs from one-dimensional graphs to 2-dimensional ones. Our attention focuses on the Abstract Cauchy problem in an infinite square lattice. A generalised diffusion equation on a square lattice corresponding to Mellin transforms of the k-path Laplacian is investigated. Similar to the one-dimensional case also for the graph embedded in two-dimensional space,we could observe superdiffusive behaviour for the Mellin transformed k-path Laplacian. In comparison to the one-dimensional case, the conclusion reached is that the asymptotic behaviour of the solution of the Cauchy problem is much subtler.A set of links and nodes are the fundamental units or components used to represent complex networks. Over the last few decades, network studies have expanded and matured, increasingly making use of complex mathematical tools. Complex networks play a significant role in the propagation of processes, which include for example the case of epidemic spreading, the diffusion process, synchronisation or the consensus process. Such dynamic processes are critically important in achieving understanding of the behaviour of complex systems at different levels of complexity - examples might be the brain and modern man-made infrastructures. Although part of the study of the diffusion of information in the dynamic processes, it is generally supposed that interactions in networks originate only from a node, spreading to its nearest neighbours, there also exist long-range interactions (LRI), which can be transmitted from a node to others not directly connected. The focus of this study is on dynamic processes on networks where nodes interact with not only their nearest neighbours but also through certain LRIs. The generalised k-path Laplacian operators (LOs) Lk, which account for the hop of a diffusive particle to its non-nearest neighbours in a graph, control this diffusive process, describing hops of nodes vi at distance k; here the distance is measured as the length of the shortest path between two nodes. In this way the introduction of the k-path LOs can facilitate conducting more precise studies of network dynamics in different applications. This thesis aims to study a generalised diffusion equation employing the transformed generalised k-path LOs for a locally finite infinite graph. This generalised diffusion equation promotes both normal and super diffusive processes on infinite graphs. Furthermore, this thesis develops a new theoretical mathematical framework for describing superdiffusion processes that use a transform of the k-path LOs defined on infinite graphs. The choice of the transform appeared to be vitally important as the probability of a long jump should be great enough. As described by other researchers the fractional diffusion equation (FDE) formed the mathematical framework employed to describe this anomalous diffusion. In this regard,it is taken that the diffusive particle is not just hopping to its nearest node but also to any other node of the network with a probability that scales according to the distance between the two places. Initially, we extend the k-path LOs above to consider a connected and locally finite infinite network with a bounded degree and investigate a number of the properties of these operators, such as their self-adjointness and boundedness. Then, three different transformations of the k-path LOs, i.e. the Laplace, Factorial and Mellin transformations as well as their properties, are studied.In addition, in order to show a number of applications of these operators and the transformed ones, the transformed k-path LOs are used to obtain a generalised diffusion process for one-dimensional and two-dimensional infinite graphs.First, the infinite path graph is studied, where it is possible to prove that when the Laplacian- and factorial-transformed operators are used in the generalised diffusion equation, the diffusive processes observed are always normal, independent of the transform parameters. It is then proven analytically that when the k-path LOs are transformed via a Mellin transform and plugged into the diffusion equation, the result is a super diffusive process for certain values of the exponent in the transform. Secondly, we generalise the results on the superdiffusive behaviour generated by transforming k-path LOs from one-dimensional graphs to 2-dimensional ones. Our attention focuses on the Abstract Cauchy problem in an infinite square lattice. A generalised diffusion equation on a square lattice corresponding to Mellin transforms of the k-path Laplacian is investigated. Similar to the one-dimensional case also for the graph embedded in two-dimensional space,we could observe superdiffusive behaviour for the Mellin transformed k-path Laplacian. In comparison to the one-dimensional case, the conclusion reached is that the asymptotic behaviour of the solution of the Cauchy problem is much subtler

From phenomenological modelling of anomalous diffusion through continuous-time random walks and fractional calculus to correlation analysis of complex systems

This document contains more than one topic, but they are all connected in ei- ther physical analogy, analytic/numerical resemblance or because one is a building block of another. The topics are anomalous diffusion, modelling of stylised facts based on an empirical random walker diffusion model and null-hypothesis tests in time series data-analysis reusing the same diffusion model. Inbetween these topics are interrupted by an introduction of new methods for fast production of random numbers and matrices of certain types. This interruption constitutes the entire chapter on random numbers that is purely algorithmic and was inspired by the need of fast random numbers of special types. The sequence of chapters is chrono- logically meaningful in the sense that fast random numbers are needed in the first topic dealing with continuous-time random walks (CTRWs) and their connection to fractional diffusion. The contents of the last four chapters were indeed produced in this sequence, but with some temporal overlap. While the fast Monte Carlo solution of the time and space fractional diffusion equation is a nice application that sped-up hugely with our new method we were also interested in CTRWs as a model for certain stylised facts. Without knowing economists [80] reinvented what physicists had subconsciously used for decades already. It is the so called stylised fact for which another word can be empirical truth. A simple example: The diffusion equation gives a probability at a certain time to find a certain diffusive particle in some position or indicates concentration of a dye. It is debatable if probability is physical reality. Most importantly, it does not describe the physical system completely. Instead, the equation describes only a certain expectation value of interest, where it does not matter if it is of grains, prices or people which diffuse away. Reality is coded and “averaged” in the diffusion constant. Interpreting a CTRW as an abstract microscopic particle motion model it can solve the time and space fractional diffusion equation. This type of diffusion equation mimics some types of anomalous diffusion, a name usually given to effects that cannot be explained by classic stochastic models. In particular not by the classic diffusion equation. It was recognised only recently, ca. in the mid 1990s, that the random walk model used here is the abstract particle based counterpart for the macroscopic time- and space-fractional diffusion equation, just like the “classic” random walk with regular jumps ±∆x solves the classic diffusion equation. Both equations can be solved in a Monte Carlo fashion with many realisations of walks. Interpreting the CTRW as a time series model it can serve as a possible null- hypothesis scenario in applications with measurements that behave similarly. It may be necessary to simulate many null-hypothesis realisations of the system to give a (probabilistic) answer to what the “outcome” is under the assumption that the particles, stocks, etc. are not correlated. Another topic is (random) correlation matrices. These are partly built on the previously introduced continuous-time random walks and are important in null- hypothesis testing, data analysis and filtering. The main ob jects encountered in dealing with these matrices are eigenvalues and eigenvectors. The latter are car- ried over to the following topic of mode analysis and application in clustering. The presented properties of correlation matrices of correlated measurements seem to be wasted in contemporary methods of clustering with (dis-)similarity measures from time series. Most applications of spectral clustering ignores information and is not able to distinguish between certain cases. The suggested procedure is sup- posed to identify and separate out clusters by using additional information coded in the eigenvectors. In addition, random matrix theory can also serve to analyse microarray data for the extraction of functional genetic groups and it also suggests an error model. Finally, the last topic on synchronisation analysis of electroen- cephalogram (EEG) data resurrects the eigenvalues and eigenvectors as well as the mode analysis, but this time of matrices made of synchronisation coefficients of neurological activity

Modeling Financial Swaps and Geophysical data Using the Barndorff-Nielsen and Shephard Model

This dissertation uses Barndoff-Nielsen and Shephard (BN-S) models to model swap, a type of financial derivative, and analyze geophysical data for estimation of major earthquakes. From empirical observation of the stock market activity and earthquake occurrence, we observe that the distributions have high kurtosis and right skewness. Consequently, such data cannot be captured by stochastic models driven by a Wiener process. Non-Gaussian processes of Ornstein-Uhlenbeck type are one of the most significant candidates for being the building blocks of models of financial economics. These models offer the possibility of capturing important distributional deviations from Gaussianity and thus these are more practical models of dependence structures. Introduced by Barndorff-Nielsen and Shephard these processes are not only convenient to model volatility in financial market, but have an independent interest for modeling stationary time series of various kinds. For the financial data we use BN-S models to price the arbitrage-free value of volatility, variance, covariance, and correlation swap. Such swaps are used in financial markets for volatility hedging and speculation. We use the S&P500 and NASDAQ index for parameter estimation and numerical analysis. We show that with this model the error estimation in fitting the delivery price is much less than the existing models with comparable parameters. For any given time interval, the earthquake magnitude data have three main properties: (1) magnitude is a non-negative stationary stochastic process; (2) for any finite interval of time there are only finite number of jumps; (3) the sample path of the magnitude of an earthquake consists of upward jumps (significant earthquake) and a gradual decrease (aftershocks). For such geophysical data we specifically use Gamma Ornstein Uhlenbeck processes driven by a Levy process to estimate a major earthquake in a certain region in California. Rigorous regression analysis is provided, and based on that, first-passage times are computed for different sets of data. Both applications demonstrate the significance of BN-S models to phenomena that follow non-Gaussian distributions

A Minimum of Stochastics for Scientists : Ten Lectures

These pages contain material I have tried to convey, in a course given occasionally, to a Caltech audience composed mostly of graduate students. The idea behind the course was to introduce students to the ideas and attitudes that underlie the statistical modeling of physical, chemical, biological systems. To provide a sufficient minimum to begin the repair of total ignorance, to get one started, to avoid embarrassing moments at an oral qualifying examination, perhaps. This little book might be titled "An Introduction to an Introduction to . . ." The student who wishes to go deeper will find much that is rich in the classic texts and essays listed under References