Dynamical Systems on Networks: A Tutorial

We give a tutorial for the study of dynamical systems on networks. We focus especially on "simple" situations that are tractable analytically, because they can be very insightful and provide useful springboards for the study of more complicated scenarios. We briefly motivate why examining dynamical systems on networks is interesting and important, and we then give several fascinating examples and discuss some theoretical results. We also briefly discuss dynamical systems on dynamical (i.e., time-dependent) networks, overview software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than original version, some reorganization and also more pointers to interesting direction

Reduction Methods in Climate Dynamics -- A Brief Review

We review a range of reduction methods that have been, or may be useful for connecting models of the Earth's climate system of differing complexity. We particularly focus on methods where rigorous reduction is possible. We aim to highlight the main mathematical ideas of each reduction method and also provide several benchmark examples from climate modelling

Quantitative uniform in time chaos propagation for Boltzmann collision processes

This paper is devoted to the study of mean-field limit for systems of indistinguables particles undergoing collision processes. As formulated by Kac \cite{Kac1956} this limit is based on the {\em chaos propagation}, and we (1) prove and quantify this property for Boltzmann collision processes with unbounded collision rates (hard spheres or long-range interactions), (2) prove and quantify this property \emph{uniformly in time}. This yields the first chaos propagation result for the spatially homogeneous Boltzmann equation for true (without cut-off) Maxwell molecules whose "Master equation" shares similarities with the one of a L\'evy process and the first {\em quantitative} chaos propagation result for the spatially homogeneous Boltzmann equation for hard spheres (improvement of the %non-contructive convergence result of Sznitman \cite{S1}). Moreover our chaos propagation results are the first uniform in time ones for Boltzmann collision processes (to our knowledge), which partly answers the important question raised by Kac of relating the long-time behavior of a particle system with the one of its mean-field limit, and we provide as a surprising application a new proof of the well-known result of gaussian limit of rescaled marginals of uniform measure on the $N$-dimensional sphere as $N$ goes to infinity (more applications will be provided in a forthcoming work). Our results are based on a new method which reduces the question of chaos propagation to the one of proving a purely functional estimate on some generator operators ({\em consistency estimate}) together with fine stability estimates on the flow of the limiting non-linear equation ({\em stability estimates})

The mean-field Limit of sparse networks of integrate and fire neurons

We study the mean-field limit of a model of biological neuron networks based on the so-called stochastic integrate-and-fire (IF) dynamics. Our approach allows to derive a continuous limit for the macroscopic behavior of the system, the 1-particle distribution, for a large number of neurons with no structural assumptions on the connection map outside of a generalized mean-field scaling. We propose a novel notion of observables that naturally extends the notion of marginals to systems with non-identical or non-exchangeable agents. Our new observables satisfy a complex approximate hierarchy, essentially a tree-indexed extension of the classical BBGKY hierarchy. We are able to pass to the limit in this hierarchy as the number of neurons increases through novel quantitative stability estimates in some adapted weak norm. While we require non-vanishing diffusion, this approach notably addresses the challenges of sparse interacting graphs/matrices and singular interactions from Poisson jumps, and requires no additional regularity on the initial distribution

Motility of Swimming Bacteria Hydrodynamics and Statistics

The present work contains original research on the field of biophysics, specifically the study of swimming bacteria. Swimming microorganisms can be modeled as active particles moving at low Reynolds number (Re ≪ 1) and subject to different sources of noise. The term “active” means that they are self-propelled, while Re ≪ 1 implies that their motion is dominated by viscous stresses, therefore relying on non-reciprocal deformations in time, in order to achieve movement. Noise arises from thermal fluctuations and the inherent stochasticity of their propelling machinery, as a result, bacteria follow random trajectories. Nevertheless, bacteria have evolved to display a number of strategies to overcome randomness and achieve directed locomotion, known as “taxis”. Here, we explore the mechanisms involved in the propulsion and navigation of swimming bacteria, using low Reynolds number flow techniques and random walks. First, we introduce the physical principles that govern the dynamics of a low Reynolds number swimmer. We pay special attention to the random walk model for the description of the swimming trajectories, since it allows to quantify motility in terms of statistical measures, such as diffusivity and drift velocity, which can be measured experimentally. After a general discussion of the model, we demonstrate its use by applying it to the dynamics of bacteria-driven microswimmers, which are active particles that use bacteria as a propulsion mechanism. We show in particular, that the diffusivity of such particles increases with the square of their size and that the microswimmers inherit the chemotactic capabilities from the bacteria that propel them. These results are in agreement with experiments and can be useful to improve the design of these artificial microswimmers. Next, we investigate the motility properties of Spiroplasma melliferum, which is special among bacteria, as it can swim without flagella. Instead, Spiroplasma can switch the handedness of its helical body and in the process, the helical domains rotate generating propulsion. Based on experimental observations, we develop a hydrodynamic model to describe Spiroplasma motility. We obtain expressions for the total linear and angular displacements of the cell body per swimming stroke. Observing that the cell body does not reorient at the end of one period, we define an effective swimming speed and a hydrodynamic efficiency. Then, we show that the helical shape that maximises speed and efficiency has a pitch angle close to that of Spiroplasma, φ ≃ 35◦, in agreement with experimental observations and with previous numerical simulations. Finally, we explore the dynamics of a low Reynolds number swimmer crossing a viscosity gradient. This is a work in collaboration with experimental groups in the National University of Mexico (UNAM) and Brown University. The experiments aim to shed light on the dynamics of the bacterium Helicobacter pylori, which inhabits the human gut and is capable of penetrating the mucus layer that protects the stomach. Experimentally, a magnetic swimmer is immersed in a stratified solution of miscible fluids with different viscosities. The swimmer consists of a helical tail and a cylindrical head that rotate at a fixed rate due to the action of an external magnetic field. As the swimmer advances, it accelerates or slows down, depending on its orientation with respect to the gradient. In general, the experimental results show that it is harder for a pusher-like swimmer to swim up the gradient, whereas for a puller-like swimmer it is the opposite. We rationalize this mathematically by assuming that the forces acting on the swimmer depend on the local viscosity that it experiences. This allows us to calculate the swimming speed as a function of the swimmer’s position along the gradient. The predictions of the model are in good agreement with the experimental observations. The results also suggest that viscotaxis is possible without viscoreceptors, and in fact governed solely by the motility pattern of the swimmer. Together, the results presented in this thesis contribute to the understanding of bacterial motility and low Reynolds number swimmers in general. Furthermore, these results may be useful for future developments in biophysics, including applications to targeted drug delivery and microrobotics.European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement 682754 to Eric Lauga

Pointwise Green's function bounds and stability of relaxation shocks

We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The techniques of this paper should have further application in the closely related case of traveling waves of systems with partial viscosity, for example in compressible gas dynamics or MHD.Comment: 120 pages. Changes since original submission. Corrected typos, esp. energy estimates of Section 7, corrected bad forward references, expanded Remark 1.17, end of introductio

Keldysh Field Theory for Driven Open Quantum Systems

Recent experimental developments in diverse areas - ranging from cold atomic gases over light-driven semiconductors to microcavity arrays - move systems into the focus, which are located on the interface of quantum optics, many-body physics and statistical mechanics. They share in common that coherent and driven-dissipative quantum dynamics occur on an equal footing, creating genuine non-equilibrium scenarios without immediate counterpart in condensed matter. This concerns both their non-thermal flux equilibrium states, as well as their many-body time evolution. It is a challenge to theory to identify novel instances of universal emergent macroscopic phenomena, which are tied unambiguously and in an observable way to the microscopic drive conditions. In this review, we discuss some recent results in this direction. Moreover, we provide a systematic introduction to the open system Keldysh functional integral approach, which is the proper technical tool to accomplish a merger of quantum optics and many-body physics, and leverages the power of modern quantum field theory to driven open quantum systems.Comment: 73 pages, 13 figure

Linear state models for volatility estimation and prediction

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis concerns the calibration and estimation of linear state models for forecasting stock return volatility. In the first two chapters I present aspects of financial modelling theory and practice that are of particular relevance to the theme of this present work. In addition to this I review the literature concerning these aspects with a particular emphasis on the area of dynamic volatility models. These chapters set the scene and lay the foundations for subsequent empirical work and are a contribution in themselves. The structure of the models employed in the application chapters 4,5 and 6 is the state-space structure, or alternatively the models are known as unobserved components models. In the literature these models have been applied in the estimation of volatility, both for high frequency and low frequency data. As opposed to what has been carried out in the literature I propose the use of these models with Gaussian components. I suggest the implementation of these for high frequency data for short and medium term forecasting. I then demonstrate the calibration of these models and compare medium term forecasting performance for different forecasting methods and model variations as well as that of GARCH and constant volatility models. I then introduce implied volatility measurements leading to two-state models and verify whether this derivative-based information improves forecasting performance. In chapter 6I compare different unobserved components models' specification and forecasting performance. The appendices contain the extensive workings of the parameter estimates' standard error calculations