Periodic measures, transitions and exit times of stochastic differential equations

Periodic measures are the time-periodic counterpart to invariant measures for dynamical systems that can characterise the long-term periodic behaviour of stochastic dynamical systems. In this thesis, sufficient conditions are given for the existence, uniqueness and geometric convergence of periodic measures of time-periodic Markovian systems on locally compact metric spaces. The results will be applied specifically to time-periodic weakly dissipative stochastic differential equations (SDEs), gradient SDEs and Langevin equations. We show that the periodic measure density sufficiently and necessarily satisfies a time-periodic Fokker-Planck equation. We will also rigorously derive that the expected exit duration of time-periodic SDEs is the time-periodic solution of a second-order linear parabolic partial differential equation (PDE). Collectively, this rigorously establishes two novel Feynman-Kac dualities for time-periodic SDEs. Casting the time-periodic solution of the PDE as a fixed point problem and a convex optimisation problem, we give sufficient conditions in which the PDE is well-posed in a weak and classical sense. With no known closed formulae, we show that these approaches can be readily implemented to compute the expected exit time numerically. Periodic measures and expected exit times are novel tools to understand physical phenomena exhibiting periodicity. Particular application towards stochastic resonance will be discussed

Expected exit time for time-periodic stochastic differential equations and applications to stochastic resonance

In this paper, we derive a parabolic partial differential equation for the expected exit time of non-autonomous time-periodic non-degenerate stochastic differential equations. This establishes a Feynman–Kac duality between expected exit time of time-periodic stochastic differential equations and time-periodic solutions of parabolic partial differential equations. Casting the time-periodic solution of the parabolic partial differential equation as a fixed point problem and a convex optimisation problem, we give sufficient conditions in which the partial differential equation is well-posed in a weak and classical sense. With no known closed formulae for the expected exit time, we show our method can be readily implemented by standard numerical schemes. With relatively weak conditions (e.g. locally Lipschitz coefficients), the method in this paper is applicable to wide range of physical systems including weakly dissipative systems. Particular applications towards stochastic resonance will be discussed

Pechukas-Yukawa approach to the evolution of the quantum state of a parametrically perturbed system

We consider the evolution of a quantum state of a Hamiltonian which is parametrically perturbed via a term proportional to the adiabatic parameter \lambda (t). Starting with the Pechukas-Yukawa mapping of the energy eigenvalues evolution on a generalised Calogero-Sutherland model of 1D classical gas, we consider the adiabatic approximation with two different expansions of the quantum state in powers of d\lambda/dt and compare them with a direct numerical simulation. We show that one of these expansions (Magnus series) is especially convenient for the description of non-adiabatic evolution of the system. Applying the expansion to the exact cover 3-satisfability problem, we obtain the occupation dynamics which provides insight on the population of states.Comment: 12 pages, 6 figure

Bogoliubov-Born-Green-Kirkwood-Yvon chain and kinetic equations for the level dynamics in an externally perturbed quantum system

Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing a new approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in case of an adiabatically slow external perturbation. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of particles with cubic repulsion. Equilibrium statistical mechanics of this Pechukas gas allows to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the Pechukas gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e. approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and inter-level transition amplitudes, they allow to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as 1/N, where N is the number of the "Pechukas gas particles" (i.e. energy levels in the system)

Pechukas-Yukawa formalism for Landau-Zener transitions in the presence of external noise

Quantum systems are prone to decoherence due to both intrinsic interactions as well as random fluctuations from the environment. Using the Pechukas-Yukawa formalism, we investigate the influence of noise on the dynamics of an adiabatically evolving Hamiltonian which can describe a quantum computer. Under this description, the level dynamics of a parametrically perturbed quantum Hamiltonian are mapped to the dynamics of one-dimensional classical gas. We show that our framework coincides with the results of the classical Landau-Zener transitions upon linearization. Furthermore, we determine the effects of external noise on the level dynamics and its impact on Landau-Zener transitions

Large expert-curated database for benchmarking document similarity detection in biomedical literature search

Document recommendation systems for locating relevant literature have mostly relied on methods developed a decade ago. This is largely due to the lack of a large offline gold-standard benchmark of relevant documents that cover a variety of research fields such that newly developed literature search techniques can be compared, improved and translated into practice. To overcome this bottleneck, we have established the RElevant LIterature SearcH consortium consisting of more than 1500 scientists from 84 countries, who have collectively annotated the relevance of over 180 000 PubMed-listed articles with regard to their respective seed (input) article/s. The majority of annotations were contributed by highly experienced, original authors of the seed articles. The collected data cover 76% of all unique PubMed Medical Subject Headings descriptors. No systematic biases were observed across different experience levels, research fields or time spent on annotations. More importantly, annotations of the same document pairs contributed by different scientists were highly concordant. We further show that the three representative baseline methods used to generate recommended articles for evaluation (Okapi Best Matching 25, Term Frequency-Inverse Document Frequency and PubMed Related Articles) had similar overall performances. Additionally, we found that these methods each tend to produce distinct collections of recommended articles, suggesting that a hybrid method may be required to completely capture all relevant articles. The established database server located at https://relishdb.ict.griffith.edu.au is freely available for the downloading of annotation data and the blind testing of new methods. We expect that this benchmark will be useful for stimulating the development of new powerful techniques for title and title/abstract-based search engines for relevant articles in biomedical research.Peer reviewe

Existence of geometric ergodic periodic measures of stochastic differential equations

Periodic measures are the time-periodic counterpart to invariant measures for dynamical systems and can be used to characterise the long-term periodic behaviour of stochastic systems. This paper gives sufficient conditions for the existence, uniqueness and geometric convergence of a periodic measure for time-periodic Markovian processes on a locally compact metric space in great generality. In particular, we apply these results in the context of time-periodic weakly dissipative stochastic differential equations, gradient stochastic differential equations as well as Langevin equations. We will establish the Fokker-Planck equation that the density of the periodic measure sufficiently and necessarily satisfies. Applications to physical problems shall be discussed with specific examples

A Neural Network Method for Retrieving Sea Surface Wind Speed for C-Band SAR

Based on the Ocean Projection and Extension neural Network (OPEN) method, a novel approach is proposed to retrieve sea surface wind speed for C-band synthetic aperture radar (SAR). In order to prove the methodology with a robust dataset, five-year normalized radar cross section (NRCS) measurements from the advanced scatterometer (ASCAT), a well-known side-looking radar sensor, are used to train the model. In situ wind data from direct buoy observations, instead of reanalysis wind data or model results, are used as the ground truth in the OPEN model. The model is applied to retrieve sea surface winds from two independent data sets, ASCAT and Sentinel-1 SAR data, and has been well-validated using buoy measurements from the National Oceanic and Atmospheric Administration (NOAA) and China Meteorological Administration (CMA), and the ASCAT coastal wind product. The comparison between the OPEN model and four C-band model (CMOD) versions (CMOD4, CMOD-IFR2, CMOD5.N, and CMOD7) further indicates the good performance of the proposed model for C-band SAR sensors. It is anticipated that the use of high-resolution SAR data together with the new wind speed retrieval method can provide continuous and accurate ocean wind products in the future.publishedVersio