Transition paths of marine debris and the stability of the garbage patches

We used transition path theory (TPT) to infer "reactive" pathways of floating marine debris trajectories. The TPT analysis was applied on a pollution-aware time-homogeneous Markov chain model constructed from trajectories produced by satellite-tracked undrogued buoys from the NOAA Global Drifter Program. The latter involved coping with the openness of the system in physical space, which further required an adaptation of the standard TPT setting. Directly connecting pollution sources along coastlines with garbage patches of varied strengths, the unveiled reactive pollution routes represent alternative targets for ocean cleanup efforts. Among our specific findings we highlight: constraining a highly probable pollution source for the Great Pacific Garbage Patch; characterizing the weakness of the Indian Ocean gyre as a trap for plastic waste; and unveiling a tendency of the subtropical gyres to export garbage toward the coastlines rather than to other gyres in the event of anomalously intense winds.Comment: Submitted to Chao

Markov-chain-inspired search for MH370

Markov-chain models are constructed for the probabilistic description of the drift of marine debris from Malaysian Airlines flight MH370. En route from Kuala Lumpur to Beijing, the MH370 mysteriously disappeared in the southeastern Indian Ocean on 8 March 2014, somewhere along the arc of the 7th ping ring around the Inmarsat-3F1 satellite position when the airplane lost contact. The models are obtained by discretizing the motion of undrogued satellite-tracked surface drifting buoys from the global historical data bank. A spectral analysis, Bayesian estimation, and the computation of most probable paths between the Inmarsat arc and confirmed airplane debris beaching sites are shown to constrain the crash site, near 25$^{\circ}$S on the Inmarsat arc.Comment: Submitted to Chao

Network Measures of Mixing

Transport and mixing processes in fluid flows can be studied directly from Lagrangian trajectory data, such as obtained from particle tracking experiments. Recent work in this context highlights the application of graph-based approaches, where trajectories serve as nodes and some similarity or distance measure between them is employed to build a (possibly weighted) network, which is then analyzed using spectral methods. Here, we consider the simplest case of an unweighted, undirected network and analytically relate local network measures such as node degree or clustering coefficient to flow structures. In particular, we use these local measures to divide the family of trajectories into groups of similar dynamical behavior via manifold learning methods

From Metastable to Coherent Sets – time-discretization schemes

Given a time-dependent stochastic process with trajectories x(t) in a space \Omega, there may be sets such that the corresponding trajectories only very rarely cross the boundaries of these sets. We can analyze such a process in terms of metastability or coherence. Metastable sets M are defined in space M \subset \Omega, coherent sets M(t) \subset \Omega are defined in space and time. Hence, if we extend the space \Omega by the time-variable t, coherent sets are metastable sets in \Omega \times [0,\infty). This relation can be exploited, because there already exist spectral algorithms for the identification of metastable sets. In this article we show that these well-established spectral algorithms (like PCCA+) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-time-discretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application

Diffusion maps tailored to arbitrary non-degenerate Ito processes

We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Ito diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling

Environmental assessment of farm sustainability – A Hungarian case study

Agricultural production is inseparable from environmental sustainability, and several methods have been developed by numerous authors and organizations worldwide to measure and evaluate it. The aim of the study is to comprehensively and accurately assess the environmental sustainability of farms. Concerning the theoretical framework, a set of 8 indicators and 23 sub-indicators was established to estimate the different aspects of environmental sustainability. The practical compliance of the indicators was assessed based on data from Hungarian agricultural enterprises; an agricultural company and three individual farmers were included in the case studies. By grouping the indicators, a new composite environmental indicator was developed to measure environmental sustainability. The results of the study show that the surveyed farms are moderately sustainable, as their composite indicators were at or close to the minimum of 0.5 points, but none of them were outstanding or at least 0.75 points. More significant efforts should be made to improve the farms’ environmental sustainability in the future

Doped carbon nanotubes as a model system of biased graphene

Albeit difficult to access experimentally, the density of states (DOS) is a key parameter in solid state systems which governs several important phenomena including transport, magnetism, thermal, and thermoelectric properties. We study DOS in an ensemble of potassium intercalated single-wall carbon nanotubes (SWCNT) and show using electron spin resonance spectroscopy that a sizeable number of electron states are present, which gives rise to a Fermi-liquid behavior in this material. A comparison between theoretical and the experimental DOS indicates that it does not display significant correlation effects, even though the pristine nanotube material shows a Luttinger-liquid behavior. We argue that the carbon nanotube ensemble essentially maps out the whole Brillouin zone of graphene thus it acts as a model system of biased graphene

Electron spin resonance signal of Luttinger liquids and single-wall carbon nanotubes

A comprehensive theory of electron spin resonance (ESR) for a Luttinger liquid (LL) state of correlated metals is presented. The ESR measurables such as the signal intensity and the line-width are calculated in the framework of Luttinger liquid theory with broken spin rotational symmetry as a function of magnetic field and temperature. We obtain a significant temperature dependent homogeneous line-broadening which is related to the spin symmetry breaking and the electron-electron interaction. The result crosses over smoothly to the ESR of itinerant electrons in the non-interacting limit. These findings explain the absence of the long-sought ESR signal of itinerant electrons in single-wall carbon nanotubes when considering realistic experimental conditions.Comment: 5 pages, 1 figur

Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds, and demonstrate $L^1$ convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.Comment: 34 page