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Bilinear Coagulation Equations
No full textBilinear Coagulation Equations
Daniel Heydecker, Robert I. A. Patterson
(Submitted on 20 Feb 2019 (v1), last revised 14 Oct 2019 (this version, v2))
We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form π(y)⋅Aπ(x) for a vector of conserved quantities π , generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time t g ∈(0,∞) , which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.
Comments: Generalises the previous version to focus on general coagulation processes of bilinear type, without restricting to the single example of the previous version. The previous results are mentioned as motivation, and all results of the previous version can be obtained from this more general version
Subjects: Probability (math.PR)
Cite as: arXiv:1902.07686 [math.PR]
(or arXiv:1902.07686v2 [math.PR] for this version)
Submission history
From: Daniel Heydecker [view email]
[v1]Wed, 20 Feb 2019 18:13:20 UTC (66 KB)
[v2] Mon, 14 Oct 2019 21:05:51 UTC (56 KB
Lagrangian coherent sets in turbulent Rayleigh-Bénard convection
No full textCoherent circulation rolls and their relevance for the turbulent heat transfer in a two-dimensional Rayleigh-Bénard convection model are analyzed. The flow is in a closed cell of aspect ratio four at a Rayleigh number
Ra=10^6 and at a Prandtl number Pr=10. Three different Lagrangian analysis techniques based on graph Laplacians (distance spectral trajectory clustering, time-averaged diffusion maps, and finite-element based dynamic Laplacian discretization) are used to monitor the turbulent fields along trajectories of massless Lagrangian particles in the evolving turbulent convection flow. The three methods are compared to each other and the obtained coherent sets are related to results from an analysis in the Eulerian frame of reference. We show that the results of these methods agree with each other and that Lagrangian and Eulerian coherent sets form basically a disjoint union of the flow domain. Additionally, a windowed time averaging of variable interval length is performed to study the degree of coherence as a function of this additional coarse graining which removes small-scale fluctuations that cause trajectories to disperse quickly. Finally, the coherent set framework is extended to study heat transport
On identification of self-similar characteristics using the Tensor Train decomposition method with application to channel turbulence flow
Get PDFA study on the application of the Tensor Train decomposition method to 3D direct numerical simulation data of channel turbulence flow is presented. The approach is validated with respect to compression rate and storage requirement. In tests with synthetic data, it is found that grid-aligned self-similar patterns are well captured, and also the application to non grid-aligned self-similarity yields satisfying results. It is observed that the shape of the input Tensor significantly affects the compression rate. Applied to data of channel turbulent flow, the Tensor Train format allows for surprisingly high compression rates whilst ensuring low relative errors
Topologies and measures on the space of functions of bounded variation taking values in a Banach or metric space
Get PDFWe study functions of bounded variation with values in a Banach or in a metric space. In finite dimensions, there are three well-known topologies; we argue that in infinite dimensions there is a natural fourth topology. We provide some insight into the structure of these four topologies. In particular, we study the meaning of convergence, duality and regularity for these topologies and provide some useful compactness criteria, also related to the classical Aubin–Lions theorem. After this we study the Bore
Approximating turbulent and non-turbulent events with the Tensor Train decomposition method
Get PDFLow-rank multilevel approximation methods are often suited to attack high-dimensional problems successfully and they allow very compact representation of large data sets. Specifically, hierarchical tensor product decomposition methods, e.g., the Tree-Tucker format and the Tensor Train format emerge as a promising approach for application to data that are concerned with cascade-of-scales problems as, e.g., in turbulent fluid dynamics. Beyond multilinear mathematics, those tensor formats are also successfully applied in e.g., physics or chemistry, where they are used in many body problems and quantum states. Here, we focus on two particular objectives, that is, we aim at capturing self-similar structures that might be hidden in the data and we present the reconstruction capabilities of the Tensor Train decomposition method tested with 3D channel turbulence flow data
Markov-chain-inspired search for MH370
Get PDFMarkov-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, 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°S on the Inmarsat arc.
Application of tools from ergodic theory on historical Lagrangian ocean data is shown to constrain the crash site of Malaysian Airlines flight MH370 given airplane debris beaching site information. The disappearance of flight MH370 constitutes one of the most enigmatic episodes in the history of commercial aviation. The tools employed have far-reaching applicability as they are particularly well suited in inverse modeling, critical, for instance, in revealing contamination sources in the ocean and the atmosphere. The only requirement for their success is sufficient spatiotemporal Lagrangian sampling
Coarse-graining molecular systems by spectral matching
No full textABSTRACT
Coarse-graining has become an area of tremendous importance within many different research fields. For molecular simulation, coarse-graining bears the promise of finding simplified models such that long-time simulations of large-scale systems become computationally tractable. While significant progress has been made in tuning thermodynamic properties of reduced models, it remains a key challenge to ensure that relevant kinetic properties are retained by coarse-grained dynamical systems. In this study, we focus on data-driven methods to preserve the rare-event kinetics of the original system and make use of their close connection to the low-lying spectrum of the system’s generator. Building on work by Crommelin and Vanden-Eijnden [Multiscale Model. Simul. 9, 1588 (2011)], we present a general framework, called spectral matching, which directly targets the generator’s leading eigenvalue equations when learning parameters for coarse-grained models. We discuss different parametric models for effective dynamics and derive the resulting data-based regression problems. We show that spectral matching can be used to learn effective potentials which retain the slow dynamics but also to correct the dynamics induced by existing techniques, such as force matchin
"Global well-posedness for the primitive equations coupled to nonlinear moisture dynamics with phase changes" by Hittmeir, Sabine; Klein, Rupert; Li, Jinkai; Titi, Edriss
No full textGlobal Well-posedness for the Primitive Equations Coupled to Nonlinear Moisture Dynamics with Phase Changes
Sabine Hittmeir, Rupert Klein, Jinkai Li, Edriss S. Titi
(Submitted on 25 Jul 2019)
In this work we study the global solvability of the primitive equations for the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. It has been used by Klein and Majda in \cite{KM} and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler \cite{Ke} and Grabowski and Smolarkiewicz \cite{GS}. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. In \cite{HKLT} we assumed the velocity field to be given and proved rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture balances coupled to the thermodynamic equation. In this paper we present the solvability of a full moist atmospheric flow model, where the moisture model is coupled to the primitive equations of atmospherical dynamics governing the velocity field. For the derivation of a priori estimates for the velocity field we thereby use the ideas of Cao and Titi \cite{CT}, who succeeded in proving the global solvability of the primitive equations
Synchronization of great subduction megathrust earthquakes: Insights from scale model analysis
No full textThe magnitude of great subduction megathrust earthquakes is controlled mainly by the number of adjacent asperities failing synchronously and the resulting rupture length. Here we investigate experimentally the long‐term recurrence behavior of a pair of asperities coupled by static stress transfer over hundreds of seismic cycles. We statistically analyze long (c. 500 ka) time series of M8‐9 analogue earthquakes simulated using a seismotectonic scale model approach with two aims: First, to constrain probabilistic measures (frequency‐size distribution, variability) useful for hazard assessment and, second, to relate them with geometric observables (coseismic slip pattern, locking pattern). We find that the number of synchronized asperity failures relative to the number of individual asperity failures as well as the coefficients of variation of recurrence intervals and seismic moment scale with the logarithm of stress coupling between the asperities. Accordingly, tighter packed asperities tend to recur more periodically and with a more characteristic magnitude while more distant asperities show clustering of more variable sized events. The probability of synchronized failures seems to be controlled to first order by geometrical relations (i.e., spacing and offset of asperities). The effects of rheological properties are evident but it remains to be explored to which extent they vary in nature and how sensitive the system is to those
Initiation of ray tracing models: Evolution of small-amplitude gravity wave packets in non-uniform background
No full textThe present study focuses on the interaction of gravity waves in the atmosphere with the tropopause. As the vertical extent of the latter is small compared to the density scale height, wave propagation is described by the Taylor-Goldstein equation as derived from the linearised Boussinesq approximation. Of particular interest in the construction of gravity wave parameterisations for the upper atmosphere are the transmission and reflection properties of the tropopause as these determine the upward fluxes of energy and momentum carried by internal waves. A method is presented that decomposes internal waves explicitly into upward and downward propagating contributions, thus giving direct access to transmission and reflection coefficients of finite regions of non-uniform stratification in a stationary atmosphere. The scheme utilizes a piecewise constant approximation for the background stratification and matches up- and downward propagating plane wave solutions in each layer through physically meaningful coupling conditions. As a result, transmission and reflection coefficients follow immediately