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Large deviations of jump process fluxes
Get PDFWe study a general class of systems of interacting particles that randomly interact to form new or different particles. In addition to the distribution of particles we consider the fluxes, defined as the rescaled number of jumps of each type that take place in a time interval. We prove a dynamic large deviations principle for the fluxes under general assumptions that include mass-action chemical kinetics. This result immediately implies a dynamic large deviations principle for the particle distribution
Homogenization theory for the random conductance model with degenerate ergodic weights and unbounded-range jumps
Get PDFWe study homogenization properties of the discrete Laplace operator with random conductances on a large domain in Z^d. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum.
We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length.
Without the long-range connections, the integrability condition on the lower tail is optimal for spectral homogenization. It coincides with a necessary condition for the validity of a local central limit theorem for the random walk among random conductances. As an application of spectral homogenization, we prove a quenched large deviation principle for the normalized and rescaled local times of the random walk in a growing box.
Our proofs are based on a compactness result for the Laplacian's Dirichlet energy, Poincare inequalities, Moser iteration and two-scale convergence
Scale interactions and anisotropy in stable boundary layers
Get PDFRegimes of interactions between motions on different time-scales are investigated in the FLOSSII dataset for nocturnal near-surface stable boundary layer (SBL) turbulence. The nonstationary response of turbulent vertical velocity variance to non-turbulent, sub-mesoscale wind velocity variability is analysed using the bounded variation, finite element, vector autoregressive factor models (FEM-BV-VARX) clustering method. Several locally stationary flow regimes are identified with different influences of sub-meso wind velocity on the turbulent vertical velocity variance. In each flow regime, we analyse multiple scale interactions and quantify the amount of turbulent variability which can be statistically explained by the individual forcing variables. The state of anisotropy of the Reynolds stress tensor in the different flow regimes is shown to relate to these different signatures of scale interactions. In flow regimes dominated by sub-mesoscale wind variability, the Reynolds stresses show a clear preference for strongly anisotropic, one-component stresses, which tend to correspond to periods in which the turbulent fluxes are against the mean gradient. These periods additionally show stronger persistence in their dynamics, compared to periods of more isotropic stresses. The analyses give insights on how the different topologies relate to non-stationary turbulence triggering by sub-mesoscale motions
Smart Speed Imaging in Digital Image Correlation: Application to Seismotectonic Scale Modeling
Get PDFAnalogue models of earthquakes and seismic cycles are characterized by strong variations in strain rate: from slow interseismic loading to fast coseismic release of elastic energy. Deformation rates vary accordingly from micrometer per second (e.g. plate tectonic motion) to meter per second (e.g. rupture propagation). Deformation values are very small over one seismic cycle, in the order of a few tens of micrometer, because of the scaled nature of such models. This cross-scale behaviour poses a major challenge to effectively monitor the experiments by means of digital image correlation techniques, i.e. at high resolution (100 Hz) during the coseismic period but without dramatically oversampling the interseismic period.
We developed a smart speed imaging tool which allows on-the-fly scaling of imaging frequency with strain rate, based on an external trigger signal and a buffer. The external trigger signal comes from a force sensor that independently detects stress drops associated with analogue earthquakes which triggers storage of a short high frequency image sequence from the buffer. After the event has passed, the system returns to a low speed mode in which image data is downsampled until the next event trigger.
Here we introduce the concept of smart speed imaging and document the necessary hard- and software. We test the algorithms in generic and real applications. A new analogue earthquake setup based on a modification of the Schulze ring-shear tester is used to verify the technique and discuss alternative trigger systems
Reactive SINDy: Discovering governing reactions from concentration data
Get PDFThe inner workings of a biological cell or a chemical reaction can be rationalized by the network of reactions, whose structure reveals the most important functional mechanisms. For complex systems, these reaction networks are not known a priori and cannot be efficiently computed with ab initio methods, therefore an important approach goal is to estimate effective reaction networks from observations, such as time series of the main species. Reaction networks estimated with standard machine learning techniques such as least-squares regression may fit the observations, but will typically contain spurious reactions. Here we extend the sparse identification of nonlinear dynamics
(SINDy) method to vector-valued ansatz functions, each describing a particular reaction process. The resulting sparse tensor regression method “reactive SINDy” is able to estimate a parsimonious reaction network. We illustrate that a gene regulation network can be correctly estimated from observed time series
Data-driven approaches to study the dynamical stability of the stably stratified boundary layer
Get PDFMany experimental or natural systems undergo critical transitions, i.e. sudden shifts from one dynamical regime to another. For example in the climate system, the lower
atmospheric layer, namely the atmospheric boundary layer, can experience sudden transitions between fully turbulent states and stable, quasi-laminar states. Such rapid transition are observed in Polar regions and at night, and have important consequences in the level of mixing with the higher levels of the atmosphere. To analyse the stable
boundary layer many approaches rely on the identification of regimes, i.e. weakly and very stable regimes. Therefore, it is crucial to detect the transitions between the regimes.
In this master thesis a combination of methods from dynamical systems and statistical modelling are applied to study these regime transitions. The analysis is based on an
indicator for the dynamical stability (i.e. the resilience to pertubations) and a conceptual model for regime transitions of near-surface temperature inversion at night as well as in Arctic conditions. A focus lies on bifurcation points in the dynamics, points in which the stability of the system changes drastically. The performance of the stability
indicator is assessed by applying it to simulated and observation data, provided from nighttime and Polar meteorological measurements
Mass-Dependent Solvent Friction of a Hydrophobic Molecule
No full textWe show by molecular dynamics simulations that the friction constant of a freely diffusing methane molecule in explicit water increases with the methane mass by a factor of up to 1.8 in the infinite mass limit compared to the massless limit. This effect is rationalized by the mass dependence of the friction memory kernel which is extracted from the simulation data by mapping on the generalized Langevin equation. On the basis of the mass-dependent memory kernels, we obtain perfect agreement between simulation results and analytic predictions for both mean-square displacements and force autocorrelation functions. The memory kernels, which account for methane interactions with the solvent, decay significantly slower with increasing methane mass. The mass-dependent friction is correlated with the mean escape time of water molecules from the first hydration shell, which increases monotonically with the methane mass by a factor of 3 over the mass range considered. Our proposed scaling expression allows the direct prediction of diffusion constants for stable methane isotopes, which will help to better understand bacterial methane isotope fractionation. Our scaling analysis suggests that mass-dependent friction will be relevant also for larger solutes in sufficiently viscous solvents
Stochastic-deterministic population balance modeling and simulation of a fluidized bed crystallizer experiment
No full textAbstract
The crystallization of potassium aluminum sulfate dodecahydrate (potash alum) in a fluidized bed crystallizer is studied both with experiments and simulations. A population balance system with three spatial coordinates and one internal coordinate (mass) is utilized as our model. The simulations are performed with a stochastic-deterministic method with novel extensions, where the fluid dynamics of the crystallizer (flow field, temperature, concentration) are solved deterministically and the particles are simulated with a stochastic method. In experiments of 30 min duration, the average crystal diameter increases by growth and agglomeration from about 130 μm to 210 μm. This observation agrees qualitatively well with our simulation results. A quantitative difference between simulation and experiment leaves room for future improvements in modeling
A quasi-Monte Carlo method for an optimal control problem under uncertainty
No full textWe study an optimal control problem under uncertainty, where the target function is the solution of an elliptic partial differential equation with random coefficients, steered by a control function. The robust formulation of the optimization problem is stated as a high-dimensional integration problem over the stochastic variables. It is well known that carrying out a high-dimensional numerical integration of this kind using a Monte Carlo method has a notoriously slow convergence rate; meanwhile, a faster rate of convergence can potentially be obtained by using sparse grid quadratures, but these lead to discretized systems that are non-convex due to the involvement of negative quadrature weights. In this paper, we analyze instead the application of a quasi-Monte Carlo method, which retains the desirable convexity structure of the system and has a faster convergence rate compared to ordinary Monte Carlo methods. In particular, we show that under moderate assumptions on the decay of the input random field, the error rate obtained by using a specially designed, randomly shifted rank-1 lattice quadrature rule is essentially inversely proportional to the number of quadrature nodes. The overall discretization error of the problem, consisting of the dimension truncation error, finite element discretization error and quasi-Monte Carlo quadrature error, is derived in detail. We assess the theoretical findings in numerical experiments
Giant Axial Dielectric Response in Water-Filled Nanotubes and Effective Electrostatic Ion–Ion Interactions from a Tensorial Dielectric Model
No full textMolecular dynamics simulations in conjunction with effective medium theory are used to investigate dielectric effects in water-filled nanotubes. The resulting effective axial dielectric constant shows a divergent increase for small nanotube radii that depends on the nanotube length, while the effective radial dielectric constant decreases significantly for thin nanotubes. By solving Poisson’s equation for an anisotropic dielectric medium in cylindrical geometry, we show that the axial ion–ion interaction depends for small separations primarily on the radial dielectric constant, not on the axial one. This means that electrostatic ion–ion interactions in thin water-filled nanotubes are on the linear dielectric level significantly enhanced due to water confinement effects at small separations, while at large separations the outside medium dominates. If the outside medium is metallic, then the ion–ion interaction decays exponentially for large ion separation