Thermodynamically Consistent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities

A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is also generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the two latter cases the classical Gibbs-Thomson equation has to be modified to include kinetic terms. Finally, we show that all sharp interface models fulfill natural energy inequalities.Comment: 34 page

Existence of positive solutions to stochastic thin-film equations

We construct martingale solutions to stochastic thin-film equations by introducing a (spatial) semidiscretization and establishing convergence. The discrete scheme allows for variants of the energy and entropy estimates in the continuous setting as long as the discrete energy does not exceed certain threshold values depending on the spatial grid size $h$. Using a stopping time argument to prolongate high-energy paths constant in time, arbitrary moments of coupled energy/entropy functionals can be controlled. Having established Hölder regularity of approximate solutions, the convergence proof is then based on compactness arguments---in particular on Jakubowski's generalization of Skorokhod's theorem---weak convergence methods, and recent tools on martingale convergence

Liquid crystal and phase-field models are related by mathematics

Ministerio de Economía y Competitivida

Evaluation strategies for isotope ratio measurements of single particles by LA-MC-ICPMS

Data evaluation is a crucial step when it comes to the determination of accurate and precise isotope ratios computed from transient signals measured by multi-collector–inductively coupled plasma mass spectrometry (MC-ICPMS) coupled to, for example, laser ablation (LA). In the present study, the applicability of different data evaluation strategies (i.e. ‘point-by-point’, ‘integration’ and ‘linear regression slope’ method) for the computation of (235)U/(238)U isotope ratios measured in single particles by LA-MC-ICPMS was investigated. The analyzed uranium oxide particles (i.e. 9073-01-B, CRM U010 and NUSIMEP-7 test samples), having sizes down to the sub-micrometre range, are certified with respect to their (235)U/(238)U isotopic signature, which enabled evaluation of the applied strategies with respect to precision and accuracy. The different strategies were also compared with respect to their expanded uncertainties. Even though the ‘point-by-point’ method proved to be superior, the other methods are advantageous, as they take weighted signal intensities into account. For the first time, the use of a ‘finite mixture model’ is presented for the determination of an unknown number of different U isotopic compositions of single particles present on the same planchet. The model uses an algorithm that determines the number of isotopic signatures by attributing individual data points to computed clusters. The (235)U/(238)U isotope ratios are then determined by means of the slopes of linear regressions estimated for each cluster. The model was successfully applied for the accurate determination of different (235)U/(238)U isotope ratios of particles deposited on the NUSIMEP-7 test samples. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1007/s00216-012-6674-3) contains supplementary material, which is available to authorized users

On A Fourth-Order Degenerate Parabolic Equation: Global Entropy Estimates, Existence, And Qualitative Behaviour Of Solutions

By means of energy and entropy estimates, we prove existence and positivity results in higher space dimensions for degenerate parabolic equations of fourth order with nonnegative initial values. We discuss their asymptotic behaviour for t !1 and give a counterexample to uniqueness. 0

Surrogate methods for spike pattern detection in non-Poisson data

In order to detect significant spatio-temporal spike patterns (STPs) at ms-precision, we developed the SPADE method[1-3]. SPADE enables the detection and evaluation of spatio-temporal patterns (STPs), i.e., spike patterns across neurons and with temporal delays. For the significance assessment of STPs, surrogates are generated to implement the null hypothesis. Here we demonstrate the requirements for appropriate surrogates.SPADE first discretizes the spike trains into bins of a few ms width. The discretization also includes clipping, i.e., if a bin is occupied by 1 or more spikes, its content is set to 1. The binarized spike trains are then mined for STPs with Frequent Itemset Mining, counting identical patterns. For the assessment of these patterns' significance, surrogata spike trains are used. The surrogate data are mined as the original data resulting in a p-value spectrum for the significance evaluation[3].Surrogate data are modifications of the original data where potential time-correlations are destroyed and thus implement the null hypothesis of independence. For that purpose, the surrogate data need to keep the statistical features of the original data as similar as possible to avoid false positives. A classical choice for a surrogate is uniform dithering (UD), which independently displaces each individual spike according to a uniform distribution. We show that UD makes the spike trains more Poisson-like and does not preserve a potential dead time after the spikes. As a consequence, more spikes are clipped away as compared to the original data. Thus, UD surrogate data reduce the expectation for the patterns.To overcome this problem, we evaluate different surrogate techniques. The first is a modification of UD that preserves the dead time. Further, we employ (joint-)ISI dithering, preserving the (joint-)ISI distribution[4]. Another surrogate is based on shuffling bins of already discretized spike data within a small window. Lastly, we evaluate trial shifting that shifts the whole spike trains against the others, trial by trial, according to a uniform distribution. To evaluate the effect of the different surrogate methods on significance assessment, we first analyze the surrogate modifications on different types of stochastic spike models, such as Poisson spike trains, Gamma spike trains but also Poisson spike trains with dead time[5]. We find that all surrogates but UD are robust to clipping. Trial shifting is the technique that preserves best the statistical features of the spike trains. Further, we analyze artificial data sets for the occurrence of false-positive patterns. These data sets were generated with non-stationary firing rates and interval statistics taken from an experimental data set but are otherwise independent. We find many false positives for UD but all other surrogates show a consistently low number of false-positive patterns. Based on these results, we conclude with a recommendation on which surrogate method to use.References1. Torre et al (2016) DOI:10.1523/JNEUROSCI.4375-15.2016.2. Quaglio et al. (2017). DOI:10.3389/fncom.2017.00041.3. Stella et al. (2019). DOI:10.1016/j.biosystems.2019.104022. 4. Gerstein (2004).5. Deger at al. (2011). DOI: 10.1007/s10827-011-0362-8