Mathematical Theory of Exchange-driven Growth

Exchange-driven growth is a process in which pairs of clusters interact and exchange a single unit of mass. The rate of exchange is given by an interaction kernel $K(j,k)$ which depends on the masses of the two interacting clusters. In this paper we establish the fundamental mathematical properties of the mean field kinetic equations of this process for the first time. We find two different classes of behaviour depending on whether $K(j,k)$ is symmetric or not. For the non-symmetric case, we prove global existence and uniqueness of solutions for kernels satisfying $K(j,k)\leq Cjk$. This result is optimal in the sense that we show for a large class of initial conditions with kernels satisfying $K(j,k)\geq Cj^{\beta}$ ($\beta>1)$ the solutions cannot exist. On the other hand, for symmetric kernels, we prove global existence of solutions for $K(j,k)\leq C(j^{\mu}k^{\nu}+j^{\nu}k^{\mu})$ ($\mu,\nu\leq2,$ $\mu+\nu\leq3),$ while existence is lost for $K(j,k)\geq Cj^{\beta}$ ($\beta>2).$ In the intermediate regime $3<\mu+\nu\leq4,$ we can only show local existence. We conjecture that the intermediate regime exhibits finite-time gelation in accordance with the heuristic results obtained for particular kernels.Comment: Mistakes in the uniqueness proofs are fixed. Some typos are corrected. Some references are adde

The role of zero-clusters in exchange-driven growth with and without input

The exchange-driven growth model describes the mean field kinetics of a population of composite particles (clusters) subject to pairwise exchange interactions. Exchange in this context means that upon interaction of two clusters, one loses a constituent unit (monomer) and the other gains this unit. Two variants of the exchange-driven growth model appear in applications. They differ in whether clusters of zero size are considered active or passive. In the active case, clusters of size zero can acquire a monomer from clusters of positive size. In the passive case they cannot, meaning that clusters reaching size zero are effectively removed from the system. The large time behaviour is very different for the two variants of the model. We first consider an isolated system. In the passive case, the cluster size distribution tends towards a self-similar evolution and the typical cluster size grows as a power of time. In the active case, we identify a broad class of kernels for which the the cluster size distribution tends to a non-trivial time-independent equilibrium in which the typical cluster size is finite. We next consider a non-isolated system in which monomers are input at a constant rate. In the passive case, the cluster size distribution again attains a self-similar profile in which the typical cluster size grows as a power of time. In the active case, a surprising new behavior is found: the cluster size distribution asymptotes to the same equilibrium profile found in the isolated case but with an amplitude that grows linearly in time

Generalized Phase Field Models with Microscopic Potentials

In this thesis we study the solidification process of systems with intrinsicanisotropy. We aim at finding a bridge between the microscopic mechanismsand macroscopic description. This is achieved by generalizing the currentphase field models in a way to incorporate microscopic physics and usingasymptotic techniques to obtain macroscopic results. Upon analysis,expressions for physically relevant quantities are obtained. Also it isfound that classical interface relations for both stationary and movinginterfaces hold. These conditions are presented in various representations.Exemplary numerical calculations are carried out to illustrate the potentialof the method as an additional tool in the study of interfaces. Furthermore,a concrete physical system with realistic parameters is considered to showhow one can use the ideas developed here in order to get results that are ofinterest to other scientific communities, e.g. materials scientists andphysicists

Large time behaviour of exchange-driven growth

Exchange-driven growth (EDG) is a model in which pairs of clusters interact by exchanging single unit with a rate given by a kernel K(j,k). Despite EDG model's common use in the applied sciences, its rigorous mathematical treatment is very recent. In this article we study the large time behaviour of EDG equations. We show two sets of results depending on the properties of the kernel (i) K(j,k)=bjak and (ii) K(j,k)=jak+bj+εβjαk. For type I kernels, under the detailed balance assumption, we show that the system admits unique equilibrium up to a critical mass ρs above which there is no equilibrium. We prove that if the system has an initial mass below ρs then the solutions converge to a unique equilibrium distribution strongly where if the initial mass is above ρs then the solutions converge to cricital equilibrium distribution in a weak sense. For type II kernels, we do not make any assumption of detailed balance and equilibrium is shown as a consequence of contraction properties of solutions. We provide two separate results depending on the monotonicity of the kernel or smallness of the total mass. For the first case we prove exponential convergence in the number of clusters norm and for the second we prove exponential convergence in the total mass norm

A Comparison of Playfulness Levels of the Secondary Students in Kayseri and Trabzon Cities

The aim of the study is to draw a comparison of 11-14 age group secondary school students playfulness level to play games which consist of physical activity. It also aims to evaluate playfulness level in terms of various parameters. For this purpose, 11-14 age group students’ playfulness level were examined according to cities, gender, and ages. A total of 523 11- 14 year-old students, including 303 from Kayseri and 220 from Trabzon, and some various schools formed a study group for the research conducted in the relational model. “Playfulness scale for playing games that consist of physical activity of children of 10-14 age group” that was developed by Hazar (2014) was used in the data collection. Based on the research, the level of playfulness of the participants was found to be "good" at 3.40-4.19. Students living in Kayseri and Trabzon were found to have "good" in terms of social adaptation, wish to play game, wish to win, and in general. Students in Kayseri "good" level had a playfulness at "medium" level in Trabzon. Due to the comparison of male and female student scores, it was found that there was a significant difference in favor of male students in game play overall, game passion, and risk taking dimensions. Girls are more likely to have social adaptation, playfulness, and a meaningful difference generally. Also, girls who live in Kayseri are more likely to be students. They wish to play game, social adaptation, wish to win, and risk taking. Meaningful differences can however be seen based on the scores of the 11 and 12 year students

Analyzing real-world accidents for test scenario generation for automated vehicles

Identification of test scenarios for Automated Driving Systems (ADSs) remains a key challenge for the Verification & Validation of ADSs. Various approaches including data based approaches and knowledge based approaches have been proposed for scenario generation. Identifying the conditions that lead to high severity traffic accidents can help us not only identify test scenarios for ADSs, but also implement measures to save lives and infrastructure resources. Taking a data based approach, in this paper, we introduce a novel accident data analysis method for generating test scenarios where we analyze UK’s Stats19 accident data to identify trends in high severity accidents for test scenario generation. This paper first focuses on the severity of the accidents with the goal of relating it to static and time-dependent internal and external factors in a comprehensive way taking into account Operational Design Domain (ODD) properties, e.g. road, environmental conditions, and vehicle properties and driver characteristics. For this purpose, the paper utilizes a data grouping strategy (coarse-graining) and builds a logistic regression approach, derived from conventional regression models, in which emerging features become more pronounced, while uninteresting features and noise weaken. The approach makes the relationship between the factors and outcome variable more visible and hence well suited for the severity analysis. The method shows superior performance as compared to ordinary logistic models measured by goodness of fit and accounting for model variance (R2=0.05 for the ordinary model, R2=0.85 for the current model). The model is then used to solve the inverse problem of constructing high-risk pre-crash conditions as test scenarios for simulation based testing of ADSs

Identification of traffic accident patterns via cluster analysis and test scenario development for autonomous vehicles

Increased safety is one of the main motivations for traffic research and planning. The arduous task has two components: (i) improving the existing traffic policies based on a good understanding of risk factors related to trends in traffic accidents, and (ii) underpinning the emerging technologies that will advance the safety of vehicles. For the latter route, the introduction of connected and automated vehicles (CAVs) is a promising option as CAVs can potentially reduce the number of accidents. However, to reap their benefits, they need to be introduced in a safe manner and tested for their ability to safely deal with risky scenarios. Unfortunately, the identification of such test scenarios remains a key challenge for the industry. This study contributes to increased safety by (i) analyzing UK’s STATS19 accident data to identify patterns in past traffic accidents, and (ii) utilizing this information to systematically generate scenarios for CAV testing. For task (i), the patterns in the accidents were identified in terms of static and time-dependent internal and external factors. For this purpose, the study employed a clustering algorithm, COOLCAT, which is particularly suitable for dealing with high-dimensional categorical data. Six different clusters emerged naturally as a result of the algorithm. To interpret the clusters, we applied a frequency analysis to each cluster. The frequency tests showed that in each cluster, certain distinct real-world situations were represented more significantly compared to the non-clustered reference case, which are the markers of each cluster. The second task (ii) complemented the first task by synthesizing the relationships between attributes. This was done by association rule mining using the market basket analysis approach. The method enabled us to develop, drawing from the characteristics of the clusters, non-trivial test scenarios that can be used in the testing of CAVs, especially in virtual testing

Laser textured surface gradients

This work demonstrates a novel technique for fabricating surfaces with roughness and wettability gradients and their subsequent applications for chemical sensors. Surface roughness gradients on brass sheets are obtained directly by nanosecond laser texturing. When these structured surfaces are exposed to air, their wettability decreases with time (up to 20 days) achieving both spatial and temporal wettability gradients. The surfaces are responsive to organic solvents. Contact angles of a series of dilute isopropanol solutions decay exponentially with concentration. In particular, a fall of 132° in contact angle is observed on a surface gradient, one order of magnitude higher than the 14° observed for the unprocessed surface, when the isopropanol concentration increased from 0 to 15.6 wt%. As the wettability changes gradually over the surface, contact angle also changes correspondingly. This effect offers multi-sensitivity at different zones on the surface and is useful for accurate measurement of chemical concentration

Laser textured superhydrophobic surfaces and their applications for homogeneous spot deposition

This work reports the laser surface modification of 304S15 stainless steel to develop superhydrophobic properties and the subsequent application for homogeneous spot deposition. Superhydrophobic surfaces, with steady contact angle of ∼154° and contact angle hysteresis of ∼4°, are fabricated by direct laser texturing. In comparison with common pico-/femto-second lasers employed for this patterning, the nanosecond fiber laser used in this work is more cost-effective, compact and allows higher processing rates. The effect of laser power and scan line separation on surface wettability of textured surfaces are investigated and optimized fabrication parameters are given. Fluid flows and transportations of polystyrene (PS) nanoparticles suspension droplets on the processed surfaces and unprocessed wetting substrates are investigated. After evaporation is complete, the coffee-stain effect is observed on the untextured substrates but not on the superhydrophobic surfaces. Uniform deposition of PS particles on the laser textured surfaces is achieved and the deposited material is confined to smaller area