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We study an inhomogeneous sparse random graph, $${\mathcal G }_N$$ G N , on $$[N]=\{1,\dots ,N\}$$ [ N ] = { 1 , ⋯ , N } as introduced in a seminal paper by Bollobás et al. (Random Struct Algorithms 31(1):3–122, 2007): vertices have a type (here in a compact metric space $${\mathcal S }$$ S ), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit $$N\rightarrow \infty $$ N → ∞ , we consider the sparse regime, where the average degree is O(1). We prove a large-deviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size $$\asymp N$$ ≍ N ). In doing so, we derive explicit logarithmic asymptotics for the probability that $${\mathcal G }_N$$ G N is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of $${\mathcal G }_N$$ G N . In particular, we recover the criterion for the existence of the phase transition given in Bollobás et al. (2007).

Let $\mathcal{G}(N,\frac 1Nt_N)$ be the Erd\H{o}s-R\'enyi graph with connection probability $\frac 1Nt_N\sim t/N$ as $N\to\infty$ for a fixed $t\in(0,\infty)$. We derive a large-deviations principle for the empirical measure of the sizes of all the connected components of $\mathcal{G}(N,\frac 1Nt_N)$, registered according to microscopic sizes (i.e., of finite order), macroscopic ones (i.e., of order $N$), and mesoscopic ones (everything in between). The rate function explicitly describes the microscopic and macroscopic components and the fraction of vertices in components of mesoscopic sizes. Moreover, it clearly captures the well known phase transition at $t=1$ as part of a comprehensive picture. The proofs rely on elementary combinatorics and on known estimates and asymptotics for the probability that subgraphs are connected. We also draw conclusions for the strongly related model of the multiplicative coalescent, the Marcus--Lushnikov coagulation model with monodisperse initial condition, and its gelation phase transition.

We 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 Borel $$\sigma $$ -algebras induced by these topologies, and we provide some results about probability measures on the space of functions of bounded variation, which can be used to study stochastic processes of bounded variation.

We 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.

In this work, a framework for modelling twin-screw granulation processes with variable screw configurations using a high-dimensional stochastic population balance method is presented. A modular compartmental approach is presented and a method for estimating residence times for model compartments based on screw element geometry is introduced. The model includes particle mechanisms for nucleation, primary particle layering, coalescence, breakage, and consolidation. A new twin-screw breakage model is introduced, which takes into account the differing breakage dynamics between two types of screw element. Additionally, a new sub-model for the layering of primary particles onto larger agglomerates is presented. The resulting model is used to simulate a twin-screw system with a number of different screw configurations and the predictive power of the model is assessed through comparison with an existing experimental data set in the literature. For most of the screw configurations simulated, the model predicts the product particle size distribution at large particle sizes with a reasonable degree of accuracy. However, the model has a tendency to over-predict the amount of fines in the final product. Nevertheless, the model qualitatively captures the reduction in fines associated with an increase in the number of kneading elements, as observed experimentally. Based on model results, a number of key areas for future model development are identified and discussed. National Research Foundation (NRF) Accepted version ADM acknowledge funding from EPSRC Grant 1486478 and AstraZeneca. ADM would also like to thank C.T. Lao for useful conversations regarding the implementation of the nucleation process. RP acknowledges support from the Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 ‘‘Scaling Cascades in Complex Systems”, Project C08. This project was partly funded by the National Research Foundation (NRF), Prime Minister’s Office, Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE) programme.

A detailed population balance model is used to perform a parametric sensitivity study on the computed optical band gap (OBG) of polycyclic aromatic hydrocarbons (PAHs) in a laminar co-flow ethylene diffusion flame. Assuming that the OBG of PAHs in soot can be estimated by the energy difference between the highest occupied and lowest unoccupied molecular orbitals in single PAHs, OBG may be correlated with the number of aromatic rings in PAHs. This allows insights into which are the key species involved in the formation of soot. PAH size distributions are computed along the centreline and in the wings of the flame. We compare our simulations with experimentally determined soot volume fraction and OBG (derived from extinction measurements) from the literature. It is shown that the model predicts reasonably well the soot volume fraction and OBG throughout the flame. We find that the computed OBG is most sensitive to the size of the smallest PAH which is assumed to contribute to the OBG. The best results are obtained accounting for PAH contribution in both gas and particle phases assuming a minimum size of ovalene (10 rings). This suggests that the extinction measurements show a significant absorption by PAHs in the gas phase at the visible wavelength that is used, which has been demonstrated by experiments in the literature. It is further shown that PAH size distributions along the centreline and in the wings are unimodal at larger heights above burner. Despite the different soot particle histories and residence times in the flame, the PAH size associated with both modes are similar which is consistent with the near-constant OBG that is observed experimentally. The simulations indicate that the transition from the gas phase to soot particles begins with PAHs with as few as 16 aromatic rings, which is consistent with experimental observations reported in the literature.

A coupled solver for population balance systems is presented, where the flow, temperature, and concentration equations are solved with finite element methods, and the particle size distribution is simulated with a stochastic simulation algorithm, a so-called kinetic Monte-Carlo method. This novel approach is applied for the simulation of an axisymmetric model of a tubular flow crystallizer. The numerical results are compared with experimental data.

We consider a stochastic particle model governed by an arbitrary binary interaction kernel. A kinetic equation for the distribution of interaction clusters is established. Under some additional assumptions a recursive representation of the solution is found. For particular choices of the interaction kernel (including the Boltzmann case) several explicit formulas are obtained. These formulas are confirmed by numerical experiments. The experiments are also used to illustrate various conjectures and open problems.

In this work we present the new PAH-PP soot model and use a data collaboration approach to determine some of its parameters. The model describes the formation, growth and oxidation of soot in laminar premixed flames. Soot particles are modelled as aggregates containing primary particles, which are built from polycyclic aromatic hydrocarbons (PAHs), the main building blocks of a primary particle (PP). The connectivity of the primary particles is stored and used to determine the rounding of the soot particles due to surface growth and condensation processes. Two neighbouring primary particles are replaced by one if the coalescence level between the two primary particles reaches a threshold. The model contains, like most of the other models, free parameters that are unknown a priori. The experimental premixed flame data from Zhao et al. [B. Zhao, Z. Yang, Z. Li, M.V. Johnston, H. Wang, Proc. Combust. Inst. 30 (2) (2005) 1441–1448] have been used to estimate the smoothing factor of soot particles, the growth factor of PAHs within particles and the soot density using a low discrepancy series method with a subsequent response surface optimisation. The optimised particle size distributions show good agreement with the experimental ones. The importance of a standardised data mining system in order to optimise models is underlined.

A detailed PAH growth model is developed, which is solved using a kinetic Monte Carlo algorithm. The model describes the structure and growth of planar PAH molecules, and is referred to as the kinetic Monte Carlo–aromatic site (KMC-ARS) model. A detailed PAH growth mechanism based on reactions at radical sites available in the literature, and additional reactions obtained from quantum chemistry calculations are used to model the PAH growth processes. New rates for the reactions involved in the cyclodehydrogenation process for the formation of 6-member rings on PAHs are calculated in this work based on density functional theory simulations. The KMC-ARS model is validated by comparing experimentally observed ensembles on PAHs with the computed ensembles for a C2H2 and a C6H6 flame at different heights above the burner. The motivation for this model is the development of a detailed soot particle population balance model which describes the evolution of an ensemble of soot particles based on their PAH structure. However, at present incorporating such a detailed model into a population balance is computationally unfeasible. Therefore, a simpler model referred to as the site-counting model has been developed, which replaces the structural information of the PAH molecules by their functional groups augmented with statistical closure expressions. This closure is obtained from the KMC-ARS model, which is used to develop correlations and statistics in different flame environments which describe such PAH structural information. These correlations and statistics are implemented in the site-counting model, and results from the site-counting model and the KMC-ARS model are in good agreement. Additionally the effect of steric hindrance in large PAH structures is investigated and correlations for sites unavailable for reaction are presented.

A new, advanced soot particle model is developed that describes soot particles by their aromatic structure, including functional site descriptions and a detailed surface chemistry mechanism. A methodology is presented for the description of polyaromatic hydrocarbon (PAH) structures by their functional sites. The model is based on statistics that describe aromatic structural information in the form of easily computed correlations, which were generated using a kinetic Monte Carlo algorithm to study the growth of single PAH molecules. A comprehensive surface reaction mechanism is presented to describe the growth and desorption of aromatic rings on PAHs. The model is capable of simulating whole particle ensembles which allows bulk properties such as soot volume fraction and number density to be found, as well as joint particle size and surface area distributions. The model is compared to the literature-standard soot model [J. Appel, H. Bockhorn, M. Frenklach, Combust. Flame 121 (2000) 122–136] in a plug-flow reactor and is shown to predict well the experimental results of soot mass, average particle size, and particle size distributions at different flow times. Finally, the carbon/hydrogen ratio and the distribution of average PAH sizes in the ensemble, as predicted by the model, are discussed.

Brief communication on the effect of preferential locations for surface reactions on flame-generated aggregate particles

We study large deviation principles for a model of wireless networks consisting of Poisson point processes of transmitters and receivers. To each transmitter we associate a family of connectable receivers whose signal-to-interference-and-noise ratio is larger than a certain connectivity threshold. First, we show a large deviation principle for the empirical measure of connectable receivers associated with transmitters in large boxes. Second, making use of the observation that the receivers connectable to the origin form a Cox point process, we derive a large deviation principle for the rescaled process of these receivers as the connection threshold tends to 0. Finally, we show how these results can be used to develop importance sampling algorithms that substantially reduce the variance for the estimation of probabilities of certain rare events such as users being unable to connect.

In a Newtonian system with localized interactions the whole set of particles is naturally decomposed into dynamical clusters, defined as finite groups of particles having an influence on each other’s trajectory during a given interval of time. For an ideal gas with short-range intermolecular force, we provide a description of the cluster size distribution in terms of the reduced Boltzmann density. In the simplified context of Maxwell molecules, we show that a macroscopic fraction of the gas forms a giant component in finite kinetic time. The critical index of this phase transition is in agreement with previous numerical results on the elastic billiard.

New bivariate models for soot particle structure are introduced to qualitatively replicate observed particle shapes and are found to offer quantitative improvements over older single-variable models. Models for the development of particle shape during surface growth and for particle collision diameters are described and implemented along with two models taken from other published work. Using a stochastic approach, bivariate soot particle distributions are calculated for the first time. Distributions calculated for the new models are found to be insensitive to the collision diameter model used for coagulation. The total mass of soot produced in a laminar premixed flame is found to vary by no more than 20% as the model for the geometric effects of chemical reactions on the surface of particles is changed. Histories of individual particles are analyzed and show the limitations of collector particle techniques for predicting the evolution of aggregate shape descriptors.

The feasibility of coupling a stochastic soot algorithm to a deterministic gas-phase chemistry solver is investigated for homogeneous combusting systems. A second-order splitting technique was used to decouple the particle population and gas phase in order to solve. A numerical convergence study is presented that demonstrates convergence with splitting step size and particle count for a batch reactor and a perfectly stirred reactor. Simulation results are presented alongside experimental data for a plug flow reactor (PFR) and are compared to a method of moments simulation of a perfectly stirred reactor. Coupling of the soot and chemistry solvers is shown to converge for both systems; however, numerical instabilities present significant challenges in the PSR case. Comparison with the experimental data for a PFR showed good agreement of the soot mass and reasonable agreement of the particle size distribution. Two different soot particle models were used to simulate the PFR: a spherical particle model and a surface–volume model that takes some account of particle shape. The results for the two models are compared. Additionally, the stochastic soot solver is used to track the evolution of the C/H ratio of individual soot particles in the PFR for the first time.

In this paper, the prediction of a soot model [J. Appel, H. Bockhorn, M. Frenklach, Combust. Flame 121 (2000) 122–136] is compared to a recently published set of highly detailed soot particle size distributions [B. Zhao, Z. Yang, Z. Li, M.V. Johnston, H. Wang, Proc. Combust. Inst. 30 (2005)]. A stochastic approach is used to obtain soot particle size distributions (PSDs). The key features of the measured and simulated particle size distributions are identified and used as a simple way of comparing PSDs. The sensitivity of the soot PSDs to the parameters defining parts of the soot model, such as soot inception, particle and PAH collision efficiency and enhancement, and surface activity is investigated. Incepting soot particle size is found to have a very significant effect on the small-size end of the PSDs, especially the position of the trough for a bimodal soot PSDs. A new model for the decay in the surface activity is proposed in which the activity of the soot particle depends only on the history of that particle and the local temperature in the flame. This is a first attempt to use local flame variables to define the surface aging which has major impact on the prediction of the large-size end of the PSDs. Using these modifications to the soot model it is possible to improve the agreement between some of the points of interest in the simulated and measured PSDs. The paper achieves the task to help advance the soot models to predict soot PSD in addition to soot volume fraction and number density, which has been the focus of the literature.

We consider a single-cell network of random transmitters and fixed relays in a bounded domain of Euclidean space. The transmitters arrive over time and select one relay according to a spatially inhomogeneous preference kernel. Once a transmitter is connected to a relay, the connection remains and the relay is occupied. If an occupied relay is selected by another transmitters with later arrival time, this transmitter becomes frustrated. We derive a large deviation principle for the space-time evolution of frustrated transmitters in the high-density regime., 24 pages, 3 figures

We analyze a model of relay-augmented cellular wireless networks. The network users, who move according to a general mobility model based on a Poisson point process of continuous trajectories in a bounded domain, try to communicate with a base station located at the origin. Messages can be sent either directly or indirectly by relaying over a second user. We show that in a scenario of an increasing number of users, the probability that an atypically high number of users experiences bad quality of service over a certain amount of time, decays at an exponential speed. This speed is characterized via a constrained entropy minimization problem. Further, we provide simulation results indicating that solutions of this problem are potentially non-unique due to symmetry breaking. Also two general sources for bad quality of service can be detected, which we refer to as isolation and screening., Comment: 28 pages, 5 figures; corrected several misprints

The convergence of stochastic particle systems representing physical advection, inflow, outflow and coagulation is considered. The problem is studied on a bounded spatial domain such that there is a general upper bound on the residence time of a particle. The laws on the appropriate Skorohod path space of the empirical measures of the particle systems are shown to be relatively compact. The paths charged by the limits are characterised as solutions of a weak equation restricted to functions taking the value zero on the outflow boundary. The limit points of the empirical measures are shown to have densities with respect to Lebesgue measure when projected on to physical position space. In the case of a discrete particle type space a strong form of the Smoluchowski coagulation equation with a delocalised coagulation interaction and an inflow boundary condition is derived. As the spatial discretisation is refined in the limit equations, the delocalised coagulation term reduces to the standard local Smoluchowski interaction.

A spatially resolved stochastic weighted particle method for inception--coagulation--advection problems is presented. Convergence to a deterministic limit is briefly studied. Numerical experiments are carried out for two problems with very different coagulation kernels. These tests show the method to be robust and confirm the convergence properties. The robustness of the weighted particle method is shown to contrast with two direct simulation algorithms which develop instabilities.

A Comment on the Letter by Rajan K. Chakrabarty et al., Phys. Rev. Lett. 102, 235504 (2009). The authors of the Letter offer a Reply.

und Metzner-Otto sind erklarbar durch die Turbulenzmodellierung fur nichtnewtonsche Fluide. Zusatzlich zu dieser Messung wurden noch in einer weiteren Anlage Messungen durchgefuhrt. Die unterschiedlichen Viskositaten (Gleichung 1) in den Anlagen sind durch den Einsatz verschiedener Substrate bedingt. Aus dem beigefugten Diagramm lassen sich die Koeffizienten der Fliesfunktion mit folgenden Werten bestimmen: n = 0,132, K = 18,67, fur die hier nicht gezeigte Anlage n = –1,205, K = 6785,8. Das Ergebnis fur K im zweiten Fall ist deutlich hoher als erwartet. Hier muss durch eine Messung bei niedrigeren Scherraten die Regression bei kleinen Drehzahlen weiter verbessert werden. Die schlechtere Regression im Diagramm wird von der insgesamt sehr inhomogenen Zusammensetzung des Substrates hervorgerufen.

We study fundamental characteristics for the connectivity of multi-hop D2D networks. Devices are randomly distributed on street systems and are able to communicate with each other whenever their separation is smaller than some connectivity threshold. We model the street systems as Poisson-Voronoi or Poisson-Delaunay tessellations with varying street lengths. We interpret the existence of adequate D2D connectivity as percolation of the underlying random graph. We derive and compare approximations for the critical device-intensity for percolation, the percolation probability and the graph distance. Our results show that for urban areas, the Poisson Boolean Model gives a very good approximation, while for rural areas, the percolation probability stays far from 1 even far above the percolation threshold., Comment: 6 pages, 7 figures, 1 table