Bei der mathematischen Modellierung vieler Vorgänge und Phänomene in Natur und Technik werden Systeme mit vielen zufälligen Teilchen und Wechselwirkungen eingesetzt; siehe das Anwendungsthema Partikelbasierte Modellierung in den Naturwissenschaften. Hierbei sprechen wir von vielen unterschiedlichen Typen von solchen Systemen, die, je nach Anwendung, bewegte oder statische Teilchen beinhalten, Interaktionen mit einander oder mit einem umgebenden (evtl. zufälligen) Medium haben, anziehende oder abstoßende Interaktionen haben etc. Auch die mathematischen Fragen, die man an das System hat, sind sehr unterschiedlich, etwa nach einer sich ausprägenden Clusterstruktur, nach Regelmäßigkeiten innerhalb der Cluster, nach Eigenschaften wie Perkolation, Kristallisation und Kondensation, nach zeitlicher Entwicklung der Kontakte der Teilchen mit einander, nach Abhängigkeiten des globalen Verhaltens von Parametern bis hin zu Phasenübergängen und mehr.

In den meisten Modellen gibt es einen oder mehrere Ordnungsparameter, an Hand derer man das makroskopische Verhalten gut beschreiben kann, wie z.B. die durchschnittliche Größe eines Clusters, die Aufenthaltswahrscheinlichkeiten der Partikel an einem gegebenen Ort zu einem gegebenen Zeitpunkt, das empirische Mittel aller Partikel etc. In manchen Fällen erfüllen diese Ordnungsparameter im betrachteten Grenzwert (meist große Anzahl oder große Zeiten) gewisse deterministische Gleichungen (etwa Differentialgleichungen), die anschließend mit analytischen oder numerischen Methoden studiert werden. In anderen Fällen werden Formeln für die freie Energie hergeleitet oder für Gesetze der großen Zahlen oder für Ergodensätze.

Zur Beantwortung dieser Fragen und Findung und Herleitung von Formeln werden am WIAS in der Regel mathematische Methoden eingesetzt sowie (wenn es diese noch nicht gibt) entwickelt. Dabei werden stochastische und analytische Methoden kombiniert und im besten Fall eine mathematische Lösung hergeleitet. Begleitende Simulationen geben dann eine Visualisierung und produzieren explizite Daten. Beispiele für eingesetzte Gebiete sind, je nach Modell, die Theorien der Gibbsmaße, der Perkolation, der stochastischen (partiellen) Differentialgleichungen, der Markovprozesse und der Großen Abweichungen auf der stochastischen Seite und die Theorien der schwachen Konvergenz, der Variationsrechnung, der konvexen Analysis und der partiellen Differentialgleichungen auf der analytischen Seite.

Beitrag des Instituts

Hier sind ein paar mathematische Errungenschaften des WIAS aus den letzten Jahren; siehe auch das Anwendungsthema Partikelbasierte Modellierung in den Naturwissenschaften.

Ein (statisches) interagierendes Vielkörpersystem ist gegeben, wenn eine große Anzahl von Punkten in einer großen Box zufällig verteilt wird, so dass sich die Teilchen nicht häufen und so dass eine gewisse Energie der Gesamtkonfiguration ein exponentielles Wahrscheinlichkeitsgewicht gibt, so dass ein Wahrscheinlichkeitsmaß auf den Konfigurationen gegeben wird. Der Energieterm trägt einen Vorfaktor, der als der Kehrwert der Temperatur interpretiert wird. In der FG5 wurde der thermodynamische Grenzwert bei tiefen Temperaturen in besonders großen Boxen studiert, so dass die Entropie (die Anteil der Wahrscheinlichkeit, der von der räumlichen Struktur der Partikelwolke herrührt) mit der Energie eine besondere Relation eingeht, so dass dieser Grenzwert mit Hilfe genau eines Parameters studiert werden kann. Interessante Phasenübergänge konnten erhalten werden. Für die Zukunft soll allerdings der übliche thermodynamische Grenzwert bei festem Verhältnis der Boxgröße zur Partikelzahl studiert werden. Für ein System, in dem jedem Teilchen eine kinetische Energie hinzugefügt wird und die Partikel einer gewissen Symmetrie unterworfen werden (charakteristisch für die Beschreibung von Bosonen), wurde ein Ansatz mit großen Abweichungen für gemittelte Verschiebungen des gesamten Partikelsystems entwickelt und dazu eingesetzt, die freie Energie des Systems in einer Variationsformel zu fassen. Damit konnte allerdings der ersehnte Effekt einer Kondensation nicht bewiesen werden, was Gegenstand weiterer Untersuchungen sein wird.

In der Telekommunikation werden die Aufenthaltsorte vieler Nutzer als die Punkte eines räumlichen Poissonprozesses modelliert, siehe das Anwendungsthema Mobile Kommunikationsnetzwerke. Je zwei von diesen Punkten interagieren, wenn ihr Abstand klein genug ist. Sutdiert wurde dir Frage, mit welcher Wahrscheinlichkeit Nachrichten, die über ein Relaissystem durch das System geschickt werden, auch wirklich ihr Ziel ereichen, d.h. eine globale Konnektivitätseigenschaft des Systems.


Jeder Punkt liegt in der Mitte eines Kreises, und Punkte kommunizieren, falls sich ihre Kreise überlappen. Eine sehr große, durchgehend verbundene Komponente wird in Grün dargestellt, dort ist die globale Konnektivität gegeben, kleinere, isolierte Teile des Netzes sind blau.

Die Konnektivitätseigenschaft wurde im Grenzwert hoher Dichten von Nutzern pro Volumen betrachtet, in welchem die betrachteten Wahrscheinlichkeiten exponentiell abfallen. Mit Hilfe der Theorie der großen Abweichungen für hochdichte Punktprozesse wurde die Abfallrate in Termen einer Entropie ausgedrückt. Anschließend wurden diejenigen Konfigurationen untersucht, die diese Entropie minimieren, denn diese haben die Interpretation der (zufälligen) Situationen, in denen die Konnektivität noch am besten ist unter den gegebenen Annahmen. Ähnliche Untersuchungen wurden für Interferenz- und Kapazitätseigenschaften gemacht. Weitergehende Untersuchungen betreffen derzeit die optimale Wahl der Trajektorien der Nachrichten bei Beachtung von Interferenz und bei Vermeidung von Überlastungen der Relais sowie die Implementierung realistischer Bewegungsmodelle für die Nutzer.

Bei dynamischen Modellen ist die Etablierung von hydrodynamischen Limes eine wichtige Aufgabe. Solche Limes werden zunächst bewiesen in konkreten Anwendungsfälle um Evolutions-Gleichungen für makroskopischen Eigenschaften des Models zu finden. Siehe die anwendungsorientierten Themen Koagulation sowie Partikelbasierte Modellierung in den Naturwissenschaften. Ein wesentlicher Teil derartigen Beweise ist die Kompaktheit in Verteilung von den Markov Prozessen, die das Model bilden. In einigen Arbeiten sind Eigenschaften der anwendungsbezogenen Probleme abstrahiert worden und die Ergebnisse erweitert, damit sie in weiteren Fällen zur Anwendung kommen können.

In der Biologie ist die Definition von geeigneten stochastischen Partikelmodellen noch lange nicht abgeschlossen, die Modellierung ist in stetem Fluss. Etablierte Modelle für Populationen und deren Bewegungen sind zum Beispiel räumliche Verzweigungsprozesse mit zufälliger Bewegung, die am WIAS in zufälliger Umgebung betrachtet werden; siehe das Mathematische Thema Spektraltheorie zufälliger Operatoren.


Publikationen

  Monografien

  • B. Jahnel, W. König, Probabilistic Methods in Telecommunications, D. Mazlum, ed., Compact Textbooks in Mathematics, Birkhäuser Basel, 2020, XI, 200 pages, (Monograph Published), DOI 10.1007/978-3-030-36090-0 .
    Abstract
    This textbook series presents concise introductions to current topics in mathematics and mainly addresses advanced undergraduates and master students. The concept is to offer small books covering subject matter equivalent to 2- or 3-hour lectures or seminars which are also suitable for self-study. The books provide students and teachers with new perspectives and novel approaches. They may feature examples and exercises to illustrate key concepts and applications of the theoretical contents. The series also includes textbooks specifically speaking to the needs of students from other disciplines such as physics, computer science, engineering, life sciences, finance.

  • W. König, Große Abweichungen, Techniken und Anwendungen, M. Brokate, A. Heinze , K.-H. Hoffmann , M. Kang , G. Götz , M. Kerz , S. Otmar, eds., Mathematik Kompakt, Birkhäuser Basel, 2020, VIII, 167 pages, (Monograph Published), DOI 10.1007/978-3-030-52778-5 .
    Abstract
    Die Lehrbuchreihe Mathematik Kompakt ist eine Reaktion auf die Umstellung der Diplomstudiengänge in Mathematik zu Bachelor- und Masterabschlüssen. Inhaltlich werden unter Berücksichtigung der neuen Studienstrukturen die aktuellen Entwicklungen des Faches aufgegriffen und kompakt dargestellt. Die modular aufgebaute Reihe richtet sich an Dozenten und ihre Studierenden in Bachelor- und Masterstudiengängen und alle, die einen kompakten Einstieg in aktuelle Themenfelder der Mathematik suchen. Zahlreiche Beispiele und Übungsaufgaben stehen zur Verfügung, um die Anwendung der Inhalte zu veranschaulichen. Kompakt: relevantes Wissen auf 150 Seiten Lernen leicht gemacht: Beispiele und Übungsaufgaben veranschaulichen die Anwendung der Inhalte Praktisch für Dozenten: jeder Band dient als Vorlage für eine 2-stündige Lehrveranstaltung

  Artikel in Referierten Journalen

  • N. Engler, B. Jahnel, Ch. Külske, Gibbsianness of locally thinned random fields, Markov Processes and Related Fields, 28 (2022), pp. 185--214, DOI 10.48550/arXiv.2201.02651 .
    Abstract
    We consider the locally thinned Bernoulli field on ℤ d, which is the lattice version of the Type-I Matérn hardcore process in Euclidean space. It is given as the lattice field of occupation variables, obtained as image of an i.i.d. Bernoulli lattice field with occupation probability p, under the map which removes all particles with neighbors, while keeping the isolated particles. We prove that the thinned measure has a Gibbsian representation and provide control on its quasilocal dependence, both in the regime of small p, but also in the regime of large p, where the thinning transformation changes the Bernoulli measure drastically. Our methods rely on Dobrushin uniqueness criteria, disagreement percolation arguments [46], and cluster expansions

  • B. Jahnel, Ch. Hirsch, E. Cali, Percolation and connection times in multi-scale dynamic networks, Stochastic Processes and their Applications, 151 (2022), pp. 490--518, DOI 10.1016/j.spa.2022.06.008 .
    Abstract
    We study the effects of mobility on two crucial characteristics in multi-scale dynamic networks: percolation and connection times. Our analysis provides insights into the question, to what extent long-time averages are well-approximated by the expected values of the corresponding quantities, i.e., the percolation and connection probabilities. In particular, we show that in multi-scale models, strong random effects may persist in the limit. Depending on the precise model choice, these may take the form of a spatial birth-death process or a Brownian motion. Despite the variety of structures that appear in the limit, we show that they can be tackled in a common framework with the potential to be applicable more generally in order to identify limits in dynamic spatial network models going beyond the examples considered in the present work.

  • A. Stephan, EDP-convergence for a linear reaction-diffusion system with fast reversible reaction, Calculus of Variations and Partial Differential Equations, 60 (2021), pp. 226/1--226/35, DOI 10.1007/s00526-021-02089-0 .
    Abstract
    We perform a fast-reaction limit for a linear reaction-diffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reaction-diffusion system as a gradient flow of the free energy in the space of probability measures equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and cosh-type functions for the reaction part. The fast-reaction limit is done on the level of the gradient structure by proving EDP-convergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slow-manifold. Moreover, the limit gradient system can be equivalently described by a coarse-grained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarse-grained slow variable.

  • S. Jansen, W. König, B. Schmidt, F. Theil, Distribution of cracks in a chain of atoms at low temperature, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 22 (2021), pp. 4131--4172, DOI 10.1007/s00023-021-01076-7 .
    Abstract
    We consider a one-dimensional classical many-body system with interaction potential of Lennard--Jones type in the thermodynamic limit at low temperature 1/β ∈ (0, ∞). The ground state is a periodic lattice. We show that when the density is strictly smaller than the density of the ground state lattice, the system with N particles fills space by alternating approximately crystalline domains (clusters) with empty domains (voids) due to cracked bonds. The number of domains is of the order of N exp(-β e surf /2) with e surf > 0 a surface energy.

  • L. Andreis, W. König, R.I.A. Patterson, A large-deviations principle for all the cluster sizes of a sparse Erdős--Rényi random graph, Random Structures and Algorithms, 59 (2021), pp. 522--553, DOI 10.1002/rsa.21007 .
    Abstract
    A large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t=1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdős-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erdős-Rényi graphs are connected.

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Phase transitions for chase-escape models on Poisson--Gilbert graphs, Electronic Communications in Probability, 25 (2020), pp. 25/1--25/14, DOI 10.1214/20-ECP306 .
    Abstract
    We present results on phase transitions of local and global survival in a two-species model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuous-time nearest-neighbor interacting particle system. The Gilbert graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.

  • S. Jansen, W. König, B. Schmidt, F. Theil, Surface energy and boundary layers for a chain of atoms at low temperature, Archive for Rational Mechanics and Analysis, 239 (2021), pp. 915--980 (published online on 21.12.2020), DOI 10.1007/s00205-020-01587-3 .
    Abstract
    We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature goes to zero. Our main results are: (1) As the temperature goes to zero and at fixed positive pressure, the Gibbs measures  for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of the surface energy functional. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in the inverse temperature.

  • J.--D. Deuschel, T. Orenshtein, Scaling limit of wetting models in 1+1 dimensions pinned to a shrinking strip, Stochastic Processes and their Applications, 130 (2020), pp. 2778--2807, DOI 10.1016/j.spa.2019.08.001 .

  • CH. Hirsch, B. Jahnel, A. Tóbiás, Lower large deviations for geometric functionals, Electronic Communications in Probability, 25 (2020), pp. 41/1--41/12, DOI 10.1214/20-ECP322 .
    Abstract
    This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson--Voronoi cells, as well as power-weighted edge lengths in the random geometric, κ-nearest neighbor and relative neighborhood graph.

  • J. Maas, A. Mielke, Modeling of chemical reaction systems with detailed balance using gradient structures, Journal of Statistical Physics, 181 (2020), pp. 2257--2303, DOI 10.1007/s10955-020-02663-4 .
    Abstract
    We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reaction-rate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailed-balance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradient-flow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailed-balance steady state. The limit of large volumes is studied in the sense of evolutionary Γ-convergence of gradient flows. Moreover, we use the gradient structures to derive hybrid models for coupling different modeling levels.

  • A. Tóbiás, B. Jahnel, Exponential moments for planar tessellations, Journal of Statistical Physics, 179 (2020), pp. 90--109, DOI 10.1007/s10955-020-02521-3 .
    Abstract
    In this paper we show existence of all exponential moments for the total edge length in a unit disc for a family of planar tessellations based on Poisson point processes. Apart from classical such tessellations like the Poisson--Voronoi, Poisson--Delaunay and Poisson line tessellation, we also treat the Johnson--Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.

  • A. Mielke, A. Stephan, Coarse-graining via EDP-convergence for linear fast-slow reaction systems, Mathematical Models & Methods in Applied Sciences, 30 (2020), pp. 1765--1807, DOI 10.1142/S0218202520500360 .
    Abstract
    We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.

  • CH. Hirsch, B. Jahnel, Large deviations for the capacity in dynamic spatial relay networks, Markov Processes and Related Fields, 25 (2019), pp. 33--73.
    Abstract
    We derive a large deviation principle for the space-time evolution of users in a relay network that are unable to connect due to capacity constraints. The users are distributed according to a Poisson point process with increasing intensity in a bounded domain, whereas the relays are positioned deterministically with given limiting density. The preceding work on capacity for relay networks by the authors describes the highly simplified setting where users can only enter but not leave the system. In the present manuscript we study the more realistic situation where users leave the system after a random transmission time. For this we extend the point process techniques developed in the preceding work thereby showing that they are not limited to settings with strong monotonicity properties.

  • L. Andreis, A. Asselah, P. Dai Pra , Ergodicity of a system of interacting random walks with asymmetric interaction, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 55 (2019), pp. 590--606.
    Abstract
    We study N interacting random walks on the positive integers. Each particle has drift delta towards infinity, a reflection at the origin, and a drift towards particles with lower positions. This inhomogeneous mean field system is shown to be ergodic only when the interaction is strong enough. We focus on this latter regime, and point out the effect of piles of particles, a phenomenon absent in models of interacting diffusion in continuous space.

  • L. Andreis, P. Dai Pra, M. Fischer, McKean--Vlasov limit for interacting systems with simultaneous jumps, Stochastic Analysis and Applications, 36 (2018), pp. 960--995, DOI 10.1080/07362994.2018.1486202 .
    Abstract
    Motivated by several applications, including neuronal models, we consider the McKean-Vlasov limit for mean-field systems of interacting diffusions with simultaneous jumps. We prove propagation of chaos via a coupling technique that involves an intermediate process and that gives a rate of convergence for the W1 Wasserstein distance between the empirical measures of the two systems on the space of trajectories D([0,T],R^d).

  • O. Gün, A. Yilmaz, The stochastic encounter-mating model, Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications, 148 (2017), pp. 71--102.

  • A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM Journal on Applied Mathematics, 77 (2017), pp. 1562--1585, DOI 10.1137/16M1102240 .
    Abstract
    We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.

  • R.I.A. Patterson, S. Simonella, W. Wagner, A kinetic equation for the distribution of interaction clusters in rarefied gases, Journal of Statistical Physics, 169 (2017), pp. 126--167.

  • J. Blath, A. González Casanova Soberón, B. Eldon, N. Kurt, M. Wilke-Berenguer, Genetic variability under the seedbank coalescent, Genetics, 200 (2015), pp. 921--934.
    Abstract
    We analyze patterns of genetic variability of populations in the presence of a large seedbank with the help of a new coalescent structure called the seedbank coalescent. This ancestral process appears naturally as a scaling limit of the genealogy of large populations that sustain seedbanks, if the seedbank size and individual dormancy times are of the same order as those of the active population. Mutations appear as Poisson processes on the active lineages and potentially at reduced rate also on the dormant lineages. The presence of "dormant" lineages leads to qualitatively altered times to the most recent common ancestor and nonclassical patterns of genetic diversity. To illustrate this we provide a Wright-Fisher model with a seedbank component and mutation, motivated from recent models of microbial dormancy, whose genealogy can be described by the seedbank coalescent. Based on our coalescent model, we derive recursions for the expectation and variance of the time to most recent common ancestor, number of segregating sites, pairwise differences, and singletons. Estimates (obtained by simulations) of the distributions of commonly employed distance statistics, in the presence and absence of a seedbank, are compared. The effect of a seedbank on the expected site-frequency spectrum is also investigated using simulations. Our results indicate that the presence of a large seedbank considerably alters the distribution of some distance statistics, as well as the site-frequency spectrum. Thus, one should be able to detect from genetic data the presence of a large seedbank in natural populations.

  Beiträge zu Sammelwerken

  • W. König, Branching random walks in random environment, in: Probabilistic Structures in Evolution, E. Baake, A. Wakolbinger, eds., Probabilistic Structures in Evolution, EMS Series of Congress Reports, European Mathematical Society Publishing House, 2021, pp. 23--41, DOI 10.4171/ECR/17-1/2 .

  • B. Jahnel, W. König, Probabilistic methods for spatial multihop communication systems, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 239--268.

  Preprints, Reports, Technical Reports

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Connectivity in mobile device-to-device networks in urban environments, Preprint no. 2952, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2952 .
    Abstract, PDF (606 kByte)
    In this article we setup a dynamic device-to-device communication system where devices, given as a Poisson point process, move in an environment, given by a street system of random planar-tessellation type, via a random-waypoint model. Every device independently picks a target location on the street system using a general waypoint kernel, and travels to the target along the shortest path on the streets with an individual velocity. Then, any pair of devices becomes connected whenever they are on the same street in sufficiently close proximity, for a sufficiently long time. After presenting some general properties of the multi-parameter system, we focus on an analysis of the clustering behavior of the random connectivity graph. In our main results we isolate regimes for the almost-sure absence of percolation if, for example, the device intensity is too small, or the connectivity time is too large. On the other hand, we exhibit parameter regimes of sufficiently large intensities of devices, under favorable choices of the other parameters, such that percolation is possible with positive probability. Most interestingly, we also show an in-and-out of percolation as the velocity increases. The rigorous analysis of the system mainly rests on comparison arguments with simplified models via spatial coarse graining and thinning approaches. Here we also make contact to geostatistical percolation models with infinite-range dependencies.

  • B. Jahnel, S.K. Jhawar, A.D. Vu, Continuum percolation in a nonstabilizing environment, Preprint no. 2943, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2943 .
    Abstract, PDF (2463 kByte)
    We prove nontrivial phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson--Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox--Boolean model. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [MR2116736].

  • B. Jahnel, J. Köppl, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, Preprint no. 2935, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2935 .
    Abstract, PDF (355 kByte)
    We consider irreversible translation-invariant interacting particle systems on the d-dimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a time-stationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translation-invariant measure implies, that the measure is Gibbs w.r.t. the same specification as the time-stationary Gibbs measure. As an application, we obtain the attractor property for irreversible interacting particle systems, which says that any weak limit point of any trajectory of translation-invariant measures is a Gibbs measure w.r.t. the same specification as the time-stationary measure. This extends previously known results to fairly general irreversible interacting particle systems.

  • A. Stephan, H. Stephan, Positivity and polynomial decay of energies for square-field operators, Preprint no. 2901, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2901 .
    Abstract, PDF (328 kByte)
    We show that for a general Markov generator the associated square-field (or carré du champs) operator and all their iterations are positive. The proof is based on an interpolation between the operators involving the generator and their semigroups, and an interplay between positivity and convexity on Banach lattices. Positivity of the square-field operators allows to define a hierarchy of quadratic and positive energy functionals which decay to zero along solutions of the corresponding evolution equation. Assuming that the Markov generator satisfies an operator-theoretic normality condition, the sequence of energies is log-convex. In particular, this implies polynomial decay in time for the energy functionals along solutions.

  • L. Andreis, W. König, H. Langhammer, R.I.A. Patterson, A large-deviations principle for all the components in a sparse inhomogeneous random graph, Preprint no. 2898, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2898 .
    Abstract, PDF (690 kByte)
    We study an inhomogeneous sparse random graph, GN, on [N] = { 1,...,N } as introduced in a seminal paper [BJR07] by Bollobás, Janson and Riordan (2007): vertices have a type (here in a compact metric space 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 → ∞ , 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 ≈ N). In doing so, we derive explicit logarithmic asymptotics for the probability that GN 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 GN. In particular, we recover the criterion for the existence of the phase transition given in [BJR07].

  • A. Stephan, Coarse-graining and reconstruction for Markov matrices, Preprint no. 2891, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2891 .
    Abstract, PDF (248 kByte)
    We present a coarse-graining (or model order reduction) procedure for stochastic matrices by clustering. The method is consistent with the natural structure of Markov theory, preserving positivity and mass, and does not rely on any tools from Hilbert space theory. The reconstruction is provided by a generalized Penrose-Moore inverse of the coarse-graining operator incorporating the inhomogeneous invariant measure of the Markov matrix. As we show, the method provides coarse-graining and reconstruction also on the level of tensor spaces, which is consistent with the notion of an incidence matrix and quotient graphs, and, moreover, allows to coarse-grain and reconstruct fluxes. Furthermore, we investigate the connection with functional inequalities and Poincaré-type constants.

  • M. Heida, B. Jahnel, A.D. Vu, Stochastic homogenization on irregularly perforated domains, Preprint no. 2880, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2880 .
    Abstract, PDF (668 kByte)
    We study stochastic homogenization of a quasilinear parabolic PDE with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies on the underlying geometry that does not allow standard homogenization techniques to be applied directly. Instead we prove homogenization on a regularized geometry and demonstrate afterwards that the form of the homogenized equation is independent from the regularization. Then we pass to the regularization limit to obtain the anticipated limit equation. Furthermore, we show that Boolean models of Poisson point processes are covered by our approach.

  • B. Jahnel, Ch. Külske, Gibbsianness and non-Gibbsianness for Bernoulli lattice fields under removal of isolated sites, Preprint no. 2878, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2878 .
    Abstract, PDF (426 kByte)
    We consider the i.i.d. Bernoulli field μ p on Z d with occupation density p ∈ [0,1]. To each realization of the set of occupied sites we apply a thinning map that removes all occupied sites that are isolated in graph distance. We show that, while this map seems non-invasive for large p, as it changes only a small fraction p(1-p)2d of sites, there is p(d) <1 such that for all p ∈ (p(d), 1) the resulting measure is a non-Gibbsian measure, i.e., it does not possess a continuous version of its finite-volume conditional probabilities. On the other hand, for small p, the Gibbs property is preserved.

  • R.I.A. Patterson, D.R.M. Renger, U. Sharma, Variational structures beyond gradient flows: A macroscopic fluctuation-theory perspective, Preprint no. 2826, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2826 .
    Abstract, PDF (522 kByte)
    Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract framework, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems -- independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models.

  • N. Nüsken, D.R.M. Renger, Stein variational gradient descent: Many-particle and long-time asymptotics, Preprint no. 2819, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2819 .
    Abstract, PDF (430 kByte)
    Stein variational gradient descent (SVGD) refers to a class of methods for Bayesian inference based on interacting particle systems. In this paper, we consider the originally proposed deterministic dynamics as well as a stochastic variant, each of which represent one of the two main paradigms in Bayesian computational statistics: emphvariational inference and emphMarkov chain Monte Carlo. As it turns out, these are tightly linked through a correspondence between gradient flow structures and large-deviation principles rooted in statistical physics. To expose this relationship, we develop the cotangent space construction for the Stein geometry, prove its basic properties, and determine the large-deviation functional governing the many-particle limit for the empirical measure. Moreover, we identify the emphStein-Fisher information (or emphkernelised Stein discrepancy) as its leading order contribution in the long-time and many-particle regime in the sense of $Gamma$-convergence, shedding some light on the finite-particle properties of SVGD. Finally, we establish a comparison principle between the Stein-Fisher information and RKHS-norms that might be of independent interest.

  • W. König, Branching random walks in random environment: A survey, Preprint no. 2779, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2779 .
    Abstract, PDF (253 kByte)
    We consider branching particle processes on discrete structures like the hypercube in a random fitness landscape (i.e., random branching/killing rates). The main question is about the location where the main part of the population sits at a late time, if the state space is large. For answering this, we take the expectation with respect to the migration (mutation) and the branching/killing (selection) mechanisms, for fixed rates. This is intimately connected with the parabolic Anderson model, the heat equation with random potential, a model that is of interest in mathematical physics because of the observed prominent effect of intermittency (local concentration of the mass of the solution in small islands). We present several advances in the investigation of this effect, also related to questions inspired from biology.

  • D. Peschka, M. Rosenau, Two-phase flows for sedimentation of suspensions, Preprint no. 2743, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2743 .
    Abstract, PDF (12 MByte)
    We present a two-phase flow model that arises from energetic-variational arguments and study its implication for the sedimentation of buoyant particles in a viscous fluid inside a Hele--Shaw cell and also compare corresponding simulation results to experiments. Based on a minimal dissipation argument, we provide a simplified 1D model applicable to sedimentation and study its properties and the numerical discretization. We also explore different aspects of its numerical discretization in 2D. The focus is on different possible stabilization techniques and their impact on the qualitative behavior of solutions. We use experimental data to verify some first qualitative model predictions and discuss these experiments for different stages of batch sedimentation.

  • B. Jahnel, A. Tóbiás, E. Cali, Phase transitions for the Boolean model of continuum percolation for Cox point processes, Preprint no. 2704, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2704 .
    Abstract, PDF (389 kByte)
    We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.

  • A. Hinsen, B. Jahnel, E. Cali, J.-P. Wary, Malware propagation in urban D2D networks, Preprint no. 2674, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2674 .
    Abstract, PDF (3133 kByte)
    We introduce and analyze models for the propagation of malware in pure D2D networks given via stationary Cox--Gilbert graphs. Here, the devices form a Poisson point process with random intensity measure λ, Λ where Λ is stationary and given, for example, by the edge-length measure of a realization of a Poisson--Voronoi tessellation that represents an urban street system. We assume that, at initial time, a typical device at the center of the network carries a malware and starts to infect neighboring devices after random waiting times. Here we focus on Markovian models, where the waiting times are exponential random variables, and non-Markovian models, where the waiting times feature strictly positive minimal and finite maximal waiting times. We present numerical results for the speed of propagation depending on the system parameters. In a second step, we introduce and analyze a counter measure for the malware propagation given by special devices called white knights, which have the ability, once attacked, to eliminate the malware from infected devices and turn them into white knights. Based on simulations, we isolate parameter regimes in which the malware survives or is eliminated, both in the Markovian and non-Markovian setting.

  Vorträge, Poster

  • A. Stephan, Gradient systems and EDP-convergence with applications to nonlinear fast-slow reaction systems (online talk), DS21: SIAM Conference on Applications of Dynamical Systems, Minisymposium 19 ``Applications of Stochastic Reaction Networks'' (Online Event), May 23 - 27, 2021, Society for Industrial and Applied Mathematics, May 23, 2021.

  • A. Stephan, Gradient systems and mulit-scale reaction networks (online talk), Limits and Control of Stochastic Reaction Networks (Online Event), July 26 - 30, 2021, American Institute of Mathematics, San Jose, USA, July 29, 2021.

  • E. Magnanini, Limit theorems for the edge density in exponential random graphs, Workshop ``Junior Female Researchers in Probability'', October 4 - 6, 2021, Stochastic Analysis in Interaction. Berlin--Oxford IRTG 2544, October 5, 2021.

  • E. Magnanini, Limit theorems for the edge density in exponential random graphs, Probability Seminar, Università degli Studi di Firenze, Dipartimento di Matematica e Informatica ``Ulisse Dini'', Italy, November 17, 2021.

  • B. Jahnel, Connectivity improvements in mobile device-to-device networks (online talk), Telecom Orange Paris, France, July 6, 2021.

  • B. Jahnel, First-passage percolation and chase-escape dynamics on random geometric graphs, Stochastic Geometry Days, November 15 - 19, 2021, Dunkerque, France, November 17, 2021.

  • B. Jahnel, Gibbsian representation for point processes via hyperedge potentials (online talk), Thematic Einstein Semester on Geometric and Topological Structure of Materials, Summer Semester 2021, Technische Universität Berlin, May 20, 2021.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes, Workshop on Randomness Unleashed Geometry, Topology, and Data, September 22 - 24, 2021, University of Groningen, Faculty of Science and Engineering, Groningen, Netherlands, September 23, 2021.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes (online talk), DYOGENE Seminar (Online Event), INRIA Paris, France, January 11, 2021.

  • B. Jahnel, Phase transitions for the Boolean model for Cox point processes (online talk), Probability Seminar Bath (Online Event), University of Bath, Department of Mathematical Sciences, UK, October 18, 2021.

  • W. König, A grid version of the interacting Bose gas (online talk), Probability Seminar (Online Event), University of Bath, Department of Mathematical Sciences, UK, February 15, 2021.

  • W. König, Cluster size distributions in a classical many-body system (online talk), Thematic Einstein Semester on Geometric and Topological Structure of Materials, Summer Semester 2021, Technische Universität Berlin, May 20, 2021.

  • W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability Meeting (Online Event), University of Oxford, Department of Statistics, UK, February 10, 2021.

  • W. van Zuijlen, Large time behaviour of the parabolic Anderson model (online talk), Probability and Statistical Physics Seminar (Online Event), The University of Chicago, Department of Mathematics, Statistics, and Computer Science, USA, February 12, 2021.

  • T. Orenshtein, Aging for the O'Conell--Yor model in intermediate disorder (online talk), Joint Israeli Probability Seminar (Online Event), Technion, Haifa, November 17, 2020.

  • T. Orenshtein, Aging for the stationary KPZ equation, The 3rd Haifa Probability School. Workshop on Random Geometry and Stochastic Analysis, February 24 - 28, 2020, Technion Israel Institute of Technology, Haifa, February 24, 2020.

  • T. Orenshtein, Aging for the stationary KPZ equation (online talk), Bernoulli-IMS One World Symposium 2020 (Online Event), August 24 - 28, 2020, A virtual one week symposium on Probability and Mathematical Statistics, August 27, 2020.

  • T. Orenshtein, Aging for the stationary KPZ equation (online talk), 13th Annual ERC Berlin--Oxford Young Researchers Meeting on Applied Stochastic Analysis (Online Event), June 8 - 10, 2020, WIAS Berlin, June 10, 2020.

  • T. Orenshtein, Aging in Edwards--Wilkinson and KPZ universality classes (online talk), Probability, Stochastic Analysis and Statistics Seminar (Online Event), University of Pisa, Italy, October 27, 2020.

  • A. Stephan, Coarse-graining via EDP-convergence for linear fast-slow reaction systems, Seminar ``Applied Analysis'', Eindhoven University of Technology, Centre for Analysis, Scientific Computing, and Applications -- Mathematics and Computer Science, Netherlands, January 20, 2020.

  • A. Stephan, On mathematical coarse-graining for linear reaction systems, 8th BMS Student Conference, February 19 - 21, 2020, Technische Universität Berlin, February 21, 2020.

  • A. Stephan, On gradient flows and gradient systems (online talk), CRC 1114 PhD Seminar (Online Event), Freie Universität Berlin, November 11, 2020.

  • A. Stephan, On gradient systems and applications to interacting particle systems (online talk), CRC 1114 PhD Seminar (Online Event), Freie Universität Berlin, November 25, 2020.

  • L. Andreis, A large deviations approach to sparse random graphs (online talk), Bernoulli--IMS One World Symposium 2020 (Online Event), August 24 - 28, 2020, A virtual one week symposium on Probability and Mathematical Statistics. Bernoulli-IMS One World Symposium 2020, August 25, 2020.

  • L. Andreis, Sparse inhomogeneous random graphs from a large deviation point of view (online talk), Probability Seminar (Online Event), University of Bath, Department of Mathematical Sciences, UK, June 1, 2020.

  • L. Andreis, The phase transition in random graphs and coagulation processes: A large deviation approach (online talk), Seminar of DISMA (Online Event), Politecnico di Torino, Department of Mathematical Sciences (DISMA), Italy, July 14, 2020.

  • A. Mielke, EDP-convergence for multiscale gradient systems with applications to fast-slow reaction systems (online talk), One World Dynamics Seminar (Online Event), Technische Universität München, November 13, 2020.

  • A. Hinsen, Data mobility in ad-hoc networks: Vulnerability and security, KEIN öffentlicher Vortrag (Orange), Telecom Orange Paris, France, December 12, 2019.

  • A. Stephan, Rigorous derivation of the effective equation of a linear reaction system with different time scales, 90th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2019), Section S14 ``Applied Analysis'', February 18 - 22, 2019, Universität Wien, Technische Universität Wien, Austria, February 21, 2019.

  • B. Jahnel, Continuum percolation in random environment, Workshop on Probability, Analysis and Applications (PAA), September 23 - October 4, 2019, African Institute for Mathematical Sciences --- Ghana (AIMS Ghana), Accra.

  • R.I.A. Patterson, Fluctuations and confidence intervals for stochastic particle simulations, First Berlin--Leipzig Workshop on Fluctuating Hydrodynamics, August 26 - 30, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, August 29, 2019.

  • R.I.A. Patterson, Flux large deviations, Workshop on Chemical Reaction Networks, July 1 - 3, 2019, Politecnico di Torino, Dipartimento di Scienze Matematiche ``G. L. Lagrange``, Italy, July 2, 2019.

  • R.I.A. Patterson, Flux large deviations, Seminar, Statistical Laboratory, University of Cambridge, Faculty of Mathematics, UK, May 7, 2019.

  • R.I.A. Patterson, Interaction clusters for the Kac process, Berlin--Leipzig Workshop in Analysis and Stochastics, January 16 - 18, 2019, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, January 18, 2019.

  • R.I.A. Patterson, Interaction clusters for the Kac process, Workshop on Effective Equations: Frontiers in Classical and Quantum Systems, June 24 - 28, 2019, Hausdorff Research Institute for Mathematics, Bonn, June 28, 2019.

  • R.I.A. Patterson, Kinetic interaction clusters, Oberseminar, Martin-Luther-Universität Halle-Wittenberg, Naturwissenschaftliche Fakultät II -- Chemie, Physik und Mathematik, April 17, 2019.

  • R.I.A. Patterson, The role of fluctuating hydrodynamics in the CRC 1114, CRC 1114 School 2019: Fluctuating Hydrodynamics, Zuse Institute Berlin (ZIB), October 28, 2019.

  • L. Taggi, Critical density in activated random walks, Horowitz Seminar on Probability, Ergodic Theory and Dynamical Systems, Tel Aviv University, School of Mathematical Sciences, Israel, May 20, 2019.

  • M. Maurelli, McKean--Vlasov SDEs with irregular drift: Large deviations for particle approximation, University of Oxford, Mathematical Institute, UK, March 5, 2018.

  • M. Maurelli , A McKean--Vlasov SDE with reflecting boundaries, CASA Colloquium, Eindhoven University of Technology, Department of Mathematics and Computer Science, Netherlands, January 10, 2018.

  • L. Andreis, A large-deviations approach to the multiplicative coagulation process, Probability Seminar, Università degli Studi di Padova, Dipartimento di Matematica ``Tullio Levi--Civita'', Italy, October 12, 2018.

  • L. Andreis, A large-deviations approach to the multiplicative coagulation process, Seminar ''Theory of Complex Systems and Neurophysics --- Theory of Statistical Physics and Nonlinear Dynamics``, Humboldt-Universität zu Berlin, Institut für Physik, October 30, 2018.

  • L. Andreis, Ergodicity of a system of interacting random walks with asymmetric interaction, 13th German Probability and Statistics Days 2018 -- Freiburger Stochastik-Tage, February 27 - March 2, 2018, Albert-Ludwigs-Universität Freiburg, Abteilung für Mathematische Stochastik, Freiburg, February 1, 2018.

  • L. Andreis, Networks of interacting components with macroscopic self-sustained periodic behavior, Neural Coding 2018, September 9 - 14, 2018, University of Torino, Department of Mathematics, Italy, September 10, 2018.

  • L. Andreis, Self-sustained periodic behavior in interacting systems, Random Structures in Neuroscience and Biology, March 26 - 29, 2018, Ludwig--Maximilians Universität München, Fakultät für Mathematik, Informatik und Statistik, Herrsching, March 26, 2018.

  • L. Andreis, System of interacting random walks with asymmetric interaction, 48th Probability Summer School, July 8 - 20, 2018, Clermont Auvergne University, Saint Flour, France, July 17, 2018.

  • W. Dreyer, Thermodynamics and kinetic theory of non-Newtonian fluids, Technische Universität Darmstadt, Mathematische Modellierung und Analysis, June 13, 2018.

  • W. Dreyer, J. Fuhrmann, P. Gajewski, C. Guhlke, M. Landstorfer, M. Maurelli, R. Müller, Stochastic model for LiFePO4-electrodes, ModVal14 -- 14th Symposium on Fuel Cell and Battery Modeling and Experimental Validation, Karlsruhe, March 2 - 3, 2017.

  • D.R.M. Renger, From large deviations to Wasserstein gradient flows in multiple dimensions, Workshop on Gradient Flows, Large Deviations and Applications, November 22 - 29, 2015, EURANDOM, Mathematics and Computer Science Department, Eindhoven, Netherlands, November 23, 2015.

  • D.R.M. Renger, The inverse problem: From gradient flows to large deviations, Workshop ``Analytic Approaches to Scaling Limits for Random System'', January 26 - 30, 2015, Universität Bonn, Hausdorff Research Institute for Mathematics, January 26, 2015.

  Preprints im Fremdverlag

  • A. Bianchi, F. Collet, E. Magnanini, Limit theorems for exponential random graphs, Preprint no. arXiv:2105.06312, Cornell University Library, arXiv.org, 2021.
    Abstract
    We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together with a complete characterization of the phase diagram of the model. We borrow tools from statistical mechanics to obtain limit theorems for the edge density. First, we determine the asymptotic distribution of this quantity, as the graph size tends to infinity, in the various phases. Then we study the fluctuations of the edge density around its average value off the critical curve and formulate conjectures about the behavior at criticality based on the analysis of a mean-field approximation of the model. Some of our results can be extended with no substantial changes to more general classes of exponential random graphs

  • D. Heydecker , R.I.A. Patterson, Kac interaction clusters: A bilinear coagulation equation and phase transition, Preprint no. arXiv:1902.07686, Cornell University Library, 2019.
    Abstract
    We consider the interaction clusters for Kac's model of a gas with quadratic interaction rates, and show that they behave as coagulating particles with a bilinear coagulation kernel. In the large particle number limit the distribution of the interaction cluster sizes is shown to follow an equation of Smoluchowski type. Using a coupling to random graphs, we analyse the limiting equation, showing well-posedness, and a closed form for the time of the gelation phase transition tg when a macroscopic cluster suddenly emerges. We further prove that the second moment of the cluster size distribution diverges exactly at tg. Our methods apply immediately to coagulating particle systems with other bilinear coagulation kernels.

  • A. González Casanova Soberón, J.C. Pardo, J.L. Perez, Branching processes with interactions: The subcritical cooperative regime, Preprint no. arXiv:1704.04203, Cornell University Library, arXiv.org, 2017.
    Abstract
    In this paper, we introduce a particular family of processes with values on the nonnegative integers that model the dynamics of populations where individuals are allow to have different types of inter- actions. The types of interactions that we consider include pairwise: competition, annihilation and cooperation; and interaction among several individuals that can be consider as catastrophes. We call such families of processes branching processes with interactions. In particular, we prove that a process in this class has a moment dual which turns out to be a jump-diffusion that can be thought as the evolution of the frequency of a trait or phenotype. The aim of this paper is to study the long term behaviour of branching processes with interac- tions under the assumption that the cooperation parameter satisfies a given condition that we called subcritical cooperative regime. The moment duality property is useful for our purposes.

  • J. Blath, E. Buzzoni, A. Casanova Soberón, M.W. Berenguer, The seed bank diffusion, and its relation to the two-island model, Preprint no. arXiv:1710.08164, Cornell University Library, arXiv.org, 2017.
    Abstract
    In this paper, we introduce a particular family of processes with values on the nonnegative integers that model the dynamics of populations where individuals are allow to have different types of inter- actions. The types of interactions that we consider include pairwise: competition, annihilation and cooperation; and interaction among several individuals that can be consider as catastrophes. We call such families of processes branching processes with interactions. In particular, we prove that a process in this class has a moment dual which turns out to be a jump-diffusion that can be thought as the evolution of the frequency of a trait or phenotype. The aim of this paper is to study the long term behaviour of branching processes with interac- tions under the assumption that the cooperation parameter satisfies a given condition that we called subcritical cooperative regime. The moment duality property is useful for our purposes.

  • K.F. Lee, M. Dosta, A.D. Mcguire, S. Mosbach, W. Wagner, S. Heinrich, M. Kraft, Development of a multi-compartment population balance model for high-shear wet granulation with Discrete Element Method, Technical report no. 170, c4e-Preprint Series, 2016.
    Abstract
    This paper presents a multi-compartment population balance model for wet granulation coupled with DEM (Discrete Element Method) simulations. Methodologies are developed to extract relevant data from the DEM simulations to inform the population balance model. First, compartmental residence times are calculated for the population balance model from DEM. Then, a suitable collision kernel is chosen for the population balance model based on particle-particle collision frequencies extracted from DEM. It is found t hat the population balance model is able to predict the trends exhibited by the experimental size and porosity distributions by utilising the information provided by the DEM simulations.