For more than a hundred years diverse processes and phenomena in the natural sciences have been modeled using random particle systems. Since the 19th century, many scientists have been modeling things such as liquids, gases, light and [solid] materials as a huge number of particles with mutual interactions. A stochastic approach is often made, i.e. the particles are subject to stochastic rules regarding their locations or their movements. Other ingredients in the models can be random environments or stochasticity in the interactions. We distinguish dynamic models in which the particles move randomly (e.g. interacting diffusions), and static models in which they do not move (e.g. Gibbs point processes)
A main task is to study the macroscopic behavior of the system and to explain it mathematically and possibly bring it into coincidence with experimental data. In many cases, the task is to develop methods or to made existing methods applicable for the description of the important order parameters and for the proof of their crucial properties. A typical example of an important order parameter is the empirical average of the particles or the whole trajectories. In the hydrodynamic limit (many particles in a fixed box) it can be described approximately by a single equation, usually a differential equation. In the thermodynamic limit (many particles in a large box with fixed positive density) often become exponential asymptotes with the help of variational formulas described and then variational equations for the description of the minimizers of the formulas. Also, the probabilities of the Particles after averaging can create interesting equations suffice. Some such equations have been studied long before they were derived from particle models. >
Particle models have become especially widespread in physics and chemistry as a good compromise between reality and tractability. For example, static, atomic many body systems are often described through an energy function, which assigns every possible configuration an energy based on the interaction between the particles and then interprets the negative exponential of the energy as proportional to the configuration probability. Such distributions are called Gibbs measures; they preferentially select configurations with low energies. An example is a salt crystal, which consists of charged particles (ions) seeking to minimize their combined electrostatic potential energy. Other systems, especially at positive temperatures contain random walks or Brownian motions, which react (e.g. coagulate) with each other when in close proximity (see the applied theme Coagulation). In this way we model, for example, the formation of soot particles in flames. A related class of stochastic particle models are families of interacting stochastic (partial) differential equations, which have recently been used in the modeling of battery charging (see the applied theme Thermodynamic models for electrochemical systems).
At the WIAS, special attention is paid to phase transitions, i.e., to the phenomenon that the behavior changes significantly when an order parameter exceeds a certain threshold. These are mostly transitions from the form of the sudden occurrence of macroscopic structures. The most important ones are percolation (e.g. in random graphs), condensation (e.g. in the Bose gas), crystallization (e.g. in atomistic point processes) and gelation (in spatial particle models with coagulation). The relevant order parameter is usually the temperature and/or the particle density.
>Particle system methods have also received tremendous attention in research in optimization and machine learning, which is the focus of the Weierstrass Group DOC. For example, modern computational algorithms often deal with big data sets in potentially high dimensions. Optimization with such data is not only computationally costly but also subject to local optima. One approach to theoretically analyzing these computational algorithms is to view them as particle gradient descent ? modeling the computational algorithms as dynamical particle systems. In this direction, we are interested in working with particle systems under the framework of optimal transport theory and gradient flow. The goal is to use the modeling insight for largescale computation, especially for robust machine learning algorithms and deep generative models.
Contribution of the Institute
Atomic, static models for interacting many body systems are described with LennardJones potentials, which cause the particles to maintain a certain amount of separation and not to collapse onto a single point. Another example is the Bosegas in which every particle has a kinetic energy in addition to its position. The work of the WIAS on the first model deals with the formation of clusters and crystallization, and for the Bosegas with condensation phenomena; see the mathematical theme Large Deviations
A realisation of a many body system showing a small crystal in the lower right corner.
Models with many random particles are also used for the description of large wireless telecommunications systems; in this case the particles
are the userdevices . When the movement of the users does not have to be considered, the modeling of the device locations is typically via a Poisson point process, but when the motion of users becomes important it is not yet clear how to model user paths especially as user behavior undergoes periodic qualitative changes (e.g. between day and night). The particle
interactions depend on their separation since a message can only be effectively transmitted when two devices are within range of each other; see the applied theme Mobile Communication Networks. The crucial phase transition is percolation, such that a message can be transmitted in principle over very long distances, because the locations are suitably close to each other on a long scale. In this connection we have used methods from the theory of large deviations to analyze the positions of the devices. By performing a constrained energy minimization we are able to characterize the most important particle distributions for which no effective network can be established.
For dynamic models a wide range of hydrodynamic limit results have been proved dealing with elastically colliding gas molecules, soot formation and chemical reactions and leading to kinetic equations (see the Mathematical Theme Nonlinear kinetic equations). For a combined generalization of soot formation and chemical reactions, a dynamic large deviations principle was derived. With additional analytic tools an entropylike free energy and its dissipation potentials were identified. Together they form a gradient structure and provide a more detailed description of the dynamics and the effect of perturbations.
While researching robustness for machine learning, we have modeled data distribution shift and perturbation as particle systems. Then, we invented algorithms based on variational approaches that can learn from noisy data robustly. We have also applied particle methods to statistical inference and sampling, such as Stein geometry used for Markov chain Monte Carlo.
Publications
Monographs

B. Jahnel, W. König, Probabilistic Methods in Telecommunications, D. Mazlum, ed., Compact Textbooks in Mathematics, Birkhäuser Basel, 2020, XI, 200 pages, (Monograph Published), DOI 10.1007/9783030360900 .
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 3hour lectures or seminars which are also suitable for selfstudy. 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 10.1007/9783030527785 .
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 2stündige Lehrveranstaltung 
P. Exner, W. König, H. Neidhardt, eds., Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, World Scientific Publishing, Singapore, 2015, xii+383 pages, (Collection Published).
Articles in Refereed Journals

M. Fradon, J. Kern, S. Rœlly, A. Zass, Diffusion dynamics for an infinite system of twotype spheres and the associated depletion effect, Stochastic Processes and their Applications, 171 (2024), 104319, DOI 10.1016/j.spa.2024.104319 .
Abstract
We consider a random diffusion dynamics for an infinite system of hard spheres of two different sizes evolving in ℝ^{d}, its reversible probability measure, and its projection on the subset of the large spheres. The main feature is the occurrence of an attractive shortrange dynamical interaction  known in the physics literature as a depletion interaction  between the large spheres, which is induced by the hidden presence of the small ones. By considering the asymptotic limit for such a system when the density of the particles is high, we also obtain a constructive dynamical approach to the famous discrete geometry problem of maximisation of the contact number of n identical spheres in ℝ^{d}. As support material, we propose numerical simulations in the form of movies. 
R.I.A. Patterson, D.R.M. Renger, U. Sharma, Variational structures beyond gradient flows: A macroscopic fluctuationtheory perspective, Journal of Statistical Physics, 191 (2024), pp. 160, DOI 10.1007/s10955024032338 .
Abstract
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 largedeviation 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 nondissipative effects. Our main contribution is an abstract framework, which for a given fluxdensity cost and a quasipotential, provides a decomposition into dissipative and nondissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems  independent copies of jump processes, zerorange processes, chemicalreaction networks in complex balance and latticegas models. 
L. Andreis, W. König, H. Langhammer, R.I.A. Patterson, A largedeviations principle for all the components in a sparse inhomogeneous random graph, Probability Theory and Related Fields, 186 (2023), pp. 521620, DOI 10.1007/s00440022011807 .
Abstract
We study an inhomogeneous sparse random graph, G_{N}, 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 largedeviations 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 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 G_{N}. In particular, we recover the criterion for the existence of the phase transition given in [BJR07]. 
O. Collin, B. Jahnel, W. König, A micromacro variational formula for the free energy of a manybody system with unbounded marks, Electronic Journal of Probability, 28 (2023), pp. 118/1118/58, DOI 10.1214/23EJP1014 .
Abstract
The interacting quantum Bose gas is a random ensemble of many Brownian bridges (cycles) of various lengths with interactions between any pair of legs of the cycles. It is one of the standard mathematical models in which a proof for the famous BoseEinstein condensation phase transition is sought for. We introduce a simplified version of the model with an organisation of the particles in deterministic boxes instead of Brownian cycles as the marks of a reference Poisson point process (for simplicity, in Z ^{d}, instead of R ^{d}). We derive an explicit and interpretable variational formula in the thermodynamic limit for the limiting free energy of the canonical ensemble for any value of the particle density. This formula features all relevant physical quantities of the model, like the microscopic and the macroscopic particle densities, together with their mutual and selfenergies and their entropies. The proof method comprises a twostep largedeviation approach for marked Poisson point processes and an explicit distinction into small and large marks. In the characteristic formula, each of the microscopic particles and the statistics of the macroscopic part of the configuration are seen explicitly; the latter receives the interpretation of the condensate. The formula enables us to prove a number of properties of the limiting free energy as a function of the particle density, like differentiability and explicit upper and lower bounds, and a qualitative picture below and above the critical threshold (if it is finite). This proves a modified saturation nature of the phase transition. However, we have not yet succeeded in proving the existence of this phase transition. 
CH. Hirsch, B. Jahnel, E. Cali, Connection intervals in multiscale infrastructureaugmented dynamic networks, Stochastic Models, 39 (2023), pp. 851877, DOI 10.1080/15326349.2023.2184832 .
Abstract
We consider a hybrid spatial communication system in which mobile nodes can connect to static sinks in a bounded number of intermediate relaying hops. We describe the distribution of the connection intervals of a typical mobile node, i.e., the intervals of uninterrupted connection to the family of sinks. This is achieved in the limit of many hops, sparse sinks and growing time horizons. We identify three regimes illustrating that the limiting distribution depends sensitively on the scaling of the time horizon. 
B. Jahnel, S.K. Jhawar, A.D. Vu, Continuum percolation in a nonstabilizing environment, Electronic Journal of Probability, 28 (2023), pp. 131/1131/38, DOI 10.1214/23EJP1029 .
Abstract
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 PoissonBoolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features longrange dependencies in the environment, leading to absence of a sharp phase transition for the associated CoxBoolean 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, The Annals of Applied Probability, 33 (2023), pp. 45704607, DOI 10.1214/22AAP1926 .
Abstract
We consider irreversible translationinvariant interacting particle systems on the ddimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a timestationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translationinvariant measure implies, that the measure is Gibbs w.r.t. the same specification as the timestationary 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 translationinvariant measures is a Gibbs measure w.r.t. the same specification as the timestationary measure. This extends previously known results to fairly general irreversible interacting particle systems. 
B. Jahnel, Ch. Külske, Gibbsianness and nonGibbsianness for Bernoulli lattice fields under removal of isolated sites, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 29 (2023), pp. 30133032, DOI 10.3150/22BEJ1572 .
Abstract
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 noninvasive for large p, as it changes only a small fraction p(1p)^{2d} of sites, there is p(d) <1 such that for all p ∈ (p(d), 1) the resulting measure is a nonGibbsian measure, i.e., it does not possess a continuous version of its finitevolume conditional probabilities. On the other hand, for small p, the Gibbs property is preserved. 
N. Djurdjevac Conrad, J. Köppl, A. Djurdjevac, Feedback loops in opinion dynamics of agentbased models with multiplicative noise, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 24 (2022), pp. e24101352/1e24101352/23, DOI 10.3390/e24101352 .
Abstract
We introduce an agentbased model for coevolving opinion and social dynamics, under the influence of multiplicative noise. In this model, every agent is characterized by a position in a social space and a continuous opinion state variable. Agents? movements are governed by positions and opinions of other agents and similarly, the opinion dynamics is influenced by agents? spatial proximity and their opinion similarity. Using numerical simulations and formal analysis, we study this feedback loop between opinion dynamics and mobility of agents in a social space. We investigate the behavior of this ABM in different regimes and explore the influence of various factors on appearance of emerging phenomena such as group formation and opinion consensus. We study the empirical distribution and in the limit of infinite number of agents we derive a corresponding reduced model given by a partial differential equation (PDE). Finally, using numerical examples we show that a resulting PDE model is a good approximation of the original ABM. 
A. Agazzi, L. Andreis, R.I.A. Patterson, D.R.M. Renger, Large deviations for Markov jump processes with uniformly diminishing rates, Stochastic Processes and their Applications, 152 (2022), pp. 533559, DOI 10.1016/j.spa.2022.06.017 .
Abstract
We prove a largedeviation principle (LDP) for the sample paths of jump Markov processes in the small noise limit when, possibly, all the jump rates vanish uniformly, but slowly enough, in a region of the state space. We further show that our assumptions on the decay of the jump rates are optimal. As a direct application of this work we relax the assumptions needed for the application of LDPs to, e.g., Chemical Reaction Network dynamics, where vanishing reaction rates arise naturally particularly the context of Mass action kinetics. 
N. Engler, B. Jahnel, Ch. Külske, Gibbsianness of locally thinned random fields, Markov Processes and Related Fields, 28 (2022), pp. 185214, DOI 10.48550/arXiv.2201.02651 .
Abstract
We consider the locally thinned Bernoulli field on ℤ ^{d}, which is the lattice version of the TypeI 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 
A.K. Giri, P. Malgaretti, D. Peschka, M. Sega, Resolving the microscopic hydrodynamics at the moving contact line, Physical Review Fluids, 7 (2022), pp. L102001/1L102001/9, DOI 10.1103/PhysRevFluids.7.L102001 .
Abstract
By removing the smearing effect of capillary waves in molecular dynamics simulations we are able to provide a microscopic picture of the region around the moving contact line (MCL) at an unprecedented resolution. On this basis, we show that the continuum character of the velocity field is unaffected by molecular layering down to below the molecular scale. The solution of the continuum Stokes problem with MCL and Navierslip matches very well the molecular dynamics data and is consistent with a sliplength of 42 Å and small contact line dissipation. This is consistent with observations of the local force balance near the liquidsolid interface. 
Z. Mokhtari, R.I.A. Patterson, F. Höfling, Spontaneous trail formation in populations of autochemotactic walkers, New Journal of Physics, 24 (2022), pp. 013012/1013012/11, DOI 10.1088/13672630/ac43ec .
Abstract
We study the formation of trails in populations of selfpropelled agents that make oriented deposits of pheromones and also sense such deposits to which they then respond with gradual changes of their direction of motion. Based on extensive offlattice computer simulations aiming at the scale of insects, e.g., ants, we identify a number of emerging stationary patterns and obtain qualitatively the nonequilibrium state diagram of the model, spanned by the strength of the agentpheromone interaction and the number density of the population. In particular, we demonstrate the spontaneous formation of persistent, macroscopic trails, and highlight some behaviour that is consistent with a dynamic phase transition. This includes a characterisation of the mass of systemspanning trails as a potential order parameter. We also propose a dynamic model for a few macroscopic observables, including the subpopulation size of trailfollowing agents, which captures the early phase of trail formation. 
B. Jahnel, A. Tóbiás, E. Cali, Phase transitions for the Boolean model of continuum percolation for Cox point processes, Brazilian Journal of Probability and Statistics, 3 (2022), pp. 2044, DOI 10.1214/21BJPS514 .
Abstract
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 nonexistence 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. 
B. Jahnel, A. Tóbiás, SINR percolation for Cox point processes with random powers, Adv. Appl. Math., 54 (2022), pp. 227253, DOI 10.1017/apr.2021.25 .
Abstract
Signaltointerference plus noise ratio (SINR) percolation is an infiniterange dependent variant of continuum percolation modeling connections in a telecommunication network. Unlike in earlier works, in the present paper the transmitted signal powers of the devices of the network are assumed random, i.i.d. and possibly unbounded. Additionally, we assume that the devices form a stationary Cox point process, i.e., a Poisson point process with stationary random intensity measure, in two or higher dimensions. We present the following main results. First, under suitable moment conditions on the signal powers and the intensity measure, there is percolation in the SINR graph given that the device density is high and interferences are sufficiently reduced, but not vanishing. Second, if the interference cancellation factor γ and the SINR threshold τ satisfy γ ≥ 1/(2τ), then there is no percolation for any intensity parameter. Third, in the case of a Poisson point process with constant powers, for any intensity parameter that is supercritical for the underlying Gilbert graph, the SINR graph also percolates with some small but positive interference cancellation factor. 
A. Stephan, EDPconvergence for a linear reactiondiffusion system with fast reversible reaction, Calculus of Variations and Partial Differential Equations, 60 (2021), pp. 226/1226/35, DOI 10.1007/s00526021020890 .
Abstract
We perform a fastreaction limit for a linear reactiondiffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reactiondiffusion 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 coshtype functions for the reaction part. The fastreaction limit is done on the level of the gradient structure by proving EDPconvergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slowmanifold. Moreover, the limit gradient system can be equivalently described by a coarsegrained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarsegrained 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. 41314172, DOI 10.1007/s00023021010767 .
Abstract
We consider a onedimensional classical manybody system with interaction potential of LennardJones 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. 
J.D. Deuschel, T. Orenshtein, N. Perkowski, Additive functionals as rough paths, The Annals of Probability, 49 (2021), pp. 14501479, DOI 10.1214/20AOP1488 .
Abstract
We consider additive functionals of stationary Markov processes and show that under KipnisVaradhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Lévy area that can be described in terms of the asymmetry (nonreversibility) of the underlying Markov process. We apply this abstract result to three model problems: First we study random walks with random conductances under the annealed law. If we consider the Itô rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. If we consider the Stratonovich rough path, then there is no correction. The second example is a nonreversible OrnsteinUhlenbeck process, while the last example is a diffusion in a periodic environment. As a technical step we prove an estimate for the pvariation of stochastic integrals with respect to martingales that can be viewed as an extension of the rough path BurkholderDavisGundy inequality for local martingale rough paths of [FV08], [CF19] and [FZ18] to the case where only the integrator is a local martingale. 
L. Andreis, W. König, R.I.A. Patterson, A largedeviations principle for all the cluster sizes of a sparse ErdősRényi random graph, Random Structures and Algorithms, 59 (2021), pp. 522553, DOI 10.1002/rsa.21007 .
Abstract
A largedeviations 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 timedependent version of the ErdősRényi 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 ErdősRényi graphs are connected. 
A. Mielke, M.A. Peletier, A. Stephan, EDPconvergence for nonlinear fastslow reaction systems with detailed balance, Nonlinearity, 34 (2021), pp. 57625798, DOI 10.1088/13616544/ac0a8a .
Abstract
We consider nonlinear reaction systems satisfying massaction kinetics with slow and fast reactions. It is known that the fastreactionrate limit can be described by an ODE with Lagrange multipliers and a set of nonlinear constraints that ask the fast reactions to be in equilibrium. Our aim is to study the limiting gradient structure which is available if the reaction system satisfies the detailedbalance condition. The gradient structure on the set of concentration vectors is given in terms of the relative Boltzmann entropy and a coshtype dissipation potential. We show that a limiting or effective gradient structure can be rigorously derived via EDP convergence, i.e. convergence in the sense of the EnergyDissipation Principle for gradient flows. In general, the effective entropy will no longer be of Boltzmann type and the reactions will no longer satisfy massaction kinetics. 
A. Hinsen, B. Jahnel, E. Cali, J.P. Wary, Phase transitions for chaseescape models on PoissonGilbert graphs, Electronic Communications in Probability, 25 (2020), pp. 25/125/14, DOI 10.1214/20ECP306 .
Abstract
We present results on phase transitions of local and global survival in a twospecies model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuoustime nearestneighbor 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 welldefinedness 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 finitedegree 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. 915980 (published online on 21.12.2020), DOI 10.1007/s00205020015873 .
Abstract
We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of LennardJones 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 semiinfinite 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. 
CH. Hirsch, B. Jahnel, A. Tóbiás, Lower large deviations for geometric functionals, Electronic Communications in Probability, 25 (2020), pp. 41/141/12, DOI 10.1214/20ECP322 .
Abstract
This work develops a methodology for analyzing largedeviation 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 PoissonVoronoi cells, as well as powerweighted edge lengths in the random geometric, κnearest 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. 22572303, DOI 10.1007/s10955020026634 .
Abstract
We consider various modeling levels for spatially homogeneous chemical reaction systems, namely the chemical master equation, the chemical Langevin dynamics, and the reactionrate equation. Throughout we restrict our study to the case where the microscopic system satisfies the detailedbalance condition. The latter allows us to enrich the systems with a gradient structure, i.e. the evolution is given by a gradientflow equation. We present the arising links between the associated gradient structures that are driven by the relative entropy of the detailedbalance steady state. The limit of large volumes is studied in the sense of evolutionary Γconvergence 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. 90109, DOI 10.1007/s10955020025213 .
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 PoissonVoronoi, PoissonDelaunay and Poisson line tessellation, we also treat the JohnsonMehl 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, Coarsegraining via EDPconvergence for linear fastslow reaction systems, Mathematical Models & Methods in Applied Sciences, 30 (2020), pp. 17651807, 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 coarsegrained 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 coshtype gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarsegrained equation again has a coshtype gradient structure. We obtain the strongest version of convergence in the sense of the EnergyDissipation Principle (EDP), namely EDPconvergence with tilting. 
A. Stephan, H. Stephan, Memory equations as reduced Markov processes, Discrete and Continuous Dynamical Systems, 39 (2019), pp. 21332155, DOI 10.3934/dcds.2019089 .
Abstract
A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we give an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as the change of the type of some quasiparticles along oneway loops. Typically, the arising Markov process does not have the detailed balance property. The method leads to a more realisitc modeling of memory equations. Moreover, it carries over the large number of investigation tools for Markov processes to memory equations, like the calculation of the equilibrium state, the asymptotic behavior and so on. The method can be used for an approximative solution of some degenerate memory equations like delay differential equations. 
CH. Hirsch, B. Jahnel, Large deviations for the capacity in dynamic spatial relay networks, Markov Processes and Related Fields, 25 (2019), pp. 3373.
Abstract
We derive a large deviation principle for the spacetime 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. 
C. Cotar, B. Jahnel, Ch. Külske, Extremal decomposition for random Gibbs measures: From general metastates to metastates on extremal random Gibbs measures, Electronic Communications in Probability, 23 (2018), pp. 112, DOI 10.1214/18ECP200 .
Abstract
The concept of metastate measures on the states of a random spin system was introduced to be able to treat the largevolume asymptotics for complex quenched random systems, like spin glasses, which may exhibit chaotic volume dependence in the strongcoupling regime. We consider the general issue of the extremal decomposition for Gibbsian specifications which depend measurably on a parameter that may describe a whole random environment in the infinite volume. Given a random Gibbs measure, as a measurable map from the environment space, we prove measurability of its decomposition measure on pure states at fixed environment, with respect to the environment. As a general corollary we obtain that, for any metastate, there is an associated decomposition metastate, which is supported on the extremes for almost all environments, and which has the same barycenter. 
G. Botirov, B. Jahnel, Phase transitions for a model with uncountable spin space on the Cayley tree: The general case, Positivity. An International Mathematics Journal Devoted to Theory and Applications of Positivity, 23 (2019), pp. 291301 (published online on 17.08.2018), DOI 10.1007/s1111701806061 .
Abstract
In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in [EsHaRo12, EsRo10, BoEsRo13, JaKuBo14, Bo17]. The potential is of nearestneighbor type and the local state space is compact but uncountable. Based on the system parameters we prove existence of a critical value θ_{ c } such that for θ≤θ _{ c } there is a unique translationinvariant splitting Gibbs measure. For θ _{ c } < θ there is a phase transition with exactly three translationinvariant splitting Gibbs measures. The proof rests on an analysis of fixed points of an associated nonlinear Hammerstein integral operator for the boundary laws. 
W. Wagner, A random walk model for the Schrödinger equation, Mathematics and Computers in Simulation, 143 (2018), pp. 138148, DOI 10.1016/j.matcom.2016.07.012 .
Abstract
A random walk model for the spatially discretized timedependent Schrödinger equation is constructed. The model consists of a class of piecewise deterministic Markov processes. The states of the processes are characterized by a position and a complexvalued weight. Jumps occur both on the spatial grid and in the space of weights. Between the jumps, the weights change according to deterministic rules. The main result is that certain functionals of the processes satisfy the Schrödinger equation. 
A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Nonequilibrium thermodynamical principles for chemical reactions with massaction kinetics, SIAM Journal on Applied Mathematics, 77 (2017), pp. 15621585, 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 nonlinear 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. 126167.

M. Erbar, M. Fathi, V. Laschos, A. Schlichting, Gradient flow structure for McKeanVlasov equations on discrete spaces, Discrete and Continuous Dynamical Systems, 36 (2016), pp. 67996833.
Abstract
In this work, we show that a family of nonlinear meanfield equations on discrete spaces, can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of Nparticle dynamics, as N goes to infinity 
S. Jansen, W. König, B. Metzger, Large deviations for cluster size distributions in a continuous classical manybody system, The Annals of Applied Probability, 25 (2015), pp. 930973.
Abstract
An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pairinteraction is given by a stable LennardJonestype potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $betain(0,infty)$ and particle density $rhoin(0,rho_rmcp)$ in the thermodynamic limit. Here $rho_rmcp >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Gamma$convergence of the rate function towards an explicit limiting rate function in the lowtemperature dilute limit $betatoinfty$, $rho downarrow 0$ such that $beta^1logrhoto nu$ for some $nuin(0,infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the decoupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $nu$. Under additional assumptions on the potential, the $Gamma$convergence along curves can be strengthened to uniform bounds, valid in a lowtemperature, lowdensity rectangle. 
M. Erbar, J. Maas, D.R.M. Renger, From large deviations to Wasserstein gradient flows in multiple dimensions, Electronic Communications in Probability, 20 (2015), pp. 112.
Abstract
We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer [ADPZ11] that this functional is asymptotically equivalent (in the sense of Gammaconvergence) to the JordanKinderlehrerOtto functional arising in the Wasserstein gradient flow structure of the FokkerPlanck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in [DLR13] relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of [ADPZ11] to arbitrary dimensions. 
M. Muminov, H. Neidhardt, T. Rasulov, On the spectrum of the lattice spinboson Hamiltonian for any coupling: 1D case, Journal of Mathematical Physics, 56 (2015), pp. 053507/1053507/24.
Abstract
A lattice model of radiative decay (socalled spinboson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling constant the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model H for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of H below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form. 
S. Simonella, M. Pulvirenti, On the evolution of the empirical measure for hardsphere dynamics, Bulletin of the Institute of Mathematics. Academia Sinica. Institute of Mathematics, Academia Sinica, Taipei, Taiwan. English. English summary., 10 (2015), pp. 171204.

A. Mielke, M.A. Peletier, D.R.M. Renger, On the relation between gradient flows and the largedeviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41 (2014), pp. 12931325.
Abstract
Motivated by the occurence in rate functions of timedependent largedeviation principles, we study a class of nonnegative functions ℒ that induce a flow, given by ℒ(z_{t},ż_{t})=0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when ℒ is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropyWasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure. 
M.H. Duong, V. Laschos, M. Renger, Wasserstein gradient flows from large deviations of manyparticle limits, ESAIM. Control, Optimisation and Calculus of Variations, 19 (2013), pp. 11661188.

M.A. Peletier, M. Renger, M. Veneroni, Variational formulation of the FokkerPlanck equation with decay: A particle approach, Communications in Contemporary Mathematics, 15 (2013), pp. 1350017/11350017/43.

S. Adams, A. Collevecchio, W. König, A variational formula for the free energy of an interacting manyparticle system, The Annals of Probability, 39 (2011), pp. 683728.
Abstract
We consider $N$ bosons in a box in $R^d$ with volume $N/rho$ under the influence of a mutually repellent pair potential. The particle density $rhoin(0,infty)$ is kept fixed. Our main result is the identification of the limiting free energy, $f(beta,rho)$, at positive temperature $1/beta$, in terms of an explicit variational formula, for any fixed $rho$ if $beta$ is sufficiently small, and for any fixed $beta$ if $rho$ is sufficiently small. The thermodynamic equilibrium is described by the symmetrised trace of $rm e^beta Hcal_N$, where $Hcal_N$ denotes the corresponding Hamilton operator. The wellknown FeynmanKac formula reformulates this trace in terms of $N$ interacting Brownian bridges. Due to the symmetrisation, the bridges are organised in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a largedeviations analysis of the stationary empirical field. The resulting variational formula ranges over random shiftinvariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process. In our proof of the lower bound for the free energy, we drop all interaction involving lq infinitely longrq cycles, and their possible presence is signalled by a loss of mass of the lq finitely longrq cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the lq finitely longrq cycles. We expect that the precise relationship between these two bounds lies at the heart of BoseEinstein condensation and intend to analyse it further in future. 
M. Aizenman, S. Jansen, P. Jung, Symmetry breaking in quasi1D Coulomb systems, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 11 (2010), pp. 14531485.
Abstract
Quasi onedimensional systems are systems of particles in domains which are of infinite extent in one direction and of uniformly bounded size in all other directions, e.g. on a cylinder of infinite length. The main result proven here is that for such particle systems with Coulomb interactions and neutralizing background, the socalled “jellium”, at any temperature and at any finitestrip width there is translation symmetry breaking. This extends the previous result on Laughlin states in thin, twodimens The structural argument which is used here bypasses the question of whether the translation symmetry breaking is manifest already at the level of the one particle density function. It is akin to that employed by Aizenman and Martin (1980) for a similar statement concerning symmetry breaking at all temperatures in strictly onedimensional Coulomb systems. The extension is enabled through bounds which establish tightness of finitevolume charge fluctuations. 
A. Collevecchio, W. König, P. Mörters, N. Sidorova, Phase transitions for dilute particle systems with LennardJones potential, Communications in Mathematical Physics, 299 (2010), pp. 603630.
Contributions to Collected Editions

A. Hinsen, B. Jahnel, E. Cali, J.P. Wary, Malware propagation in urban D2D networks, in: IEEE 18th International Symposium on on Modeling and Optimization in Mobile, ad Hoc, and Wireless Networks, (WiOpt), Volos, Greece, Institute of Electrical and Electronics Engineers (IEEE), 2020, pp. 19.
Abstract
We introduce and analyze models for the propagation of malware in pure D2D networks given via stationary CoxGilbert 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 edgelength measure of a realization of a PoissonVoronoi 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 nonMarkovian 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 nonMarkovian setting. 
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. 239268.

M. Kantner, U. Bandelow, Th. Koprucki, H.J. Wünsche, Multiscale modelling and simulation of singlephoton sources on a device level, in: EuroTMCS II  Theory, Modelling & Computational Methods for Semiconductors, 7th  9th December 2016, Tyndall National Institute, University College Cork, Ireland, E. O'Reilly, S. Schulz, S. Tomic, eds., Tyndall National Institute, 2016, pp. 65.
Preprints, Reports, Technical Reports

E. Bolthausen, W. König, Ch. Mukherjee, Selfrepellent Brownian bridges in an interacting Bose gas, Preprint no. 3110, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3110 .
Abstract, PDF (478 kByte)
We consider a model of ddimensional interacting quantum Bose gas, expressed in terms of an ensemble of interacting Brownian bridges in a large box and undergoing the influence of all the interactions between the legs of each of the Brownian bridges. We study the thermodynamic limit of the system and give an explicit formula for the limiting free energy and a necessary and sufficient criterion for the occurrence of a condensation phase transition. For d ≥ 5 and sufficiently small interaction, we prove that the condensate phase is not empty. The ideas of proof rely on the similarity of the interaction to that of the selfrepellent random walk, and build on a lace expansion method conducive to treating paths undergoing mutual repellence within each bridge. 
B. Jahnel, L. Lüchtrath, M. Ortgiese, Cluster sizes in subcritical soft Boolean models, Preprint no. 3106, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3106 .
Abstract, PDF (435 kByte)
We consider the soft Boolean model, a model that interpolates between the Boolean model and longrange percolation, where vertices are given via a stationary Poisson point process. Each vertex carries an independent Paretodistributed radius and each pair of vertices is assigned another independent Pareto weight with a potentially different tail exponent. Two vertices are now connected if they are within distance of the larger radius multiplied by the edge weight. We determine the tail behaviour of the Euclidean diameter and the number of points of a typical maximally connected component in a subcritical percolation phase. For this, we present a sharp criterion in terms of the tail exponents of the edgeweight and radius distributions that distinguish a regime where the tail behaviour is controlled only by the edge exponent from a regime in which both exponents are relevant. Our proofs rely on fine pathcounting arguments identifying the precise order of decay of the probability that faraway vertices are connected. 
J. Köppl, N. Lanchier, M. Mercer, Survival and extinction for a contact process with a densitydependent birth rate, Preprint no. 3103, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3103 .
Abstract, PDF (860 kByte)
To study later spatial evolutionary games based on the multitype contact process, we first focus in this paper on the conditions for survival/extinction in the presence of only one strategy, in which case our model consists of a variant of the contact process with a densitydependent birth rate. The players are located on the ddimensional integer lattice, with natural birth rate λ and natural death rate one. The process also depends on a payoff a_{11} = a modeling the effects of the players on each other: while players always die at rate one, the rate at which they give birth is given by λ times the exponential of a times the fraction of occupied sites in their neighborhood. In particular, the birth rate increases with the local density when a > 0, in which case the payoff a models mutual cooperation, whereas the birth rate decreases with the local density when a < 0, in which case the payoff a models intraspecific competition. Using standard coupling arguments to compare the process with the basic contact process (the particular case a = 0 ), we prove that, for all payoffs a , there is a phase transition from extinction to survival in the direction of λ. Using various block constructions, we also prove that, for all birth rates λ, there is a phase transition in the direction of a. This last result is in sharp contrast with the behavior of the nonspatial deterministic meanfield model in which the stability of the extinction state only depends on λ . This underlines the importance of space (local interactions) and stochasticity in our model. 
B. Jahnel, U. Rozikov, Gibbs measures for hardcoreSOS models on Cayley trees, Preprint no. 3100, WIAS, Berlin, 2024.
Abstract, PDF (420 kByte)
We investigate the finitestate psolidonsolid model, for p=∞, on Cayley trees of order k ≥ 2 and establish a system of functional equations where each solution corresponds to a (splitting) Gibbs measure of the model. Our main result is that, for three states, k=2,3 and increasing coupling strength, the number of translationinvariant Gibbs measures behaves as 1→3 →5 →6 →7. This phase diagram is qualitatively similar to the one observed for threestate pSOS models with p>0 and, in the case of k=2, we demonstrate that, on the level of the functional equations, the transition p → ∞ is continuous. 
P.P. Ghosh, B. Jahnel, S.K. Jhawar, Large and moderate deviations in Poisson navigations, Preprint no. 3096, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3096 .
Abstract, PDF (318 kByte)
We derive large and moderatedeviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this nonMarkovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizonal displacement as well as renewalprocess arguments. 
B. Jahnel, J. Köppl, Timeperiodic behaviour in one and twodimensional interacting particle systems, Preprint no. 3092, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3092 .
Abstract, PDF (311 kByte)
We provide a class of examples of interacting particle systems on $Z^d$, for $din1,2$, that admit a unique translationinvariant stationary measure, which is not the longtime limit of all translationinvariant starting measures, due to the existence of timeperiodic orbits in the associated measurevalued dynamics. This is the first such example and shows that even in low dimensions, not every limit point of the measurevalued dynamics needs to be a timestationary measure. 
L. Lüchtrath, Ch. Mönch, The directed agedependent random connection model with arc reciprocity, Preprint no. 3090, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3090 .
Abstract, PDF (304 kByte)
We introduce a directed spatial random graph model aimed at modelling certain aspects of social media networks. We provide two variants of the model: an infinite version and an increasing sequence of finite graphs that locally converge to the infinite model. Both variants have in common that each vertex is placed into Euclidean space and carries a birth time. Given locations and birth times of two vertices, an arc is formed from younger to older vertex with a probability depending on both birth times and the spatial distance of the vertices. If such an arc is formed, a reverse arc is formed with probability depending on the ratio of the endpoints' birth times. Aside from the local limit result connecting the models, we investigate degree distributions, two different clustering metrics and directed percolation. 
B. Jahnel, U. Rozikov, Threestate $p$SOS models on binary Cayley trees, Preprint no. 3089, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3089 .
Abstract, PDF (640 kByte)
We consider a version of the solidonsolid model on the Cayley tree of order two in which vertices carry spins of value 0,1 or 2 and the pairwise interaction of neighboring vertices is given by their spin difference to the power p>0. We exhibit all translationinvariant splitting Gibbs measures (TISGMs) of the model and demonstrate the existence of up to seven such measures, depending on the parameters. We further establish general conditions for extremality and nonextremality of TISGMs in the set of all Gibbs measures and use them to examine selected TISGMs for a small and a large p. Notably, our analysis reveals that extremality properties are similar for large p compared to the case p=1, a case that has been explored already in previous work. However, for the small p, certain measures that were consistently nonextremal for p=1 do exhibit transitions between extremality and nonextremality. 
A. Erhardt, D. Peschka, Ch. Dazzi, L. Schmeller, A. Petersen, S. Checa, A. Münch, B. Wagner, Modeling cellular selforganization in strainstiffening hydrogels, Preprint no. 3076, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3076 .
Abstract, PDF (14 MByte)
We develop a threedimensional mathematical model framework for the collective evolution of cell populations by an agentbased model (ABM) that mechanically interacts with the surrounding extracellular matrix (ECM) modeled as a hydrogel. We derive effective twodimensional models for the geometrical setup of a thin hydrogel sheet to study cellcell and cellhydrogel mechanical interactions for a range of external conditions and intrinsic material properties. We show that without any stretching of the hydrogel sheets, cells show the wellknown tendency to form long chains with varying orientations. Our results further show that external stretching of the sheet produces the expected nonlinear strainsoftening or stiffening response, with, however, little qualitative variation of the overall cell dynamics for all the materials considered. The behavior is remarkably different when solvent is entering or leaving from strain softening or stiffening hydrogels, respectively. 
CH. Hirsch, B. Jahnel, S.K. Jhawar, P. Juhász, Poisson approximation of fixeddegree nodes in weighted random connection models, Preprint no. 3057, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3057 .
Abstract, PDF (474 kByte)
We present a processlevel Poissonapproximation result for the degree$k$ vertices in a highdensity weighted random connection model with preferentialattachment kernel in the unit volume. Our main focus lies on the impact of the left tails of the weight distribution for which we establish general criteria based on their smallweight quantiles. To illustrate that our conditions are broadly applicable, we verify them for weight distributions with polynomial and stretched exponential left tails. The proofs rest on truncation arguments and a recently established quantitative Poisson approximation result for functionals of Poisson point processes. 
B. Jahnel, Ch. Külske, A. Zass, Locality properties for discrete and continuum WidomRowlinson models in random environments, Preprint no. 3054, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3054 .
Abstract, PDF (606 kByte)
We consider the WidomRowlinson model in which hard disks of two possible colors are constrained to a hardcore repulsion between particles of different colors, in quenched random environments. These random environments model spatially dependent preferences for the attach ment of disks. We investigate the possibility to represent the joint process of environment and infinitevolume WidomRowlinson measure in terms of continuous (quasilocal) Papangelou inten sities. We show that this is not always possible: In the case of the symmetric WidomRowlinson model on a nonpercolating environment, we can explicitly construct a discontinuity coming from the environment. This is a new phenomenon for systems of continuous particles, but it can be understood as a continuousspace echo of a simpler nonlocality phenomenon known to appear for the diluted Ising model (Griffiths singularity random field [ EMSS00]) on the lattice, as we explain in the course of the proof. 
B. Jahnel, A.D. Vu, A longrange contact process in a random environment, Preprint no. 3047, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3047 .
Abstract, PDF (3735 kByte)
We study survival and extinction of a longrange infection process on a diluted onedimensional lattice in discrete time. The infection can spread to distant vertices according to a Pareto distribution, however spreading is also prohibited at random times. We prove a phase transition in the recovery parameter via block arguments. This contributes to a line of research on directed percolation with longrange correlations in nonstabilizing random environments. 
L. Andreis, T. Iyer, E. Magnanini, Gelation, hydrodynamic limits and uniqueness in cluster coagulation processes, Preprint no. 3039, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3039 .
Abstract, PDF (627 kByte)
We consider the problem of gelation in the cluster coagulation model introduced by Norris [Comm. Math. Phys., 209(2):407435 (2000)]; this model is general enough to incorporate various inhomogenieties in the evolution of clusters, for example, their shape, or their location in space. We derive general, sufficient criteria for stochastic gelation in this model, and for trajectories associated with this process to concentrate among solutions of a generalisation of the Flory equation; thus providing sufficient criteria for the equation to have gelling solutions. As particular cases, we extend results related to the classical MarcusLushnikov coagulation process and Smoluchowski coagulation equation, showing that reasonable 'homogenous' coagulation processes with exponent γ larger than 1 yield gelation. In another special case, we prove a law of large numbers for the trajectory of the empirical measure of the stochastic cluster coagulation process, by means of a uniqueness result for the solution of the aforementioned generalised Flory equation. Finally, we use coupling arguments with inhomogeneous random graphs to deduce sufficient criterion for strong gelation (the emergence of a particle of size O(N)). 
B. Jahnel, J. Köppl, B. Lodewijks, A. Tóbiás, Percolation in lattice kneighbor graphs, Preprint no. 3028, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3028 .
Abstract, PDF (437 kByte)
We define a random graph obtained via connecting each point of ℤ ^{d} independently to a fixed number 1 ≤ k ≤ 2d of its nearest neighbors via a directed edge. We call this graph the emphdirected kneighbor graph. Two natural associated undirected graphs are the emphundirected and the emphbidirectional kneighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed kneighbor graph between them in at least one, respectively precisely two, directions. In these graphs we study the question of percolation, i.e., the existence of an infinite selfavoiding path. Using different kinds of proof techniques for different classes of cases, we show that for k=1 even the undirected kneighbor graph never percolates, but the directed one percolates whenever k≥ d+1, k≥ 3 and d ≥5, or k ≥4 and d=4. We also show that the undirected 2neighbor graph percolates for d=2, the undirected 3neighbor graph percolates for d=3, and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the knearestneighbor graphs studied in continuum percolation, and our results support this interpretation. 
W. König, N. Pétrélis, R. Soares Dos Santos, W. van Zuijlen, Weakly selfavoiding walk in a Paretodistributed random potential, Preprint no. 3023, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3023 .
Abstract, PDF (604 kByte)
We investigate a model of continuoustime simple random walk paths in ℤ ^{d} undergoing two competing interactions: an attractive one towards the large values of a random potential, and a selfrepellent one in the spirit of the wellknown weakly selfavoiding random walk. We take the potential to be i.i.d. Paretodistributed with parameter α > d, and we tune the strength of the interactions in such a way that they both contribute on the same scale as t → ∞. Our main results are (1) the identification of the logarithmic asymptotics of the partition function of the model in terms of a random variational formula, and, (2) the identification of the path behaviour that gives the overwhelming contribution to the partition function for α > 2d: the randomwalk path follows an optimal trajectory that visits each of a finite number of random lattice sites for a positive random fraction of time. We prove a law of large numbers for this behaviour, i.e., that all other path behaviours give strictly less contribution to the partition function.The joint distribution of the variational problem and of the optimal path can be expressed in terms of a limiting Poisson point process arising by a rescaling of the random potential. The latter convergence is in distribution?and is in the spirit of a standard extremevalue setting for a rescaling of an i.i.d. potential in large boxes, like in KLMS09. 
B. Jahnel, J. Köppl, On the longtime behaviour of reversible interacting particle systems in one and two dimensions, Preprint no. 3004, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3004 .
Abstract, PDF (287 kByte)
By refining Holley's free energy technique, we show that, under quite general assumptions on the dynamics, the attractor of a (possibly nontranslationinvariant) interacting particle system in one or two spatial dimensions is contained in the set of Gibbs measures if the dynamics admits a reversible Gibbs measure. In particular, this implies that there can be no reversible interacting particle system that exhibits timeperiodic behaviour and that every reversible interacting particle system is ergodic if and only if the reversible Gibbs measure is unique. In the special case of nonattractive stochastic Ising models this answers a question due to Liggett. 
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 translationinvariant interacting particle systems on the ddimensional cubic lattice with finite local state space, which admit at least one Gibbs measure as a timestationary measure. Under some mild degeneracy conditions on the rates and the specification we prove, that zero relative entropy loss of a translationinvariant measure implies, that the measure is Gibbs w.r.t. the same specification as the timestationary 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 translationinvariant measures is a Gibbs measure w.r.t. the same specification as the timestationary measure. This extends previously known results to fairly general irreversible interacting particle systems.
Talks, Poster

J. Köppl, Dynamical Gibbs variational principles for irreversible interacting particle systems and applications to attractor properties, Analysis and Probability Seminar Passau, Universität Passau, Fakultät für Informatik und Mathematik, January 17, 2023.

J. Köppl, Dynamical Gibbs variational principles for irreversible interacting particle systems with applications to attractor properties, 16th German Probability and Statistics Days (GPSD) 2023, March 7  10, 2023, Universität DuisburgEssen, March 9, 2023.

J. Köppl, The longtime behaviour of interacting particle systems: A Lyapunov functional approach, In Search of Model Structures for Nonequilibrium Systems, Münster, April 24  28, 2023.

J. Köppl, The longtime behaviour of interacting particle systems: A Lyapunov functional approach, In search of model structures for nonequilibrium systems, April 24  28, 2023, Westfälische WilhelmsUniversität Münster, Fachbereich Mathematik und Informatik, April 25, 2023.

J. Köppl, The longetime behavior of interacting particle systems: A Lyapunov functional approach, Mathematics of Random Systems: Summer School 2023, September 11  15, 2023, Kyoto University, Research Institute for Mathematical Sciences, Kyoto, Japan, November 13, 2023.

D. Peschka, Moving contact lines for sliding droplets, 93rd Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2023), Session 11 ``Interfacial Flows'', May 30  June 2, 2023, Technische Universität Dresden, June 1, 2023.

A. Zass, Diffusion dynamics for an system of twotype speres and the associated depletion effect, Workshop MathMicS 2023: Mathematics and microscopic theory for random Soft Matter systems, February 13  15, 2023, HeinrichHeineUniversität Düsseldorf, Institut für Theoretische Physik II  Soft Matter, February 14, 2023.

A. Zass, The statistical mechanics of the interlacement point process, Second Annual Conference of the SPP2265, March 27  30, 2023, Deutsches Zentrum für Luft und Raumfahrt (DLR), Köln, March 30, 2023.

B. Jahnel, Stochastische Methoden für Kommunikationsnetzwerke, Seminar der Fakultät Informatik, Hochschule Reutlingen, October 6, 2023.

B. Jahnel, Stochastische Methoden für Kommunikationsnetzwerke, Orientierungsmodul der Technischen Universität Braunschweig, Institut für Mathematische Stochastik, November 2, 2023.

B. Jahnel, Stochastische Methoden für Kommunikationsnetzwerke, Orientierungsmodul der Technischen Universität Braunschweig, Institut für Mathematische Stochastik, January 30, 2023.

B. Jahnel, Subcritical percolation phases for generalized weightdependent random connection models, 21st INFORMS Applied Probability Society Conference, June 28  30, 2023, Centre Prouvé, Nancy, France, June 29, 2023.

B. Jahnel, Subcritical percolation phases for generalized weightdependent random connection models, DMV Annual Meeting 2023, Minisymposium MS 12 ``Random Graphs and Statistical Network Analysis'', September 25  28, 2023, Technische Universität Ilmenau, September 25, 2023.

B. Jahnel, The statistical mechanics of the interlacement point process, Second Annual Conference of the SPP 2265, March 27  30, 2023, Deutsches Zentrum für Luft und Raumfahrt (DLR), Köln, March 29, 2023.

W. König, The statistical mechanics of the interlacement point process, Second Annual Conference of the SPP 2265, March 27  30, 2023, Deutsches Zentrum für Luft und Raumfahrt (DLR), Köln, March 30, 2023.

L. Lüchtrath, The emergence of a giant component in onedimensional inhomogeneous networks with longrange effects, 18th Workshop on Algorithms and Models for Web Graphs, May 23  26, 2023, The Fields Institute for Research in Mathematical Sciences, Toronto, Canada, May 25, 2023.

A. Stephan, Positivity and polynomial decay of energies for squarefield operators, Variational and Geometric Structures for Evolution, October 9  13, 2023, Centro Internazionale per la Ricerca Matematica (CIRM), Levico Terme, Italy, October 13, 2023.

A. Stephan, Fastslow chemical reaction systems: Gradient systems and EDPconvergence, Oberseminar Dynamics, Technische Universität München, Department of Mathematics, April 17, 2023.

S. Schindler, Convergence to selfsimilar profiles for a coupled reactiondiffusion system on the real line, CRC 910: Workshop on Control of SelfOrganizing Nonlinear Systems, Wittenberg, September 26  28, 2022.

S. Schindler, Energy approach for a coupled reactiondiffusion system on the real line (online talk), SFB 910 Symposium ``Pattern formation and coherent structure in dissipative systems'' (Online Event), Technische Universität Berlin, January 14, 2022.

S. Schindler, On asymptotic selfsimilar behavior of solutions to parabolic systems (hybrid talk), SFB910: International Conference on Control of SelfOrganizing Nonlinear Systems (Hybrid Event), November 23  26, 2022, Technische Universität Berlin, Potsdam, November 25, 2022.

A. Stephan, EDPconvergence for a linear reactiondiffusion systems with fast reversible reaction (online talk), SIAM Conference on Analysis of Partial Differential Equations (PD22) (Online Event), Minisymposium MS11: ``Bridging Gradient Flows, Hypocoercivity and ReactionDiffusion Systems'', March 14  18, 2022, March 14, 2022.

B. Jahnel, Malware propagation in mobile devicetodevice networks (online talk), Joint H2020 AI@EDGE and INSPIRE5G Project Workshop  Platforms and Mathematical Optimization for Secure and Resilient Future Networks (Online Event), Paris, France, November 8  9, 2022, November 8, 2022.

R.I.A. Patterson, Large deviations with vanishing reactant concentrations, Workshop on Chemical Reaction Networks, July 6  8, 2022, Politecnico di Torino, Department of Mathematical Sciences ``G. L. Lagrange'', Torino, Italy, July 7, 2022.

A. Stephan, EDPconvergence for a linear reactiondiffusion system with fast reversible reaction, Mathematical Models for Biological Multiscale Systems (Hybrid Event), September 12  14, 2022, WIAS Berlin, September 12, 2022.

A. Stephan, EDPconvergence for gradient systems and applications to fastslow chemical reaction systems, Block Course ``Multiscale Problems and Homogenization'' at Freie Universität Berlin from Nov. 10 to Dec. 15, 2022, Berlin Mathematical School & Berlin Mathematics Research Center MATH+, November 24, 2022.

S. Schindler, Selfsimilar diffusive equilibration for a coupled reactiondiffusion system with massaction kinetics, SFB910: International Conference on Control of SelfOrganizing Nonlinear Systems (Hybrid Event), August 29  September 2, 2021, Technische Universität Berlin, Potsdam, September 1, 2021.

A. Stephan, Gradient systems and EDPconvergence with applications to nonlinear fastslow 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 mulitscale 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.

A. Stephan, Coarsegraining via EDPconvergence for linear fastslow reactiondiffusion systems (online talk), 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (Online Event), Section S14 ``Applied Analysis'', March 15  19, 2021, Universität Kassel, March 17, 2021.

B. Jahnel, Firstpassage percolation and chaseescape 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 (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.

B. Jahnel, Stochastic geometry for epidemiology (online talk), Monday Biostatistics Roundtable, Institute of Biometry and Clinical Epidemiology (Online Event), Campus Charité, June 14, 2021.

T. Orenshtein, Aging for the O'ConellYor 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), BernoulliIMS 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 BerlinOxford Young Researchers Meeting on Applied Stochastic Analysis (Online Event), June 8  10, 2020, WIAS Berlin, June 10, 2020.

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

A. Stephan, EDPconvergence for nonlinear fastslow reactions, Workshop ``Variational Methods for Evolution'', September 13  19, 2020, Mathematisches Forschungsinstitut Oberwolfach, September 18, 2020.

A. Stephan, On mathematical coarsegraining 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.

A. Stephan, Coarsegraining for gradient systems with applications to reaction systems (online talk), Thematic Einstein Semester on Energybased Mathematical Methods for Reactive Multiphase Flows: Student Compact Course ``Variational Methods for Fluids and Solids'' (Online Event), October 12  23, 2020, WIAS Berlin, October 15, 2020.

A. Stephan, EDPconvergence for nonlinear fastslow reaction systems (online talk), Annual Workshop of the GAMM Activity Group on Analysis of PDEs (Online Event), September 30  October 2, 2020, Institute of Science and Technology Austria (IST Austria), Klosterneuburg, October 1, 2020.

R.I.A. Patterson, Interpreting LDPs without detailed balance, Workshop ``Variational Methods for Evolution'', September 13  19, 2020, Mathematisches Forschungsinstitut Oberwolfach, September 15, 2020.

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, A novel simulation method for stochastic particle systems, Seminar, Department of Chemical Engineering and Biotechnology, University of Cambridge, Faculty of Mathematics, UK, May 9, 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.

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.

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

M. Kantner, Multiscale modeling and numerical simulation of singlephoton emitters, Matheon Workshop9th Annual Meeting ``Photonic Devices", Zuse Institut, Berlin, March 3, 2016.

M. Kantner, Multiscale modelling and simulation of singlephoton sources on a device level, EuroTMCS II Theory, Modelling & Computational Methods for Semiconductors, Tyndall National Institute and University College Cork, Cork, Ireland, December 9, 2016.

A. Mielke, On entropic gradient structures for classical and quantum Markov processes with detailed balance, Pure Analysis and PDEs Seminar, Imperial College London, Department of Mathematics, UK, May 11, 2016.

A. Mielke, Chemical Master Equation: Coarse graining via gradient structures, Kolloquium des SFB 1114 ``Scaling Cascades in Complex Systems'', Freie Universität Berlin, Fachbereich Mathematik, Berlin, June 4, 2015.

A. Mielke, Geometric approaches at and for theoretical and applied mechanics, Phil Holmes Retirement Celebration, October 8  9, 2015, Princeton University, Mechanical and Aerospace Engineering, New York, USA, October 8, 2015.

A. Mielke, The Chemical Master Equation as a discretization of the FokkerPlanck and Liouville equation for chemical reactions, Colloquium of Collaborative Research Center/Transregio ``Discretization in Geometry and Dynamics'', Technische Universität Berlin, Institut für Mathematik, Berlin, February 10, 2015.

A. Mielke, The FokkerPlanck and Liouville equations for chemical reactions as largevolume approximations of the Chemical Master Equation, Workshop ``Stochastic Limit Analysis for Reacting Particle Systems'', December 16  18, 2015, WIAS Berlin, Berlin, December 18, 2015.

R.I.A. Patterson, Approximation errors for Smoluchowski simulations, 10 th IMACS Seminar on Monte Carlo Methods, July 6  10, 2015, Johannes Kepler Universität Linz, Austria, July 7, 2015.

A. Mielke, Generalized gradient structures for reactiondiffusion systems, Applied Mathematics Seminar, Università di Pavia, Dipartimento di Matematica, Italy, June 17, 2014.

R.I.A. Patterson, Monte Carlo simulation of nanoparticle formation, University of Technology Eindhoven, Institute for Complex Molecular Systems, Netherlands, September 5, 2013.

S. Jansen, Large deviations for interacting manyparticle systems in the Saha regime, BerlinLeipzig Seminar on Analysis and Probability Theory, July 8, 2011, Technische Universität Clausthal, Institut für Mathematik, July 8, 2011.

W. König, Eigenvalue order statistics and mass concentration in the parabolic Anderson model, BerlinLeipzig Seminar on Analysis and Probability Theory, Technische Universität Clausthal, Institut für Mathematik, July 8, 2011.

W. König, Phase transitions for dilute particle systems with LennardJones potential, University of Bath, Department of Mathematical Sciences, UK, April 14, 2010.

W. König, Phase transitions for dilute particle systems with LennardJones potential, Workshop on Mathematics of Phase Transitions: Past, Present, Future, November 12  15, 2009, University of Warwick, Coventry, UK, November 15, 2009.
External Preprints

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 wellposedness, 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.