In the mathematical modeling of many processes and phenomena in the Sciences and Technology one employs systems with many random particles and interactions; see the Application Theme Particlebased modelling in the Sciences. Here we mean many different types of such systems, which, depending on the application, contain moving or static particles, have interactions with each other or with a (possibly random) surrounding medium, have attracting or repelling forces, etc. Also the mathematical questions that we have about the system are quite diverse, e.g., for an emerging cluster structure, for regularities within the clusters, for properties like percolation, crystallization or condensation, for temporal development of the contacts of the particles with each other, for dependencies of the global behavior on parameters, to the point of phase transitions, and more.
In most of the models considered there is one or more order parameters, in terms of which one well expresses the macroscopic behavior, like the average size of a cluster, the occupation probabilities of the particles at a given site at a given time, the empirical mean of all the particles, etc. In some cases, these order parameters satisfy deterministic equations (e.g. differential equations) in the limit considered (often large number of particles or late times), which can afterwards be studied with the help of analytic or numeric methods. In other cases, formulae are derived for the limiting free energy or for laws of large numbers or for ergodic theorems.
For answering these questions and for finding and derivation of formulae, at WIAS mathematical tools and concepts are usually employed and (if not yet available) developed. Stochastic and analytic methods are combined, and in the best case a mathematical solution is derived. Accompanying simulations give visualizations and produce explicit data. Examples of mathematical theories employed are, depending on the model, Gibbs measures, percolation, stochastic (partial) differential equations, Markov processes, and large deviations on the stochastic side, and weak convergence, calculus of variations, convex analysis and partial differential equations on the analytic side.
Contribution of the Institute
Here is a selection of some of the mathematical achievements of the WIAS, see also the Application Theme Particlebased modelling in the Sciences.
A (static) interacting manybody system is given if a large number of points are randomly distributed in a large box, such that they do not accumulate and such that a certain energy is given to the configuration as an exponential probability weight. In this way, a probability measure on configurations is given. The energy term carries a prefactor, the inverse temperature. In the FG5, the thermodynamic limit at low temperatures in particularly large boxes was studied, such that the entropy (the part of the probability that comes from the spatial distribution of the particle cloud) has a particular relation with the energy and the free energy depends on just one parameter. Interesting phase transitions could be obtained. In the future, however, the usual thermodynamic limit will be studied, where the box volume has a fixed relation to the particle number. For a system in which each particle also carries a kinetic energy and the cloud is subject to some symmetry condition, an ansatz with large deviations for a meanfield version of the entire particle system was developed and was used for expressing the free energy in terms of a variational formula. However, the desired effect of a condensation could not be proved in this way, and future investigations will be made.
In telecommunication, the locations of many users are usually modeled as the points of a spatial Poisson point process, see the Application Theme Mobile Communication Networks. Each two of these points interact if their distance is small enough. The question of the probability with which many messages, which are sent through the system via a system of relays, actually reach their intended receiver was studied. That is, a global connectivity property for the system was studied.
The connectivity property was studied in the limit of a high spatial density of the users per volume, in which case the probabilities decay exponentially. With the help of the theory of large deviations for highly dense point processes, the decay rate was expressed in term of an entropy. Afterwards, those configurations were analyzed that minimize the entropy, as these carry the interpretation of the (random) situations of best connectivity under the assumptions made. Similar investigations were made for interference and capacity properties. Further research currently concerns optimal trajectories of the messages, subject to interference and under avoidance of congestion of the relays, as well as the implementation of realistic movement schemes for the users.
A key task in the case of dynamic models is the establishment of a hydrodynamic limit. Such limits are typically first proved for complete applications in order to find an evolution equation for the macroscopic model properties. For examples see the application oriented themes "Coagulation" and Particlebased modelling in the Sciences. A fundamental part of these proofs is the demonstration of compactness in distribution of the Markov Processes that make up the model. In a number of works, properties of application specific problems have been abstracted and the results generalised so that they can be used in further applications.
In Biology the definition of useful stochastic models is an active topic of research that is far from complete. Established models for populations and their movements include spatial branching processes with random motions, which the WIAS studies in random environments; see the mathematical theme Spectral theory of random operators.
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
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. 
A. Hinsen, B. Jahnel, E. Cali, J.P. Wary, Connectivity in mobile devicetodevice networks in urban environments, IEEE Transactions on Information Theory, 69 (2023), pp. 71327148, DOI 10.1109/TIT.2023.3298278 .
Abstract
In this article we setup a dynamic devicetodevice communication system where devices, given as a Poisson point process, move in an environment, given by a street system of random planartessellation type, via a randomwaypoint 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 multiparameter system, we focus on an analysis of the clustering behavior of the random connectivity graph. In our main results we isolate regimes for the almostsure 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 inandout 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 infiniterange dependencies. 
T. Iyer, Degree distributions in recursive trees with fitnesses, Advances in Applied Probability, 55 (2023), pp. 407443, DOI 10.1017/apr.2022.40 .
Abstract
We study a general model of recursive trees where vertices are equipped with independent weights and at each timestep a vertex is sampled with probability proportional to its fitness function (a function of its weight and degree) and connects to ? newcoming vertices. Under a certain technical assumption, applying the theory of CrumpModeJagers branching processes, we derive formulas for the almost sure limiting distribution of the proportion of vertices with a given degree and weight, and proportion of edges with endpoint having a certain weight. As an application of this theorem, we prove rigorously observations of Bianconi related to the evolving Cayley tree in [Phys.Rev.E66, 036116 (2002)]. We also study the process in depth when the technical condition can fail in the particular case when the fitness function is affine, a model we call "generalised preferential attachment with fitness". We show that this model can exhibit condensation where a positive proportion of edges accumulate around vertices with maximal weight, or, more drastically, have a degenerate limiting degree distribution where the entire proportion of edges accumulate around these vertices. Finally, we prove stochastic convergence for the degree distribution under a different assumption of a strong law of large numbers for the partition function associated with the process. 
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. 
A. Hinsen, B. Jahnel, E. Cali, J.P. Wary, Chaseescape in dynamic devicetodevice networks, Journal of Applied Probability, published online in August 2023, DOI 10.1017/jpr.2023.47 .
Abstract
The present paper features results on global survival and extinction of an infection in a multilayer network of mobile agents. Expanding on a model first presented in CHJW22, we consider an urban environment, represented by linesegments in the plane, in which agents move according to a random waypoint model based on a Poisson point process. Whenever two agents are at sufficiently close proximity for a sufficiently long time the infection can be transmitted and then propagates into the system according to the same rule starting from a typical device. Inspired by wireless network architectures, the network is additionally equipped with a second class of agents that is able to transmit a patch to neighboring infected agents that in turn can further distribute the patch, leading to a chaseescape dynamics. We give conditions for parameter configurations that guarantee existence and absence of global survival as well as an inandout of the survival regime, depending on the speed of the devices. We also provide complementary results for the setting in which the chaseescape dynamics is defined as an independent process on the connectivity graph. The proofs mainly rest on percolation arguments via discretization and multiscale analysis. 
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. 
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 
N. Fountoulakis, T. Iyer, C. Mailler, H. Sulzbach, Dynamical models for random simplicial complexes, The Annals of Applied Probability, 32 (2022), pp. 28602913, DOI 10.1214/21AAP1752 .
Abstract
We study a general model of random dynamical simplicial complexes and derive a formula for the asymptotic degree distribution. This asymptotic formula encompasses results for a number of existing models, including random Apollonian networks and the weighted random recursive tree. It also confirms results on the scalefree nature of Complex Quantum Network Manifolds in dimensions d>2, and special types of Network Geometry with Flavour models studied in the physics literature by Bianconi, Rahmede [Sci.Rep.5, 13979 (2015) and Phys.Rev.E93, 032315 (2016)]. 
N. Fountoulakis, T. Iyer, Condensation phenomena in preferential attachment trees with neighbourhood influence, Electronic Journal of Probability, 27 (2022), pp. 76/176/49, DOI 10.1214/22EJP787 .
Abstract
We introduce a model of evolving preferential attachment trees where vertices are assigned weights, and the evolution of a vertex depends not only on its own weight, but also on the weights of its neighbours. We study the distribution of edges with endpoints having certain weights, and the distribution of degrees of vertices having a given weight. We show that the former exhibits a condensation phenomenon under a certain critical condition, whereas the latter converges almost surely to a distribution that resembles a power law distribution. Moreover, in the absence of condensation, we prove almostsure setwise convergence of the related quantities. This generalises existing results on the BianconiBarabÃ¡si tree as well as on an evolving tree model introduced by the second author. 
B. Jahnel, Ch. Hirsch, E. Cali, Percolation and connection times in multiscale dynamic networks, Stochastic Processes and their Applications, 151 (2022), pp. 490518, DOI 10.1016/j.spa.2022.06.008 .
Abstract
We study the effects of mobility on two crucial characteristics in multiscale dynamic networks: percolation and connection times. Our analysis provides insights into the question, to what extent longtime averages are wellapproximated by the expected values of the corresponding quantities, i.e., the percolation and connection probabilities. In particular, we show that in multiscale models, strong random effects may persist in the limit. Depending on the precise model choice, these may take the form of a spatial birthdeath 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. 
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. 
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. 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. 
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. 27782807, 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. 41/141/12, 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. 
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. 
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. 590606.
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, McKeanVlasov limit for interacting systems with simultaneous jumps, Stochastic Analysis and Applications, 36 (2018), pp. 960995, DOI 10.1080/07362994.2018.1486202 .
Abstract
Motivated by several applications, including neuronal models, we consider the McKeanVlasov limit for meanfield 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 encountermating model, Acta Applicandae Mathematicae. An International Survey Journal on Applying Mathematics and Mathematical Applications, 148 (2017), pp. 71102.

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.

J. Blath, A. González Casanova Soberón, B. Eldon, N. Kurt, M. WilkeBerenguer, Genetic variability under the seedbank coalescent, Genetics, 200 (2015), pp. 921934.
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 WrightFisher 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 sitefrequency 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 sitefrequency spectrum. Thus, one should be able to detect from genetic data the presence of a large seedbank in natural populations.
Contributions to Collected Editions

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. 2341, DOI 10.4171/ECR/171/2 .

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. 239268.
Preprints, Reports, Technical Reports

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. 
M. Heida, B. Jahnel, A.D. Vu, An ergodic and isotropic zeroconductance model with arbitrarily strong local connectivity, Preprint no. 3095, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3095 .
Abstract, PDF (377 kByte)
We exhibit a percolating ergodic and isotropic lattice model in all but at least two dimensions that has zero effective conductivity in all spatial directions and for all nontrivial choices of the connectivity parameter. The model is based on the socalled randomly stretched lattice where we additionally elongate layers containing few open edges. 
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. 
L. Andreis, W. König, H. Langhammer, R.I.A. Patterson, Spatial particle processes with coagulation: Gibbsmeasure approach, gelation and Smoluchowski equation, Preprint no. 3086, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3086 .
Abstract, PDF (651 kByte)
We study a spatial Markovian particle system with pairwise coagulation, a spatial version of the MarcusLushnikov process: according to a coagulation kernel K, particle pairs merge into a single particle, and their masses are united. We introduce a statisticalmechanics approach to the study of this process. We derive an explicit formula for the empirical process of the particle configuration at a given fixed time T in terms of a reference Poisson point process, whose points are trajectories that coagulate into one particle by time T. The noncoagulation between any two of them induces an exponential pairinteraction, which turns the description into a manybody system with a Gibbsian pairinteraction. Based on this, we first give a largedeviation principle for the joint distribution of the particle histories (conditioning on an upper bound for particle sizes), in the limit as the number N of initial atoms diverges and the kernel scales as 1/N K. We characterise the minimiser(s) of the rate function, we give criteria for its uniqueness and prove a law of large numbers (unconditioned). Furthermore, we use the unique minimiser to construct a solution of the Smoluchowski equation and give a criterion for the occurrence of a gelation phase transition. endabstract 
M. Eigel, Ch. Miranda, J. Schütte, D. Sommer, Approximating Langevin Monte Carlo with ResNetlike neural network architectures, Preprint no. 3077, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3077 .
Abstract, PDF (795 kByte)
We sample from a given target distribution by constructing a neural network which maps samples from a simple reference, e.g. the standard normal distribution, to samples from the target. To that end, we propose using a neural network architecture inspired by the Langevin Monte Carlo (LMC) algorithm. Based on LMC perturbation results, we show approximation rates of the proposed architecture for smooth, logconcave target distributions measured in the Wasserstein2 distance. The analysis heavily relies on the notion of subGaussianity of the intermediate measures of the perturbed LMC process. In particular, we derive bounds on the growth of the intermediate variance proxies under different assumptions on the perturbations. Moreover, we propose an architecture similar to deep residual neural networks and derive expressivity results for approximating the sample to target distribution map. 
W. König, Q. Vogel, A. Zass, Offdiagonal longrange order for the free Bose gas via the FeynmanKac formula, Preprint no. 3067, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3067 .
Abstract, PDF (416 kByte)
We consider the pathintegral representation of the ideal Bose gas under various boundary conditions. We show that BoseEinstein condensation occurs at the famous critical density threshold, by proving that its $1$particlereduced density matrix exhibits offdiagonal longrange order above that threshold, but not below. Our proofs are based on the wellknown FeynmanKac formula and a representation in terms of a crucial Poisson point process. Furthermore, in the condensation regime, we derive a law of large numbers with strong concentration for the number of particles in short loops. In contrast to the situation for free boundary conditions, where the entire condensate sits in just one loop, for all other boundary conditions we obtain the limiting PoissonDirichlet distribution for the collection of the lengths of all long loops. Our proofs are new and purely probabilistic (a part from a standard eigenvalue expansion), using elementary tools like Markov's inequality, Poisson point processes, combinatorial formulas for cardinalities of particular partition sets and asymptotics for random walks with Paretodistributed steps. 
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. 
J. Kern, Exponential equivalence for misanthrope processes in contact with weak reservoirs and applications to totally asymmetric exclusion processes, Preprint no. 3051, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3051 .
Abstract, PDF (200 kByte)
We provide a short proof for the exponential equivalence between misanthrope processes in contact with weak reservoirs and those with impermeable boundaries. As a consequence, we can derive both the hydrodynamic limit and the large deviations of the totally asymmetric exclusion process (TASEP) in contact with weak reservoirs. This extends a recent result which proved the hydrodynamic behaviour of a vanishing viscocity approximation of the TASEP in contact with weak reservoirs. Furthermore, applications to a class of asymmetric exclusion processes with long jumps is discussed. 
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, L. Lüchtrath, Existence of subcritical percolation phases for generalised weightdependent random connection models, Preprint no. 2993, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.2993 .
Abstract, PDF (299 kByte)
We derive a sufficient condition for the existence of a subcritical percolation phase for a wide range of continuum percolation models where each vertex is embedded into Euclidean space and carries an independent weight. In contrast to many established models, the presence of an edge is not only allowed to depend on the distance and weights of its end vertices but can also depend on the surrounding vertex set. Our result can be applied in particular to models combining heavytailed degree distributions and longrange effects, which are typically well connected. Moreover, we establish bounds on the taildistribution of the number of points and the diameter of the subcritical component of a typical point. The proofs rest on a multiscale argument. 
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

W. König, Offdiagonal longrange order for the free Bose gas via the FeynmanKac formula, Forschungsseminar Analysis, FernUniversit ¨at in Hagen , Fakultät für Mathematik und Informatik, online, null, April 24, 2024.

T. Iyer, Properties of recursive trees with independent fitnesses (online talk), Seminar Complex Systems, Queen Mary University of London, School of Mathematical Sciences, London, UK, February 14, 2023.

J. Kern, Mini Course: A unified approach to nongradient systems, Seminar Interacting Random Systems (Hybrid Event), May 11  15, 2023, WIAS Berlin.

J. Kern, Young measures and the hydrodynamic limit of asymmetric exclusion processes, Seminar Stochastics, Technical University of Lisbon, Lisbon, Portugal, July 10, 2023.

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, April 24  28, 2023, Westfälische WilhelmsUniversität Münster, Fachbereich Mathematik und Informatik, April 25, 2023.

J. Köppl, The longtime behaviour of interacting particle systems, Stochastic Processes and Related Fields, Kyoto, September 4  8, 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.

A.D. Vu, Discrete contact process in random environment, Mathematics of Random Systems: Summer School 2023, September 11  15, 2023, Kyoto University, Research Institute for Mathematical Sciences, Kyoto, Japan, September 15, 2023.

A.D. Vu, Percolation on the Manhattan grid, Stochastic Processes and Related Fields, Kyoto, Japan, September 4  8, 2023.

A.D. Vu, Percolation on the Manhattan grid, 16th German Probability and Statistics Days (GPSD) 2023, March 7  10, 2023, Universität DuisburgEssen, March 9, 2023.

A.D. Vu, Percolation on the Manhattan grid, 18. Doktorand:innentreffen der Stochastik 2023, August 21  23, 2023, Universität Heidelberg, Fakultät für Mathematik und Informatik, August 23, 2023.

C. ZarcoRomero, Wilson's current idea using Markov Chains (online talk), 56th National Congress of the Mexican Mathematical Society, Autonomous University of San Luis Potosí, San Luis Potosí, Mexico, October 9, 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.

E. Magnanini, Gelation and hydrodynamic limits in a spatial MarcusLushnikov process, In Search of Model Structures for Nonequilibrium Systems, Münster, April 24  28, 2023.

E. Magnanini, Gelation and hydrodynamic limits in a spatial MarcusLushnikov process, Workshop ``In search of model structures for nonequilibrium systems'', April 24  28, 2023, Westfälische WilhelmsUniversität Münster, Fachbereich Mathematik und Informatik, April 25, 2023.

E. Magnanini, Gelation in a spatial MarcusLushnikov process, Workshop MathMicS 2023: Mathematics and Microscopic Theory for Random Soft Matter Systems, Düsseldorf, February 13  15, 2023.

E. Magnanini, Gelation in a spatial MarcusLushnikov process, 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.

E. Magnanini, Spatial coagulation and gelation, SPP2265ReviewerKolloquium, Köln, August 29, 2023.

B. Jahnel, Continuum percolation in random environment, Oberseminar zur Stochastik, OttovonGuerickeUniversität Magdeburg, Fakultät für Mathematik, June 22, 2023.

B. Jahnel, Percolation, Oberseminar, Technische Universität Braunschweig, Institut für Mathematische Stochastik, November 8, 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, 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.

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.

W. König, Survey of 1st phase of the SPP2265, SPP2265ReviewerKolloquium, August 29, 2023, Deutsches Zentrum für Luft und Raumfahrt, Köln, August 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, Euclidean diameter of the soft Boolean model, 29th Nordic Congress of Mathematicians with EMS, July 3  7, 2023, Aalborg University, Department of Mathematical Sciences, Denmark, July 6, 2023.

L. Lüchtrath, Finite percolation thresholds in one dimensional inhomogeneous random graphs, 16th German Probability and Statistics Days (GPSD) 2023, March 7  10, 2023, Universität DuisburgEssen, March 7, 2023.

L. Lüchtrath, Percolation in weightdependent random connection models, Workshop on Random Discrete Structures, Münster, March 20  24, 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.

L. Lüchtrath, The random cluster graph, Probability and Financical Mathematics Seminar, University of Leeds, School of Mathematics, Leeds, UK, November 23, 2023.

L. Lüchtrath, The random cluster graph, Probability Seminar, University of Sheffield, School of Mathematics and Statistics, Sheffield, UK, November 15, 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.

W. van Zuijlen, Weakly self avoiding walk in a random potential, iPOD Seminar, Leiden University, Institute of Mathematics, Leiden, Netherlands, May 4, 2023.

W. van Zuijlen, Weakly self avoiding walk in a random potential, Probability Seminar, University of Warwick, Mathematics Institute, UK, June 14, 2023.

W. van Zuijlen, Weakly self avoiding walk in a random potential, Seminar Dipartimento di Matematica e Applicazioni, Università degli Studi di MilanoBicocca, Milano, Italy, September 7, 2023.

W. van Zuijlen, Weakly selfavoiding walk in a random potential (part I and II), Workshop ``Polymers and selfavoiding walks'', May 31  June 2, 2023, Henri Lebesgue, Centre de Mathematiques, Nantes, France.

T. Iyer, Preferential attachment trees with neighbourhood influence, Summer School: Mathematics of Large Networks, May 30  June 3, 2022, Erdős Center, Budapest, Hungary, May 31, 2022.

T. Iyer, Preferential attachment trees with neighbourhood influence, Probability Seminar, Universität zu Köln, Department Mathematik/Informatik, April 29, 2022.

T. Iyer, The influence of competition on genealogical trees associated with explosive agedependent branching processes, Oberseminar Stochastik, Westfälische WilhelmsUniversität Münster, Fachbereich Mathematik und Informatik, November 16, 2022.

E. Magnanini, Gelation in a Spatial MarcusLushnikov Process, Conference for Junior Female Researchers in Probability 2022, October 5  7, 2022, Stochastic Analysis in Interaction. BerlinOxford IRTG 2544, October 4, 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.

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.

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. BerlinOxford 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 devicetodevice networks (online talk), Telecom Orange Paris, France, July 6, 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, 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 manybody 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'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, Coarsegraining via EDPconvergence for linear fastslow 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 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.

L. Andreis, A large deviations approach to sparse random graphs (online talk), BernoulliIMS One World Symposium 2020 (Online Event), August 24  28, 2020, A virtual one week symposium on Probability and Mathematical Statistics. BernoulliIMS 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, EDPconvergence for multiscale gradient systems with applications to fastslow reaction systems (online talk), One World Dynamics Seminar (Online Event), Technische Universität München, November 13, 2020.

A. Hinsen, Data mobility in adhoc 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 BerlinLeipzig Workshop on Fluctuating Hydrodynamics, August 26  30, 2019, MaxPlanckInstitut 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, BerlinLeipzig Workshop in Analysis and Stochastics, January 16  18, 2019, MaxPlanckInstitut 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, MartinLutherUniversität HalleWittenberg, 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, McKeanVlasov SDEs with irregular drift: Large deviations for particle approximation, University of Oxford, Mathematical Institute, UK, March 5, 2018.

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

L. Andreis, A largedeviations approach to the multiplicative coagulation process, Probability Seminar, Università degli Studi di Padova, Dipartimento di Matematica ``Tullio LeviCivita'', Italy, October 12, 2018.

L. Andreis, A largedeviations approach to the multiplicative coagulation process, Seminar ''Theory of Complex Systems and Neurophysics  Theory of Statistical Physics and Nonlinear Dynamics``, HumboldtUniversität 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 StochastikTage, February 27  March 2, 2018, AlbertLudwigsUniversität Freiburg, Abteilung für Mathematische Stochastik, Freiburg, February 1, 2018.

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

L. Andreis, Selfsustained periodic behavior in interacting systems, Random Structures in Neuroscience and Biology, March 26  29, 2018, LudwigMaximilians 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 nonNewtonian 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 LiFePO4electrodes, 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.
External Preprints

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 edgetriangle model, a twoparameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the socalled 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 meanfield 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 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. 
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 jumpdiffusion 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 twoisland 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 jumpdiffusion 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 multicompartment population balance model for highshear wet granulation with Discrete Element Method, Technical report no. 170, c4ePreprint Series, 2016.
Abstract
This paper presents a multicompartment 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 particleparticle 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.