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
Articles in Refereed Journals

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.
Preprints, Reports, Technical Reports

M. Mittnenzweig, Hydrodynamic limit and large deviations of reactiondiffusion master equations, Preprint no. 2521, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2521 .
Abstract, PDF (389 kByte)
We derive the hydrodynamic limit of a reactiondiffusion master equation, that combines an exclusion process with a reversible chemical master equation expression for the reaction rates. The crucial assumption is that the associated macroscopic reaction network has a detailed balance equilibrium. The hydrodynamic limit is given by a system of reactiondiffusion equations with a modified mass action law for the reaction rates. We provide the upper bound for large deviations of the empirical measure from the hydrodynamic limit. 
P. Dupuis, V. Laschos, K. Ramanan, Exit time risksensitive stochastic control problems related to systems of cooperative agents, Preprint no. 2407, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2407 .
Abstract, PDF (447 kByte)
We study sequences, parametrized by the number of agents, of exit time stochastic control problems with risksensitive costs structures generate by unbounded costs. We identify a fully characterizing assumption, under which, each of them corresponds to a riskneutral stochastic control problem with additive cost, and also to a riskneutral stochastic control problem on the simplex, where the specific information about the state of each agent can be discarded. We finally prove that, under some additional assumptions, the sequence of value functions converges to the value function of a deterministic control problem.
Talks, Poster

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.

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. 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.