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 Particle-based 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 Particle-based modelling in the Sciences.
A (static) interacting many-body 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 pre-factor, 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 mean-field 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 Particle-based 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.
J. Blath, A. González Casanova Soberón, B. Eldon, N. Kurt, M. Wilke-Berenguer, Genetic variability under the seedbank coalescent, Genetics, 200 (2015) pp. 921--934.
We analyze patterns of genetic variability of populations in the presence of a large seedbank with the help of a new coalescent structure called the seedbank coalescent. This ancestral process appears naturally as a scaling limit of the genealogy of large populations that sustain seedbanks, if the seedbank size and individual dormancy times are of the same order as those of the active population. Mutations appear as Poisson processes on the active lineages and potentially at reduced rate also on the dormant lineages. The presence of "dormant" lineages leads to qualitatively altered times to the most recent common ancestor and nonclassical patterns of genetic diversity. To illustrate this we provide a Wright-Fisher model with a seedbank component and mutation, motivated from recent models of microbial dormancy, whose genealogy can be described by the seedbank coalescent. Based on our coalescent model, we derive recursions for the expectation and variance of the time to most recent common ancestor, number of segregating sites, pairwise differences, and singletons. Estimates (obtained by simulations) of the distributions of commonly employed distance statistics, in the presence and absence of a seedbank, are compared. The effect of a seedbank on the expected site-frequency spectrum is also investigated using simulations. Our results indicate that the presence of a large seedbank considerably alters the distribution of some distance statistics, as well as the site-frequency spectrum. Thus, one should be able to detect from genetic data the presence of a large seedbank in natural populations.
L. Avena, O. Gün, M. Hesse, The parabolic Anderson model on the hypercube, Preprint no. 2319, WIAS, Berlin, 2016, DOI 10.20347/WIAS.PREPRINT.2319 .
Abstract, PDF (240 kByte)
We consider the parabolic Anderson model (PAM) on the n-dimensional hypercube with random i.i.d. potentials. We parametrize time by volume and study the solution at the location of the k-th largest potential. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales it behaves like a system without diffusion, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the Anderson operator, for which we give precise asymptotics. Moreover, the transition time depends only on the difference between the largest and k-th largest potential. One of our main motivations in this article is to investigate the mutation-selection model of population genetics on a random fitness landscape, which is given by the ratio of the solution of PAM to its total mass, with the field corresponding to the fitness landscape. We show that the phase transition of the solution translates to the mutation-selection model as follows: a population initially concentrated at the site of the k-th best fitness value moves completely to the site of the best fitness on time scales where the transition of growth rates happens. The class of potentials we consider involve the Random Energy Model (REM) of statistical physics which is studied as one of the main examples of a random fitness landscape.
A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, Preprint no. 2165, WIAS, Berlin, 2015, DOI 10.20347/WIAS.PREPRINT.2165 .
Abstract, PDF (363 kByte)
We study stochastic interacting particle systems that model chemical reaction networks on the micro scale, converging to the macroscopic Reaction Rate Equation. One abstraction level higher, we study the ensemble of such particle systems, converging to the corresponding Liouville transport equation. For both systems, we calculate the corresponding large deviations and show that under the condition of detailed balance, the large deviations induce a non-linear relation between thermodynamic fluxes and free energy driving force.
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.
K.F. Lee, M. Dosta, A.D. Mcguire, S. Mosbach, W. Wagner, S. Heinrich, M. Kraft, Development of a multi-compartment population balance model for high-shear wet granulation with Discrete Element Method, Technical report no. 170, c4e-Preprint Series, 2016.
This paper presents a multi-compartment population balance model for wet granulation coupled with DEM (Discrete Element Method) simulations. Methodologies are developed to extract relevant data from the DEM simulations to inform the population balance model. First, compartmental residence times are calculated for the population balance model from DEM. Then, a suitable collision kernel is chosen for the population balance model based on particle-particle collision frequencies extracted from DEM. It is found t hat the population balance model is able to predict the trends exhibited by the experimental size and porosity distributions by utilising the information provided by the DEM simulations.