Kinetic equations describe the rate at which a system or mixture changes its chemical properties, for example a melting snowman turns into water, or a potato becomes soft during cooking, or wood in a fire turns into gas. Such equations are often nonlinear, because interactions in the material are complex and the speed of change is dependent on the system size as well as the strength of the external influences. For example, doubling the mass of a snowman will not generally double the rate at which water is produced when it stands in the sun, because the surface, the only part that is likely to melt, only increases by a factor of 1.6.
The institute works on more complex problems than the melting of snowmen: In order to advance our understanding of the physical world, gases made up of many small molecules, clouds of droplets and collections of charged particles are considered. The kinetic equations for these problems are generally of a very high dimensional and complex nature. As a consequence the existence, uniqueness and regularity of solutions to these kinetic equations is far from obvious.
Two classic kinetic equations are the Boltzmann equation for gas molecules and the Smoluchowski equations for coagulating particles. Both models provide a continuum description for systems of colliding particles, in the Boltzmann case the particles rebound elastically on collision, in the Smoluchowski case they stick together.
Contribution of the Insitute
At the institute, kinetic equations are studied using analytic methods, such as fixed point techniques and via approximating stochastic particle systems. These particle systems generally reflect in some approximate way the discrete nature of matter on a sufficiently small scale. This perspective has been extensively exploited for numerical purposes and a wide range of results have been published showing that convergence of stochastic particle approximations to solutions of nonlinear kinetic equations.
The WIAS also derives new kinetic equations as hydrodynamic limits of more detailed physical models. For example, detailed charging phenomena for new battery technologies are studied in this way, see the applied research theme Phase transition and hysteresis in the context of storage problems.
Publications
Monographs

S. Rjasanow, W. Wagner, Stochastic Numerics for the Boltzmann Equation, 37 of Springer Series in Computational Mathematics, Springer, Berlin, 2005, xiii+256 pages, (Monograph Published).

W. Wagner, Stochastic models and Monte Carlo algorithms for Boltzmann type equations, in: Monte Carlo and QuasiMonte Carlo Methods 2002, H. Niederreiter, ed., Springer, New York, 2004, pp. 129153, (Chapter Published).
Articles in Refereed Journals

W. Dreyer, P. Friz, P. Gajewski, C. Guhlke, M. Maurelli, Stochastic manyparticle model for LFP electrodes, Continuum Mechanics and Thermodynamics, 30 (2018), pp. 593628, DOI 10.1007/s0016101806297 .
Abstract
In the framework of nonequilibrium thermodynamics we derive a new model for porous electrodes. The model is applied to LiFePO4 (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithiumpoor to a lithiumrich phase within LFP electrodes is controlled by surface fluctuations leading to a system of stochastic differential equations. The model is capable to derive an explicit relation between battery voltage and current that is controlled by thermodynamic state variables. This voltagecurrent relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltagecharge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates. 
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.

R.I.A. Patterson, Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries, Journal of Evolution Equations, 16 (2016), pp. 261291.
Abstract
Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of $d$dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semilinear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semigroups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, Stoch. Anal. Appl. 31, 2013] to a unique and differentiable limit. 
R.I.A. Patterson, W. Wagner, Cell size error in stochastic particle methods for coagulation equations with advection, SIAM Journal on Numerical Analysis, 52 (2014), pp. 424442.
Abstract
The paper studies the approximation error in stochastic particle methods for spatially inhomogeneous population balance equations. The model includes advection, coagulation and inception. Sufficient conditions for second order approximation with respect to the spatial discretization parameter (cell size) are provided. Examples are given, where only first order approximation is observed. 
O. Muscato, V. Di Stefano, W. Wagner, A variancereduced electrothermal Monte Carlo method for semiconductor device simulation, Computers & Mathematics with Applications. An International Journal, 65 (2013), pp. 520527.
Abstract
This paper is concerned with electron transport and heat generation in semiconductor devices. An improved version of the electrothermal Monte Carlo method is presented. This modification has better approximation properties due to reduced statistical fluctuations. The corresponding transport equations are provided and results of numerical experiments are presented. 
W. Wagner, Some properties of the kinetic equation for electron transport in semiconductors, Kinetic and Related Models, 6 (2013), pp. 955967.
Abstract
The paper studies the kinetic equation for electron transport in semiconductors. New formulas for the heat generation rate are derived by analyzing the basic scattering mechanisms. In addition, properties of the steady state distribution are discussed and possible extensions of the deviational particle Monte Carlo method to the area of electron transport are proposed. 
M. Sander, R.I.A. Patterson, A. Braumann, A. Raj, M. Kraft, Developing the PAHPP soot particle model using process informatics and uncertainty propagation, Proceedings of the Combustion Institute, 33 (2011), pp. 675683.
Abstract
n this work we present the new PAHPP soot model and use a data collaboration approach to determine some of its parameters. The model describes the formation, growth and oxidation of soot in laminar premixed flames. Soot particles are modelled as aggregates containing primary particles, which are built from polycyclic aromatic hydrocarbons (PAHs), the main building blocks of a primary particle (PP). The connectivity of the primary particles is stored and used to determine the rounding of the soot particles due to surface growth and condensation processes. Two neighbouring primary particles are replaced by one if the coalescence level between the two primary particles reaches a threshold. The model contains, like most of the other models, free parameters that are unknown a priori. The experimental premixed flame data from Zhao et al. [B. Zhao, Z. Yang, Z. Li, M.V. Johnston, H. Wang, Proc. Combust. Inst. 30 (2) (2005) 1441?1448] have been used to estimate the smoothing factor of soot particles, the growth factor of PAHs within particles and the soot density using a low discrepancy series method with a subsequent response surface optimisation. The optimised particle size distributions show good agreement with the experimental ones. The importance of a standardised data mining system in order to optimise models is underlined. 
O. Muscato, W. Wagner, V. Di Stefano, Properties of the steady state distribution of electrons in semiconductors, Kinetic and Related Models, 4 (2011), pp. 809829.
Abstract
This paper studies a Boltzmann transport equation with several electronphonon scattering mechanisms, which describes the charge transport in semiconductors. The electric field is coupled to the electron distribution function via Poisson's equation. Both the parabolic and the quasiparabolic band approximations are considered. The steady state behaviour of the electron distribution function is investigated by a Monte Carlo algorithm. More precisely, several nonlinear functionals of the solution are calculated that quantify the deviation of the steady state from a Maxwellian distribution with respect to the wavevector. On the one hand, the numerical results illustrate known theoretical statements about the steady state and indicate possible directions for future studies. On the other hand, the nonlinear functionals provide tools that can be used in the framework of Monte Carlo algorithms for detecting regions in which the steady state distribution has a relatively simple structure, thus providing a basis for domain decomposition methods. 
W. Wagner, Stochastic models in kinetic theory, Physics of Fluids, 23 (2011), pp. 030602/1030602/14.
Abstract
The paper is concerned with some aspects of stochastic modelling in kinetic theory. First, an overview of the role of particle models with random interactions is given. These models are important both in the context of foundations of kinetic theory and for the design of numerical algorithms in various engineering applications. Then, the class of jump processes with a finite number of states is considered. Two types of such processes are studied, where particles change their states either independently of each other (monomolecular processes), or via binary interactions (bimolecular processes). The relationship of these processes with corresponding kinetic equations is discussed. Equations are derived both for the average relative numbers of particles in a given state and for the fluctuations of these numbers around their averages. The simplicity of the models makes several aspects of the theory more transparent. 
O. Muscato, V. Di Stefano, W. Wagner, Numerical study of the systematic error in Monte Carlo schemes for semiconductors, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), pp. 10491068.
Abstract
The paper studies the convergence behavior of Monte Carlo schemes for semiconductors. A detailed analysis of the systematic error with respect to numerical parameters is performed. Different sources of systematic error are pointed out and illustrated in a spatially onedimensional test case. The error with respect to the number of simulation particles occurs during the calculation of the internal electric field. The time step error, which is related to the splitting of transport and electric field calculations, vanishes sufficiently fast. The error due to the approximation of the trajectories of particles depends on the ODE solver used in the algorithm. It is negligible compared to the other sources of time step error, when a second order RungeKutta solver is used. The error related to the approximate scattering mechanism is the most significant source of error with respect to the time step. 
W. Wagner, Deviational particle Monte Carlo for the Boltzmann equation, Monte Carlo Methods and Applications, 14 (2008), pp. 191268.
Abstract
The paper describes the deviational particle Monte Carlo method for the Boltzmann equation. The approach is an application of the general “control variates” variance reduction technique to the problem of solving a nonlinear equation. The deviation of the solution from a reference Maxwellian is approximated by a system of positive and negative particles. Previous results from the literature are modified and extended. New algorithms are proposed that cover the nonlinear Boltzmann equation (instead of a linearized version) with a general interaction model (instead of hard spheres). The algorithms are obtained as procedures for generating trajectories of Markov jump processes. This provides the framework for deriving the limiting equations, when the number of particles tends to infinity. These equations reflect the influence of various numerical approximation parameters. Detailed simulation schemes are provided for the variable hard sphere interaction model. 
S. Rjasanow, W. Wagner, Time splitting error in DSMC schemes for the spatially homogeneous inelastic Boltzmann equation, , 45 (2007), pp. 5467.

I.M. Gamba, S. Rjasanow, W. Wagner, Direct simulation of the uniformly heated granular Boltzmann equation, Mathematical and Computer Modelling, 42 (2005), pp. 683700.

O. Muscato, W. Wagner, Time step truncation in direct simulation Monte Carlo for semiconductors, COMPEL. The International Journal for Computation and Mathematics in Electrical and Electronic Engineering. Emerald, Bradford, West Yorkshire. English, English abstracts., 24 (2005), pp. 13511366.

O. Muscato, W. Wagner, Time step truncation in direct simulation Monte Carlo for semiconductors, .

A.L. Garcia, W. Wagner, Direct simulation Monte Carlo method for the UehlingUhlenbeckBoltzmann equation, Phys. Rev. E (3), 68, 056703 (2003), 11.

I. Matheis, W. Wagner, Convergence of the stochastic weighted particle method for the Boltzmann equation, , 24 (2003), pp. 15891609.
Preprints, Reports, Technical Reports

D. Heydecker, R.I.A. Patterson, Bilinear coagulation equations, Preprint no. 2637, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2637 .
Abstract, PDF (453 kByte)
We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form π(y) · Aπ (x) for a vector of conserved quantities π, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time t_{g} ∈ (0,∞), which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time. 
L. Andreis, W. König, R.I.A. Patterson, A largedeviations approach to gelation, Preprint no. 2568, WIAS, Berlin, 2019, DOI 10.20347/WIAS.PREPRINT.2568 .
Abstract, PDF (338 kByte)
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.
Talks, Poster

R.I.A. Patterson, A novel simulation method for stochastic particle systems, Seminar, Department of Chemical Engineering and Biotechnology, University of Cambridge, Faculty of Mathematics, UK, May 9, 2019.

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

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

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.

R.I.A. Patterson, Coagulation  Transport Simulations with Stochastic Particles, CIMWIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6  8, 2017, University of Lisbon, International Center for Mathematics, Lisboa, Portugal, December 7, 2017.

M. Maurelli, Regularization by noise for scalar conservation laws, Mathematical Finance and Stochastic Analysis Seminars, University of York, UK, October 26, 2016.

CH. Heinemann, On elastic CahnHilliard systems coupled with evolution inclusions for damage processes, 86th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2015), Young Researchers' Minisymposium 2, March 23  27, 2015, Lecce, Italy, March 23, 2015.

R. Soares Dos Santos, Random walk on random walks, YEP XII: Workshop on Random Walk in Random Environment, March 23  27, 2015, Technical University of Eindhoven, EURANDOM, Netherlands, March 24, 2015.

R.I.A. Patterson, Uniqueness and regularity for coagulationadvection problems, Workshop on Theory and Numerics of Kinetic Equations, June 1  4, 2015, Universität Saarbrücken, June 2, 2015.

W. Wagner, Stochastic weighted algorithms for population balance equations with multidimensional type space, 10th IMACS Seminar on Monte Carlo Methods, July 6  10, 2015, Johannes Kepler Universität Linz, Linz, Austria, July 6, 2015.

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

W. Wagner, Kinetic equations and Markov jump processes, Isaac Newton Institute for Mathematical Sciences, Programme: Partial Differential Equations in Kinetic Theories, Cambridge, UK, November 29, 2010.

W. Wagner, Stochastische Modellierung für die BoltzmannGleichung, WIASDay, Berlin, February 24, 2006.
External Preprints

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