For more than a hundred years diverse processes and phenomena in the natural sciences have been modelled using random particle systems. Starting in the 19th century scientist have become used to the idea of regarding things like fluids, gases, light and solid materials as enormous collections of interacting particles. As a result ever more models that specify a large number of particles and rules for how they interact have been both proposed and mathematically analysed. In such cases one often make a stochastic assumption, that is the the locations or motions of the particles are somehow stochastic, additional model components may also be specified stochastically, for example the environment and the interactions. We distinguish between dynamic models where the time evolution of a particle system is described and static systems where the particles do not move.

The task is then to describe the macroscopic behaviour of the complete system, to explain it mathematically and where possible to relate it to experimental data. In many cases this task becomes ones of finding or adapting methods to describe important order parameters and for proving how the qualitative aspects of the system behaviour depends on these parameters. A typical example of an order parameter is the empirical average measure of the particles, which, under an appropriate rescaling, allows an approximate description of the overall system by a single equation, in most cases a differential equation. This procedure is often carried out in the thermodynamic limit where many particles are considered in an expanding large box, but where the concentration of particles is also increased so as to keep the concentration constant. A second important setting is the hydrodynamic limit where the box stays constant, but the number of particles increases while their effective size decreases. Residence probabilities for the particles after averaging can also satisfy interesting equations. Some of these equations where studied long before they were derived from particle models!

Particle models have become especially widespread in physics and chemistry as a good compromise between reality and tractability. For example, static, atomic many body systems are often described through an energy function, which assigns every possible configuration an energy based on the interaction between the particles and then interprets the negative exponential of the energy as proportional to the configuration probability. Such distributions are called Gibbs measures; they preferentially select configurations with low energies. An example is a salt crystal, which consists of charged particles (ions) seeking to minimise their combined electrostatic potential energy. Other systems, especially at positive temperatures contain random walks or Brownian motions, which react (e.g. coagulate) with each other when in close proximity (see the applied theme Coagulation). In this way we model, for example, the formation of soot particles in flames. A related class of stochastic particle models are families of interacting stochastic (partial) differential equations, which have recently been used in the modelling of battery charging (see the applied theme Thermodynamic models for electrochemical systems).

Contribution of the Institute

Atomic, static models for interacting many body systems are described with Lennard-Jones potentials, which cause the particles to maintain a certain amount of separation and not to collapse onto a single point. Another example is the Bose-gas in which every particle has a kinetic energy in addition to its position. The work of the WIAS on the first model deals with the formation of clusters and crystallisation, and for the Bose-gas with condensation phenomena; see the mathematical theme Large Deviations

A mixture of crystallised and disorganised matter
A realisation of a many body system showing a small crystal in the lower right corner.

Models with many random particles are also used for the description of large wireless telecommunications systems; in this case the particles are the user-devices . When the movement of the users does not have to be considered, the modelling of the device locations is typically via a Poisson point process, but when the motion of users becomes important it is not yet clear how to model user paths especially as user behaviour undergoes periodic qualitative changes (e.g. between day and night). The particle interactions depend on their separation since a message can only be effectively transmitted when two devices are within range of each other; see the applied theme Mobile Communication Networks. In this connection we have used methods from the theory of large deviations to analyse the positions of the devices. By performing a constrained energy minimisation we are able to characterise the most important particle distributions for which no effective network can be established.

For dynamic models a wide range of hydrodynamic limit results have been proved dealing with elastically colliding gas molecules, soot formation and chemical reactions and leading to kinetic equations (see the Mathematical theme Nonlinear kinetic equations). For a combined generalisation of soot formation and chemical reactions, a dynamic large deviations principle was derived. With additional analytic tools an entropy-like free energy and its dissipation potentials were identified. Together they form a gradient structure and provide a more detailed description of the dynamics and the effect of perturbations.

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. Further biological models can be found in the Applied Theme Stochastic biological evolution.


Publications

  Monographs

  • P. Exner, W. König, H. Neidhardt, eds., Mathematical Results in Quantum Mechanics. Proceedings of the QMath12 Conference, World Scientific Publishing, Singapore, 2015, xii+383 pages, (Collection Published).

  Articles in Refereed Journals

  • A. Mielke, R.I.A. Patterson, M.A. Peletier, D.R.M. Renger, Non-equilibrium thermodynamical principles for chemical reactions with mass-action kinetics, SIAM Journal on Applied Mathematics, 77 (2017) pp. 1562--1585, DOI 10.20347/WIAS.PREPRINT.2165 .
    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 non-linear relation between thermodynamic fluxes and free energy driving force.

  • S. Jansen, W. König, B. Metzger, Large deviations for cluster size distributions in a continuous classical many-body system, The Annals of Applied Probability, 25 (2015) pp. 930--973.
    Abstract
    An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $betain(0,infty)$ and particle density $rhoin(0,rho_rmcp)$ in the thermodynamic limit. Here $rho_rmcp >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Gamma$-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit $betatoinfty$, $rho downarrow 0$ such that $-beta^-1logrhoto nu$ for some $nuin(0,infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the de-coupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $nu$. Under additional assumptions on the potential, the $Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.

  • M. Erbar, J. Maas, D.R.M. Renger, From large deviations to Wasserstein gradient flows in multiple dimensions, Electronic Communications in Probability, 20 (2015) pp. 1--12.
    Abstract
    We study the large deviation rate functional for the empirical distribution of independent Brownian particles with drift. In one dimension, it has been shown by Adams, Dirr, Peletier and Zimmer [ADPZ11] that this functional is asymptotically equivalent (in the sense of Gamma-convergence) to the Jordan-Kinderlehrer-Otto functional arising in the Wasserstein gradient flow structure of the Fokker-Planck equation. In higher dimensions, part of this statement (the lower bound) has been recently proved by Duong, Laschos and Renger, but the upper bound remained open, since the proof in [DLR13] relies on regularity properties of optimal transport maps that are restricted to one dimension. In this note we present a new proof of the upper bound, thereby generalising the result of [ADPZ11] to arbitrary dimensions.

  • M. Muminov, H. Neidhardt, T. Rasulov, On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case, Journal of Mathematical Physics, 56 (2015) pp. 053507/1--053507/24.
    Abstract
    A lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling constant the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model H for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of H below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form.

  • S. Simonella, M. Pulvirenti, On the evolution of the empirical measure for hard-sphere dynamics, Bulletin of the Institute of Mathematics. Academia Sinica. Institute of Mathematics, Academia Sinica, Taipei, Taiwan. English. English summary., 10 (2015) pp. 171--204.

  • A. Mielke, M.A. Peletier, D.R.M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Analysis, 41 (2014) pp. 1293--1325.
    Abstract
    Motivated by the occurence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions ℒ that induce a flow, given by ℒ(zt,żt)=0. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when ℒ is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.

  • M.H. Duong, V. Laschos, M. Renger, Wasserstein gradient flows from large deviations of many-particle limits, ESAIM. Control, Optimisation and Calculus of Variations, 19 (2013) pp. 1166--1188.

  • M.A. Peletier, M. Renger, M. Veneroni, Variational formulation of the Fokker--Planck equation with decay: A particle approach, Communications in Contemporary Mathematics, 15 (2013) pp. 1350017/1--1350017/43.

  • S. Adams, A. Collevecchio, W. König, A variational formula for the free energy of an interacting many-particle system, The Annals of Probability, 39 (2011) pp. 683--728.
    Abstract
    We consider $N$ bosons in a box in $R^d$ with volume $N/rho$ under the influence of a mutually repellent pair potential. The particle density $rhoin(0,infty)$ is kept fixed. Our main result is the identification of the limiting free energy, $f(beta,rho)$, at positive temperature $1/beta$, in terms of an explicit variational formula, for any fixed $rho$ if $beta$ is sufficiently small, and for any fixed $beta$ if $rho$ is sufficiently small. The thermodynamic equilibrium is described by the symmetrised trace of $rm e^-beta Hcal_N$, where $Hcal_N$ denotes the corresponding Hamilton operator. The well-known Feynman-Kac formula reformulates this trace in terms of $N$ interacting Brownian bridges. Due to the symmetrisation, the bridges are organised in an ensemble of cycles of various lengths. The novelty of our approach is a description in terms of a marked Poisson point process whose marks are the cycles. This allows for an asymptotic analysis of the system via a large-deviations analysis of the stationary empirical field. The resulting variational formula ranges over random shift-invariant marked point fields and optimizes the sum of the interaction and the relative entropy with respect to the reference process. In our proof of the lower bound for the free energy, we drop all interaction involving lq infinitely longrq cycles, and their possible presence is signalled by a loss of mass of the lq finitely longrq cycles in the variational formula. In the proof of the upper bound, we only keep the mass on the lq finitely longrq cycles. We expect that the precise relationship between these two bounds lies at the heart of Bose-Einstein condensation and intend to analyse it further in future.

  • M. Aizenman, S. Jansen, P. Jung, Symmetry breaking in quasi-1D Coulomb systems, Annales Henri Poincare. A Journal of Theoretical and Mathematical Physics, 11 (2010) pp. 1453--1485.
    Abstract
    Quasi one-dimensional systems are systems of particles in domains which are of infinite extent in one direction and of uniformly bounded size in all other directions, e.g. on a cylinder of infinite length. The main result proven here is that for such particle systems with Coulomb interactions and neutralizing background, the so-called ``jellium'', at any temperature and at any finite-strip width there is translation symmetry breaking. This extends the previous result on Laughlin states in thin, two-dimens The structural argument which is used here bypasses the question of whether the translation symmetry breaking is manifest already at the level of the one particle density function. It is akin to that employed by Aizenman and Martin (1980) for a similar statement concerning symmetry breaking at all temperatures in strictly one-dimensional Coulomb systems. The extension is enabled through bounds which establish tightness of finite-volume charge fluctuations.

  • A. Collevecchio, W. König, P. Mörters, N. Sidorova, Phase transitions for dilute particle systems with Lennard--Jones potential, Communications in Mathematical Physics, 299 (2010) pp. 603--630.

  Contributions to Collected Editions

  • M. Kantner, U. Bandelow, Th. Koprucki, H.-J. Wünsche, Multi-scale modelling and simulation of single-photon sources on a device level, in: Euro-TMCS II -- Theory, Modelling & Computational Methods for Semiconductors, 7th -- 9th December 2016, Tyndall National Institute, University College Cork, Ireland, E. O'Reilly, S. Schulz, S. Tomic, eds., Tyndall National Institute, 2016, pp. 65.

  Preprints, Reports, Technical Reports

  • M. Erbar, M. Fathi, V. Laschos, A. Schlichting, Gradient flow structure for McKean--Vlasov equations on discrete spaces, Preprint no. 2219, WIAS, Berlin, 2016.
    Abstract, PDF (393 kByte)
    In this work, we show that a family of non-linear mean-field equations on discrete spaces, can be viewed as a gradient flow of a natural free energy functional with respect to a certain metric structure we make explicit. We also prove that this gradient flow structure arises as the limit of the gradient flow structures of a natural sequence of N-particle dynamics, as N goes to infinity

  • 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.

  • W. Wagner, A random walk model for the Schrödinger equation, Preprint no. 2109, WIAS, Berlin, 2015.
    Abstract, PDF (143 kByte)
    A random walk model for the spatially discretized time-dependent Schrödinger equation is constructed. The model consists of a class of piecewise deterministic Markov processes. The states of the processes are characterized by a position and a complex-valued weight. Jumps occur both on the spatial grid and in the space of weights. Between the jumps, the weights change according to deterministic rules. The main result is that certain functionals of the processes satisfy the Schrödinger equation.

  Talks, Poster

  • M. Kantner, Multi-scale modeling and numerical simulation of single-photon emitters, Matheon Workshop--9th Annual Meeting ``Photonic Devices", Zuse Institut, Berlin, March 3, 2016.

  • M. Kantner, Multi-scale modelling and simulation of single-photon sources on a device level, Euro--TMCS II Theory, Modelling & Computational Methods for Semiconductors, Tyndall National Institute and University College Cork, Cork, Ireland, December 9, 2016.

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

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

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

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

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

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

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

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

  • S. Jansen, Large deviations for interacting many-particle systems in the Saha regime, Berlin-Leipzig Seminar on Analysis and Probability Theory, July 8, 2011, Technische Universität Clausthal, Institut für Mathematik, July 8, 2011.

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

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

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