Kolloquium Wintersemester 2016/17

18.10.2016

Variational Mean Field Games

Prof. Dr. Filippo Santambrogio, Laboratoire de Mathématiques d'Orsay, Université Paris-Sud

I will give a brief introduction to the the emerging topic of Mean Field Games, introduced by J-M Lasry and P-L Lions some years ago as a model for the Nash equilibrium of a population of agents each selecting his own evolution, taking into account the density of the other agents that he meets, in the form of a congestion charge. This gives rise to a coupled system of PDEs, a continuity equation where the density moves according to the gradient of a value function, and a Hamilton-Jacobi equation solved by the value function, where the density also appears.
I will mainly deal with the case where this equilibrium problem may be seen as optimality conditions of a convex variational problem, and give the main results in this framework. I will also present some recent regularity results which allow to give a precise mathematical meaning to the equilibrium condition. At the end of the talk I will present an interesting variant, where the congestion cost is replaced by a capacity constraint. In this case an extra unknown appears, which plays both the role of a pressure in the fluid mechanics point of view, and of a price in the economical interpretation.

(Einladender: Prof. Dr. B. Wirth)

Nonlinear dynamics of bacterial and active colloidal suspensions

Prof. Dr. Hartmut Löwen, Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf

This talk addresses the nonlinear dynamical aspects of active matter for both biological and
artificial colloidal microswimmers. 
[1]. A number of collective phenomena in active matter will be proposed. In detail, we shall discuss active turbulence in bacterial systems [2] and its ability to steer the transport of shuttles [3]. Then we turn to kinetic phase separation in artificial swimmers [4]. The late time kinetics of phase separation can be mapped on the Cahn-Hilliard model [5]. Moreover we put forward the concept of negative interfacial tension [6] in nonequilibrium phase-separated swimmer systems and that of the swim pressure at system boundaries [7].

References:

[1] For a recent review, see: C. Bechinger, R. di Leonardo, H. Lowen, C. Reichhardt, G. Volpe, G. Volpe,
  Active Brownian motion in complex and crowded environments, arXiv: 1602.00081

[2] H. H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher,  R. E. Goldstein,  H. Lowen, J. M. Yeomans, PNAS  109, 14308  (2012).

[3] A. Kaiser, A. Peshkow, A. Sokolov, B. ten Hagen, H. Lowen, I. S. Aranson, Physical Review Letters 112, 158101  (2014).

[4] I. Buttinoni, J. Bialke, F. Kummel, H. Lowen, C. Bechinger, T. Speck, Physical Review Letters 110, 238301  (2013).

[5] T. Speck, J. Bialke, A. M. Menzel, H. Lowen, Physical Review Letters 112, 218304 (2014).

[6] J. Bialk'e, J. T. Siebert, H. Lowen, T. Speck,Physical Review Letters  115, 098301  (2015).

[7] F. Smallenburg, H. Lowen, Physical Review E 92, 032304 (2015).

(Einladender: Prof. Dr. M. Burger)

Physical modeling of cellular motility

PD Dr. Falko Ziebert, Physikalisches Institut Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg

Substrate-based crawling motility of eukaryotic cells is essential for many biological functions, both in developing and mature organisms. Motility dysfunctions are involved in several life-threatening pathologies such as cancer and metastasis. Motile cells are also a natural realization of active, self-propelled ‘particles’, a popular research topic in nonequilibrium physics. Finally, from the materials perspective, assemblies of motile cells and evolving tissues constitute a class of adaptive self-healing materials that respond to the topography, elasticity, and surface chemistry of the environment and react to external stimuli.
Although a comprehensive understanding of substrate-based cell motility remains elusive, progress has been achieved recently in its modeling on the whole cell level. I will survey our recent advances in computational approaches to cell movement on structured substrates (with modulated adhesion or stiffness), as well as on collective cell migration.

Ziebert, F. & Aranson, I. S. Effects of adhesion dynamics and substrate compliance on the shape and motility of crawling cells. PLoS ONE 8, e64511 (2013).

Löber, J., Ziebert, F. & Aranson, I. S. Modeling crawling cell movement on soft engineered substrates. Soft Matt. 10, 1365–1373 (2014).

Löber, J., Ziebert, F. & Aranson, I. S. Collisions of deformable cells lead to collective migration. Sci. Rep. 5, 9172 (2015).

Ziebert, F., Löber, J. & Aranson, I. S. Macroscopic model of substrate-based cell motility. in Physical Models of Cell motility, Ed. I. S. Aranson, Springer (Switzerland) 1–67 (2016).

(Einladender: Prof. Dr. U. Thiele)

The Fibonacci family of dynamical universality classes

Prof. Dr. Gunter M. Schütz, Theoretical Soft Matter and Biophysics, Forschungszentrum Jülich GmbH

We use the theory of nonlinear fluctuating hydrodynamics to study stochastic transport far from thermal equilibrium in terms of the dynamical structure function which is universal at low frequencies and for large times and which encodes whether transport is diffusive or anomalous. For generic quasi one-dimensional systems we predict [1] that transport of mass, energy and other locally conserved quantities is governed by mode-dependent dynamical universality classes with dynamical exponents z which are Kepler ratios of neighboring Fibonacci numbers, starting with z = 2 (corresponding to a diffusive mode) or z = 3/2 (Kardar-Parisi-Zhang (KPZ) mode). If neither a diffusive nor a KPZ mode are present, all modes have as dynamical exponent the golden mean z=(1+\sqrt 5)/2. The universal scaling functions of the higher Fibonacci modes are Lévy distributions. The theoretical predictions are confirmed by Monte-Carlo simulations of a three-lane asymmetric simple exclusion process.

[1] V. Popkov, A. Schadschneider, J. Schmidt, and G. M. Schütz, PNAS Early Edition (2015), www.pnas.org/cgi/doi/10.1073/pnas.1512261112

(Einladender: Prof. Dr. S. Dereich)

Asymptotic homogenization for fluid and drug transport in vascularized tumors: theory and applications

Dr. Raimondo Penta, Departamento de Mecanica de los Medios Continuos y T. Estructuras,
E.T.S. de caminos, canales y puertos, Universidad Politecnica de Madrid

A vascularized tumor is characterized by a sharp length scale separation between the intercapillary distance (microscale) and the average tumor size (macroscale). Application of the asymptotic homogenization technique leads to the macroscale double Darcy, double advection-diffusion-reaction type model for fluid and drug transport in the tumor, vessels, and across the capillaries’ walls. We recover microscopic information solving cell problems representative of the microvessels’ geometries. We present the theory [1], and applications concerning the role of the vessels’ geometry on blood convection [2] and diffusion/consumption of anticancer drugs [3].

[1] R. Penta, D. Ambrosi, and A. Quarteroni. Multiscale homogenization for fluid and drug transport in vascularized malignant tissues. Mathematical Models and Methods in Applied Sciences, 25(1):79–108, 2015.

[2] R. Penta and D. Ambrosi. The role of the microvascular tortuosity in tumor transport phenomena. Journal of theoretical biology, 364:80–97, 2015.

[3] P.Mascheroni and R.Penta. The role of the angiogenic network structure on diffusion and consumption of anti-cancer drugs, Int. J. Numer. Meth. Biomed. Engng, submitted (2016).

(Einladender: Prof. Dr. M. Burger)

Bending, Buckling and Bifurcations

(Einladender: Prof. Dr. U. Thiele)

Quantifying and controlling the buckling and bending of solid materials has applications from molecular scales (DNA) to large-scale natural and artificial structures. In general, buckling can be interpreted as a consequence of bifurcation phenomena in nonlinear systems. In this talk, I shall present results from our recent investigation of a new buckling mode that can arise after introducing internal structure, a line of holes, into a simple column [Soft Matter (2016), 12, pp 7112--7118]. This simple geometric modification introduces a smaller lengthscale into the system and has a profound effect on the solution structure, which becomes increasing complex as the number of holes increases.

Controlling caustics: Weaving catastrophic structures in Airy-, Pearcey-, and swallowtail beams

Catastrophe science is a branch of bifurcation theory in the study of dynamical systems; it is also a special case of more general singularity physics. Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behaviour arising from small changes in circumstances, analysing how the qualitative nature of solutions depends on control parameters. In optics, these dramatic changes manifest as geometrically stable caustics, which, as natural phenomena, are associated with the arcs close to rainbows, or may occur as high-intensity networks on the floor of shallow waters. Similar to their formation behind refractive index lenses with imperfections, the formation of corresponding structures has been observed for numerous kinds of lenses with importance in optics, astrophysics and surface analytics.
The most prominent representative of catastrophe that manifests as wave package is the fold catastrophe that has been realized as paraxial Airy beam, and shows a form-invariant and accelerated propagation. Here, we present higher-order catastrophes as paraxial light, discuss the mapping of corresponding cusp and swallowtail catastrophes to these beams and demonstrate their unique propagation effects. Their often auto-focusing propagation of curved trajectories is well suited to optically induce photonic structures, waveguides to nonlinear photosensitive materials.
Alessandro Zannotti, AG Denz, Institut für Angewandte Physik

Delay-induced Dynamics in a Swift-Hohenberg Model with Spatial Inhomogeneities

We study pattern formation in a Swift-Hohenberg model that, aside from many other applications, describes the evolution of the electric field envelope in the transverse plane of a bistable optical cavity pumped by an external injection beam.
Typical solutions of the Swift-Hohenberg equation are homogeneous, periodic and localized states. Here, we focus on the destabilization of stable localized solutions by time-delayed feedback, which can be implemented by using an external cavity in optical applications. We show that by varying the delay time and the delay strength, one can induce a variety of different dynamics including the drift or the annihilation of the localized solutions as well as travelling wave solutions.
A special focus lies on the introduction of spatial inhomogeneities into the system, e.g., by introducing a spatially inhomogeneous injection beam, which changes the delay-induced dynamics drastically. We report on pinned localized solutions, oscillating solutions and depinning due to timedelayed feedback. The competition of a pinning inhomogeneity and a destabilizing time-delayed feedback is studied both analytically and numerically.
Felix Tabbert, AG Thiele, Institut für Theoretische Physik

Einladender: Dr. O. Kamps

Rational design of stimuli-responsive nanoreactors

Prof. Dr. Joe Dzubiella, HU Berlin und Helmholtz Zentrum Berlin

(Einladender: Prof. Dr. A. Heuer)

The catalysis by metal nanoparticles is one of the fastest growing fields in nanoscience. However, the optimal control of catalytic activity and selectivity in nanoparticle catalysis remains a grand scientific challenge. Here, we describe our ongoing efforts how to theoretically derive design rules for the optimization of nanoparticle catalysis (in the fluid phase) by means of thermosensitive yolk-shell and core-shell carrier systems [1-3]. In the latter, nanoparticles are stabilized in solution by an encapsulating, thermosensitive hydrogel shell. The latter contains and shelters the reaction. The physicochemical properties, in particular the permeability, of this polymeric 'nanogate' react to stimuli in the environment and thus permit the reactant transport and with that the catalytic reaction to be switched and tuned, e.g., by the temperature [1-3], salt concentration, or solvent composition. Hence, the novel hybrid character of these emerging 'nanoreactors' opens up unprecedented ways for the control of nanocatalysis due to new designable degrees of freedom, if theoretical understanding and rational design principles are available.

References

[1] S. Wu, J. Dzubiella, J. Kaiser, M. Drechsler, X. Guo, M. Ballauff, Y. Lu, Angewandte Chemie 51, 2229 (2012).

[2] P. Herves, M. Perez-Lorenzo, L. M. Liz-Marzan, J. Dzubiella. Y. Lu, and M. Ballauff, Chem. Soc. Rev. 41, 5577 (2012).

[3] S. Angioletti-Uberti, Y. Lu, M. Ballauff, and J. Dzubiella, J. Phys. Chem. C 119, 15723 (2015).

Temporal Localized Structures and Light Bullets in Passively mode-locked Lasers

Dr. Julien Javaloyes, Departament de Fisica, Edifici Mateu Orfila, Universitat de les Illes Baleares

Localized structures (LS) are nonlinear states of dissipative extended systems characterized by a correlation range much shorter than the size of the system, thus allowing for individual addressing. They appear ubiquitously in nature and they are very appealing in optical systems for applications to information processing, especially in semiconductor lasers which are fast, scalable and cheap devices. We investigate the relationship between passive mode-locking and the formation of temporal localized structures in the output intensity of a laser coupled to a saturable absorber, in the framework of time delayed dynamical systems. We present experimental and theoretical evidences [1] regarding how the mode-locked pulses transform into lasing localized structures, allowing for individual addressing and arbitrary low repetition rates. Our analysis reveals that this occurs when i) the cavity round-trip is much larger than the slowest medium timescale, namely the gain recovery time and ii) the mode-locked solution coexists with the stable off solution. These conditions enable the coexistence of a large quantity of stable solutions, each of them being characterized by a different number of pulses per round-trip and with different arrangements. A modulation of the bias current allows controlling the number and the location of the pulses traveling within the cavity [2, 3]. These results formed the basis for a very recent prediction [4] as we theoretically demonstrated the existence of three dimensional dissipative localized structures in the output of a broad area passively mode-locked laser coupled to a saturable absorber in self-imaging conditions. These phase invariant light bullets are individually addressable and can be envisioned for three dimensional optical information storage. An effective theory provides for an intuitive picture and allows to relate their formation to static auto-solitons.

[1] M. Marconi, J. Javaloyes, S. Balle, and M. Giudici. How lasing localized structures evolve out of passive mode locking. Phys. Rev. Lett., 112:223901, Jun 2014.

[2] M. Marconi, J. Javaloyes, P. Camelin, D.C. Gonzalez, S. Balle, and M. Giudici. Control and generation of localized pulses in passively mode-locked semiconductor lasers. Selected Topics in Quantum Electronics, IEEE Journal of, 21(6):1-10, Nov 2015.

[3] J. Javaloyes, P. Camelin, M. Marconi, and M. Giudici. Dynamics of localized structures in systems with broken parity symmetry. Phys. Rev. Lett., 116:133901, Mar 2016.

[4] J. Javaloyes. Cavity light bullets in passively mode-locked semiconductor lasers. Phys. Rev. Lett., 116:043901, Jan 2016.

(Einladende: Dr. S. Gurevich)

Inhomogeneous Boltzmann-type equations modelling opinion leadership and political segregation

Dr. Bertram Düring, Department of Mathematics, University of Sussex

In recent years different kinetic models to describe opinion formation have been proposed. Such models successfully use mathematical tools from statistical mechanics to describe the behaviour of a large number of interacting individuals in a society. This leads to generalizations of the classical Boltzmann equation for gas dynamics. Most approaches in the literature assume a homogeneous society. To model additional sociologic effects in real societies, e.g. the influence opinion leaders, one needs to consider inhomogeneous models.
In this talk we discuss such kinetic models for opinion formation, where the opinion formation process depends on an additional independent variable, e.g. a leadership or a spatial variable. More specifically, we consider:
(i) opinion dynamics under the effect of opinion leadership, where each individual is characterised not only by its opinion, but also by another independent variable which quantifies leadership qualities;
(ii) opinion dynamics modelling political segregation in `The Big Sort', a phenomenon that US citizens increasingly prefer to live in neighbourhoods with politically like-minded individuals. Based on microscopic opinion consensus dynamics such models lead to inhomogeneous Boltzmann-type equations for the opinion distribution. We derive macroscopic Fokker-Planck-type equations in a quasi-invariant opinion limit and present results of numerical experiments.

(Einladender: Prof. Dr. M. Burger)