Workshop MathMicS 2023:
Mathematics and microscopic theory for random Soft Matter systems

SPP2265: Random Geometric Systems      February 13-15, 2023


   Monday, 13.2.2023    Tuesday, 14.2.2023   Wednesday, 15.2.2023  
09:00 Sabine Klapp
Colloids under time-delayed feedback: from single-particle dynamics to collective motion
Robert Evans
Sum Rules and their Repercussions for the Statistical Mechanics of Liquids: What did/does Classical DFT contribute?
09:30 Philipp Maass
Solitons in driven overdamped Brownian motion
Andreas Härtel
Screening in electrolytes
10:00 Coffee Break Tobias Kuna
Convergence of the density expansion and Ornstein-Zernike equation
10:30 Umberto Marini Bettolo Marconi
Spontaneous velocity alignment and solid-like behavior in Active systems
Coffee Break
11:00 Tanniemola Liverpool
The mathematics of active matter
Till Kranz
Dynamics of the Fredrickson-Andersen Model on intermediate time scales
11:30 Posters & Discussion with Coffee Wolfgang König
Spatial particle processes with coagulation: large deviations and gelation
12:00 Registration & Snacks Closing & Lunch Snack
12:30 Registration, Snacks & Welcome Lunch Break Lunch Snack
13:00 Klaus Mecke
Finite projective geometry as mathematical frame for physics
13:30 Günter Last
On the uniqueness of certain Gibbs measures
Sabine Jansen
Large deviations and distribution of cracks in a chain of atoms at low temperature
14:00 Michael Klatt
Foundations of dynamical density functional theory: on the uniqueness of density-potential mappings
Patrick Farrell
A phenomenological Q-tensor model for smectic liquid crystals
14:30 Coffee Break Matthias Schmidt
What is liquid, from Noether´s perspective?
15:00 Cornelia Pokalyuk
Invasion of cooperative parasites in spatially structured host populations
Coffee Break
15:30 Karin Jacobs
Integral geometry to characterize patterns or roughness on (soft or hard) interfaces
Gerd Schröder-Turk
Random packings of bi-disperse ellipsoid mixtures
16:00 Coffee Break Myfanwy Evans
Geometric simulation of tubular structures
16:30 Mini Talks Posters & Discussion with Beer or Coffee
17:30 Posters & Discussion with Beer Free evening for Altstadt visit
19:00 Workshop dinner
Topics: GeometryStaticsDynamicsActivityVariousRecreation

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Monday 12:00-12:50 - Registration & Snacks

Monday 12:50-13:00 - Welcome

Monday 13:00-13:30 - Klaus Mecke: Finite projective geometry as mathematical frame for physics

Since the introduction of differential calculus at the beginning of modern physics in the 17th century the use of integrals and limits are ubiquitous in science. For instance, the standard models of spacetime and elementary particles are based on differentiable manifolds and measure theory is in the heart of statistical physics. Although the elegance and practicality of analysis is unquestionable in physics there are epistemological and ontological doubts about the role of real numbers in nature. Since in measurements only rational numbers are reachable and the existence of Cauchy sequences is questionable in reality, the mathematics relevant for fundamental physics is presumably different from differential calculus. Finite projective geometry provides a simple starting point to develop the mathematical tools necessary in physics and to elucidate the specific assumptions used in physical theories. In the first part of the talk, the axioms and main theorems are introduced and their relation to continuous concepts are discussed. The second part points out possible applications on random geometric systems, integral geometry and statistical physics. In the third part, finite projective analogs of classical mechanics (time dependence), electrodynamics (spatial fields) and quantum mechanics (randomness) are formulated and their equivalence to the standard (analytical) theories is shown in the continuum limit. The origin of such important concepts as Legendre transform, gauge symmetry and commutator relations in the basic features of finite projective geometry is elucidated. Finally, the possibility of a unified theory of general relativity and elementary particle theory is outlined.

Monday 13:30-14:00 - Günter Last: On the uniqueness of certain Gibbs measures

We consider a Gibbs process on a general phase space with interactions governed by a non-negative pair potential. Such a point process is closely related to the random connection model driven by a Poisson process, a well-studied random geometric graph. Whenever the latter does not percolate the (infinite volume) Gibbs measure is uniquely determined. Our proof is based on a general version of the so-called disagreement coupling of finite Gibbs processes, which is of independent interest.

Joint work with Steffen Betsch.

Monday 14:00-14:30 - Michael Klatt: Foundations of dynamical density functional theory: on the uniqueness of density-potential mappings

Classical density functional theory (DFT) is grounded on the rigorously proven existence of a one-to-one mapping between the density of the fluid and the external potential acting on the particles. In this talk, we present rigorous and explicit conditions under which this fundamental relation also holds between the density of diffusing interacting particles (out of equilibrium) and a time-dependent external potential. This unique mapping between the density and the external potential is a cornerstone of classical dynamical DFT (DDFT), for which we here establish a rigorous mathematical background.

Joint work with René Wittmann and Hartmut Löwen.

Monday 14:30-15:00 - Coffee Break

Monday 15:00-15:30 - Cornelia Pokalyuk: Invasion of cooperative parasites in spatially structured host populations

Certain defense mechanisms of phages against the immune system of their bacterial host rely on cooperation of phages. Motivated by this example we analysed in [BP] the spread of cooperative parasites in host populations that were structured according to a configuration model. Building on these results we consider now the case of a host population which is (genuinely spatially) structured according to a random geometric graph. We identify the spatial scale at which invasion of parasites turns from being an unlikely to a highly probable event and give in the critical regime upper and lower bounds on the invasion probability.

[BP] V. Brouard and C. Pokalyuk. Invasion of parasites in moderately structured host populations, Stoch. Proc. Appl., 153, pp 221-263, 2022

Joint work in progress with Vianney Brouard and Marco Seiler.

Monday 15:30-16:00 - Karin Jacobs: Integral geometry to characterize patterns or roughness on (soft or hard) interfaces

The characterization of recurring structures on surfaces or interfaces is well established, whether in 2d or 3d. However, when it comes to higher order correlations between objects, noisy structures on surfaces, or surface roughness without structure, the toolbox for quantitative characterization shrinks drastically. For example, the root mean square roughness (rms) is often used to characterize surface roughness, but it is very "non-specific". In our current project, we use Minkowski measures to comprehensively describe the roughness of substrate surfaces like semiconductors or implants whose topography we record by atomic force microscopy.

Joint work with Klaus Mecke, Michael Klatt and Jens Uwe Neurohr.

Monday 16:00-16:30 - Coffee Break

Monday 16:30-17:30 - Mini Talks

Monday 17:30-19:00 - Posters & Discussion with Beer

Monday 19:00-21:00 - Workshop dinner (Schloss Mickeln)

Tuesday 09:00-09:30 - Sabine Klapp: Colloids under time-delayed feedback: from single-particle dynamics to collective motion

Within the last years, feedback (closed-loop) control of fluctuating systems, such as colloids in a thermal bath, has become a focus of growing interest. In this talk I will discuss recent results for colloids under repulsive, nonlinear feedback with time delay. Starting with the dynamics of single particles I will show by analytical methods that, at appropriate values of the feedback parameters and in the absence of noise, a steady state with a constant, non-zero velocity emerges whose direction is constant as well. In the presence of noise, the direction of motion becomes randomized at long times but the (numerically obtained) mean-squared displacement still reveals some persistence of motion, resembling the behavior of an active particle. I will then discuss simulation results for the collective behavior of many particles interacting via a repulsive (WCA) potential, each being subject to time-delayed repulsive feedback. Intriguingly, one observes a state with spontaneous, large scale alignment of the velocity vectors. We discuss this delay-induced phenomenon in the light of recent results for active particles.

Tuesday 09:30-10:00 - Philipp Maass: Solitons in driven overdamped Brownian motion

In systems with inertia, solitons are waves whose dispersion is suppressed by nonlinear effects. We show that solitons occur also in the absence of inertia in driven Brownian motion of hard spheres through periodic potentials at high densities [1,2]. The solitons manifest themselves as periodic sequences of particle clusters, which propagate even in the limit of zero noise, where single particles cannot surmount potential barriers. These clusters form at special hard-sphere diameters that are rational fractions of the wavelength of the periodic potential. They consist of several particles in contact, which are held together during their motion by the external forces. At low temperatures, the solitons give rise to particle currents in narrow intervals around the special hard-sphere diameters, where the intervals broaden with the driving force. At high temperatures, particle currents occur for all hard-sphere diameters, and the variation of the current magnitudes with particle size and driving force reflects the inherent soliton formation.

[1] A. P. Antonov, A. Ryabov, and P. Maass, Phys. Rev. Lett. 129, 080601 (2022).
[2] A. P. Antonov, D. Voráč, A. Ryabov, and P. Maass, New J. Phys. 24, 093020 (2022).

Joint work with Alexander P. Antonov and Artem Ryabov.

Tuesday 10:00-10:30 - Coffee Break

Tuesday 10:30-11:00 - Umberto Marini Bettolo Marconi: Spontaneous velocity alignment and solid-like behavior in Active systems

In the last decade, significant progress has been made in the study of the collective behavior of active particles, which comprise bacteria, cell assemblies, active colloidal suspensions, vibrated granular particles, autonomous micromotors, bird flocks, etc. A minimal theoretical model, the so-called Active Brownian Particles and its companion, the Active Ornstein-Uhlenbeck particle model satisfactorily describe many aspects of dry active matter. They successfully explain its steady-state behavior and in particular, the Mobility induced phase separation (MIPS), a novel type of non-equilibrium transition. Recently, we have shown that not only the density field undergoes an ordering process, but also the velocity field displays a non-trivial behavior, i.e. spatial velocity correlations. We have shown that the observed velocity ordering is due to the interplay between persistent active forces and repulsive interactions. Particles arrange into aligned or vortex-like domains. Their sizes increase as the persistence of the self-propulsion grows, an effect that is quantified by studying the spatial correlation function of the velocities. We build a phase diagram as a function of packing fraction and persistence time to compare the structural properties of the system (i.e. liquid, hexatic and solid phases) with the emergent velocity order (i.e. the correlation length of the spatial velocity correlations). For active solids, our findings are corroborated by a microscopic theory while, for active liquids, we developed a hydrodynamic theory, derived from the microscopic model under suitable approximations.

Tuesday 11:00-11:30 - Tanniemola Liverpool: TBA


Tuesday 11:30-12:30 - Posters & Discussion with Coffee

Tuesday 12:30-13:30 - Lunch Break

Tuesday 13:30-14:00 - Sabine Jansen: Large deviations and distribution of cracks in a chain of atoms at low temperature

The talk presents results on the low-temperature asymptotics for a one-dimensional chain of atoms. The latter can be modelled as a one-dimensional Gibbs point process or as a Gibbs measure on sequences of interparticle spacings. As the temperature goes to zero at fixed density, either (1) the measure is well approximated by a Gaussian measure in the vicinity of the periodic energy minimizer, or (2) points fill space by alternating approximately periodic patterns with stretches of empty space (gaps). In case (1) we prove Gaussian limit laws and large deviations principles for both bulk behavior and boundary layers. In case (2) we map the system to an effective gas of defects and analyze in detail the length of empty and approximately crystalline domains as a function of defect energy.

Based on joint works with Wolfgang König, Bernd Schmidt and Florian Theil.

Tuesday 14:00-14:30 - Patrick Farrell: A phenomenological Q-tensor model for smectic liquid crystals

Nematic liquid crystals are a form of soft matter. They are fluid in nature, but possess additional order in molecular orientation: their molecules locally line up in the same direction, like matches in a matchbox. This gives them anisotropic elastic and optical properties. Smectics are a class of liquid crystals where, in addition to the nematic ordering, the molecules line up in layers that are able to slide over one another. In this talk we propose a new phenomenological free energy functional to describe smectic liquid crystals. The free energy builds on the Landau-de Gennes theory for describing nematics. We discuss the analysis of this functional, including existence proofs of minimizers and convergence of a finite element discretization. We then demonstrate how minimizers of the functional successfully reproduce canonical smectic structures such as oily streaks and focal conic domains.

Tuesday 14:30-15:00 - Matthias Schmidt: What is liquid, from Noether´s perspective?

The structure of liquids carries deep imprints of an inherent thermal invariance against a spatial transformation of the underlying classical many-body Hamiltonian. At first order in the transformation field the Noether theorem yields the local force balance. Three distinct two-body correlation functions emerge at second order, i.e., the standard two-body density, the local force autocorrelation function, and the local mean potential energy curvature. Exact self and distinct sum rules interrelate these correlation functions. Many-body computer simulations of the Lennard-Jones liquid, of monatomic water and of a three-body colloidal gel former demonstrate the fundamental role that these correlators play in the formation of spatial structure in soft matter.

Joint work with Florian Sammüller, Sophie Hermann and Daniel de las Heras.

Tuesday 15:00-15:30 - Coffee Break

Tuesday 15:30-16:00 - Gerd Schröder-Turk: Random packings of bi-disperse ellipsoid mixtures

The structure and spatial statistical properties of amorphous ellipsoid assemblies have profound scientific and industrial significance in many systems, from cell assays to granular materials. This talk describes research that uses a fundamental theoretical relationship for mixture distributions to explain the observations of an extensive X-ray computed tomography study of granular ellipsoidal packings. We study a size-bi-disperse mixture of two types of ellipsoids of revolutions that have the same aspect ratio of α ≈ 0.57 and differ in size, by about 10% in linear dimension, and compare these to mono-disperse systems of ellipsoids with the same aspect ratio. Jammed configurations with a range of packing densities are achieved by employing different tapping protocols. We numerically interrogate the final packing configurations by analyses of the local packing fraction distributions calculated from the Voronoi diagrams. Our main finding is that the bi-disperse ellipsoidal packings studied here can be interpreted as a mixture of two uncorrelated mono-disperse packings, insensitive to the compaction protocol. Our results are consolidated by showing that the local packing fraction shows no correlation beyond their first shell of neighbours in the binary mixtures. We propose a model of uncorrelated binary mixture distribution that describes the observed experimental data with high accuracy. This analysis framework will enable future studies to test whether the observed mean-field behaviour is specific to the particular granular system or the specific parameter values studied here or if it is observed more broadly in other bi-disperse non-spherical particle systems.

Tuesday 16:00-16:30 - Myfanwy Evans: Geometric simulation of tubular structures

The morphometric approach to computing the solvation free energy of a system allows the transformation of a physical problem into a geometric one. An interesting question is; can we use this idea to simulate solutes with complicated shapes? I will present the simulation of long flexible tubular structures, including loops, knots and open strands, and their equilibrium configurations under different solvent conditions. This is joint work with Rhoslyn Coles at the TU Berlin.

Tuesday 16:30-17:30 - Posters & Discussion with Beer/Coffee

Tuesday from 17:30 - Free evening for Altstadt visit

Wednesday 09:00-09:30 - Robert Evans: Sum Rules and their Repercussions for the Statistical Mechanics of Liquids: What did/does Classical DFT contribute?

Statistical mechanical sum rules relate the structure of liquids, i.e. correlation functions, to thermodynamic properties. Whilst several sum rules were derived directly from the partition function, the method of functional differentiation employed within the formal framework of classical density functional theory (DFT) has provided a plethora of important results, many of which date back to the 1980´s and the work of J.R. Henderson and others. We recall some of the key results, which pertain to an inhomogeneous fluid adsorbed at a planar wall or confined between two walls and discuss the repercussions of these for surface phase transitions such as wetting and drying and for confinement induced transitions such as capillary condensation and evaporation. Explicit (non-local) DFT approximations satisfy the majority of the sum rules, and we describe results of some recent calculations that employ these. We argue that, guided by sum rules, modern DFT is now well-equipped to determine the subtleties of two- body correlation functions that underlie rich interfacial phenomena.

Wednesday 09:30-10:00 - Andreas Härtel: Screening in electrolytes

Electrolytes consist of mobile ionic charge carriers and a solvent and, in bulk, are overall neutral. While we will discuss how charges are physically screened in these systems, our particular interest lies in underscreening, a situation where mobile charges screen less efficiently than expected. The framework of liquid state theory further allows us to derive general mathematical statements for the complex electrolyte system which we will use as a starting point for further joint discussions on site.

Wednesday 10:00-10:30 - Tobias Kuna: Convergence of the density expansion and Ornstein-Zernike equation

Graphical expansion gives a full description of gases in the low density high temperature regime, however being perturbative expansions around the ideal gas, the density expansion are not expected to be valid in the liquid regime. In order to develop a theory of classical fluids not based on expansion, one derives effective equations for the density, often based on curtailing the graphical expansion in a suitable way. Starting with works of Penrose and Groeneveld in the 60s the convergence of the density expansion of the free energy has been established. Though absolute convergent, hence analytic, in the density, the graphical expansion only converges conditionally. Studying in more details the underlying inversion of the density-activity relation and using methods to directly control the convergence of the expansion, we hope to get, at least in the low density-high temperature regime, an insight which terms are most relevant and hence have to be represented in an effective theory.

Wednesday 10:30-11:00 - Coffee Break

Wednesday 11:00-11:30 - Till Kranz: Dynamics of the Fredrickson-Andersen Model on intermediate time scales

We consider the Fredrickson-Anderen (FA) model on the Bethe lattice, i.e., a kinetically constraint spin model. Here, non-interacting Ising spins are biased to a spin-down ground state and follow a Metropolis dynamics constraint to having two or more down-spin neighbors. The long time asymptotic value of spin-spin correlation functions can be calculated analytically from bootstrap percolation (BP). Depending on the initial fraction of up-spins p, there is a transition at p = pc from a ergodic to a non-ergodic state with finite asymptotic correlations q > 0. For p towards pc from below, correlation functions are known to feature a two-step decay with a long plateau near q. I propose a analytic continuation of BP in a Frechet space that allows to determine correlation functions around q. I will discuss the proposed approach and some results on scaling behavior and time scales that can be derived from it.

Wednesday 11:30-12:00 - Mauro Sellitto: Fluctuation-induced forces in boundary-driven systems with a diffusivity anomaly

I discuss nonequilibrium Casimir-like forces in boundary-driven kinetically constrained particle systems and show that in the presence of a bulk diffusivity anomaly they are enhanced by cooperative dynamical effects and can be made locally attractive or repulsive depending on the boundary densities. Theoretical predictions based on mean-field arguments and explicit evaluation in the fluctuating hydrodynamics framework are supported by Monte Carlo simulation of a two-dimensional exclusion process with selective kinetic constraints.

Wednesday 12:00-12:30 - Wolfgang König: Spatial particle processes with coagulation: large deviations and gelation

We consider a random spatial particle process with coagulation, a variant of the Marcus–Lushnikov process. We derive an asymptotic formula for the joint distribution of all particles sizes at a given fixed time in the limit of many particles in terms of a large-deviation principle. Based on an analysis of the rate function, we discuss criteria under which we can deduce a gelation phase transition (emergence of a particle with macroscopic size). In a special case (which can be mapped on the inhomogeneous Erdős–Rényi graph), we can give a complete answer. (joint work with Luisa Andreis (Milano), Heide Langhammer and Robert Patterson (Berlin))

Wednesday 12:30-12:40 - Closing

Wednesday 12:40-13:30 - Lunch Snack

Last modified: 14 February 2023 11:01