• Thematic program: Mean-field transportation models in the life and social sciences (2015/2016)
• Event: Working Group on "Multiscale transport of particles" (2015)

• Thematic program: Mean-field transportation models in the life and social sciences (2015/2016)
• Event: Working Group on "Multiscale transport of particles" (2015)

• Thematic program: Mean-field transportation models in the life and social sciences (2015/2016)
• Event: Working Group on "Multiscale transport of particles" (2015)

The nature is dominated by systems composed of many individuals with a collective behavior. Examples include wildlife populations, biological cell dynamics, and tumor growth. There is a fast growing interest in multispecies systems both in theoretical biology and applied mathematics, but because of their enormous complexity, the scientific understanding is still very poor. Instead of calculating the trajectories of all individuals, it is computationally much simpler to describe the dynamics of the individuals on a macroscopic level by averaged quantities such as population densities. This leads to systems of highly nonlinear partial differential equations with cross diffusion, which may reveal surprising effects such as uphill diffusion and diffusioninduced instabilities, seemingly contradicting our intuition on diffusion. Major difficulties of the mathematical analysis of the crossdiffusion equations are their highly nonlinear structure and the lack of positive definiteness of the diffusion matrix. In this talk, a method inspired from nonequilibrium thermodynamics is proposed, which allows for a mathematical theory of some classes of such systems. It is based on a transformation of entropy variables which make the diffusion matrix positive definite. This property is a purely algebraic condition and may be shown by computer algebra systems. We explain the technique for systems modeling populations and transport through ion channels.

• Thematic program: Mean-field transportation models in the life and social sciences (2015/2016)
• Event: Working Group on "Multiscale transport of particles" (2015)

When diffusing agents interact with each other their motions are correlated, and the configuration space is of very high dimension. Often an equation for the marginal distribution function of one particle (the “concentration”) is sought by “integrating out” the positions of all the others. This leads to the classic problem of closure, since the equation for the concentration so derived depends on the two-point correlation function. A common closure is to assume independence at this stage, leading to some form of nonlinear (drift) diffusion equation. Such an approach works well for long range interactions (such as electric fields), but fails for short range interactions (such as steric effects). Here we consider an alternative approach using matched asymptotic expansions, in which the approximation is entirely systematic. We show how information about correlations can be recovered from the concentration. Finally we consider some of the difficulties when both long and short range forces are present.

• Thematic program: Mean-field transportation models in the life and social sciences (2015/2016)
• Event: Working Group on "Multiscale transport of particles" (2015)

Metastable transitions are rare events, such as bistable switching, that occur under weak noise conditions, causing dramatic shifts in the expression of a gene. Within a gene circuit, one or more genes randomly switch between regulatory states, each having a different mRNA transcription rate. The circuit is self regulating when the proteins it produces affect the rate of switching between gene regulatory states. Under weak noise conditions, the deterministic forces are much stronger than fluctuations from gene switching and protein synthesis. A general tool used to describe metastability is the quasi stationary analysis (QSA). A large deviation principle is der ived so that the QSA can explicitly account for random gene switching without using an adiabatic limit or diffusion approximation, which are unreliable and inaccurate for metastable events.This allows the existing asymptotic and numerical methods that have been developed for continuous Markov processes to be used to analyze the full model.

• Thematic program: Mean-field transportation models in the life and social sciences (2015/2016)
• Event: Working Group on "Multiscale transport of particles" (2015)

Ordinary materials are "passive" in the sense that their constituents are typically made by inert particles which are subjected to thermal fluctuations, internal interactions and external fields but do not move on their own. Living systems, like schools of fish, swarms of birds, pedestrians and swimming microbes are called "active matter" since they are composed of self-propelled constituents. Active matter is intrinsically in nonequilibrium and exhibits a plethora of novel phenomena as revealed by a recent combined effort of statistical theory, hydrodynamics and real-space experiments. The talk provides an introduction into the modelling of active matter focussing on biological and artificial microswimmers as key examples of active systems. A number of single-particle and collective phenomena in active matter will be addressed ranging from the most disordered state of matter (turbulence) to the purely kinetic phase separation in active systems.

• Thematic program: Mean-field transportation models in the life and social sciences (2015/2016)
• Event: Working Group on "Multiscale transport of particles" (2015)

I will present examples of modelling collective movement via robot experiments. In the first part I will focus on an investigation on how two communicating individuals can most efficiently navigate a corridor without external sensory input. The second part of my talk will be about robot swarms and their strategies for target finding in an unknown environment. These studies where performed via a combination of robot experiments and numerical simulations.

• Thematic program: Mean-field transportation models in the life and social sciences (2015/2016)
• Event: Working Group on "Multiscale transport of particles" (2015)

Until recently, the (Poisson-)Nernst-Planck equations have been the standard model for the description of ion transport in diluted electrolyte solutions, although it was known that they fail to reasonably limit the ion concentration in diffuse double layers. This weakness can be remedied by a thermodynamic consistent coupling to the momentum balance and introducing an appropriate elastic law, rather than by a mere modification of the entropy of mixing. In many electrochemical applications, the Debye length --that controls the width of the diffuse layers-- is typically very small compared to the macroscopic dimensions of the system. In these situations a spacial resolution of the layers is often not necessary. By the method of formal asymptotic analysis we derive a reduced model that is locally electric neutral and does not resolve the layers but incorporates all relevant features of the layers into a new set of interface equations. Nanopores typically have a strongly anisotropic geometry where the diameter is close to the Debye length but the length in axial direction is larger by at least one order of magnitude. We discuss the scaling to dimensionless quantities and present a reduced 1d-model for arbitrary geometries with rotational symmetry. Multi-dimensional solutions that resolve boundary layers can be recovered from the lower-dimensional solution.

• Thematic program: Mean-field transportation models in the life and social sciences (2015/2016)
• Event: Working Group on "Multiscale transport of particles" (2015)

I will present the WED variational approach to gradient-flow evolution in metric spaces. A reference application is to densities and empirical measures. In the linear-space case, the WED strategy entails in an elliptic-in-time regularization of the problem. The picture in the metric case is confined to the variational level and the discussion relies on a Pontyagin-type principle. This is joint work with Riccarda Rossi (Brescia), Giuseppe Savar' (Pavia), and Antonio Segatti (Pavia).

• Thematic program: Mean-field transportation models in the life and social sciences (2015/2016)
• Event: Working Group on "Multiscale transport of particles" (2015)

In this talk we will discuss two different cross-diffusion models. The first model is used in population dynamics in biology and can be derived from a lattice in the case when we are not taking into account any volume-filling effects. We will present recent results concerning the existence of global weak solutions under the assumption that the system possesses a formal gradient-flow structure using ideas of [A. Jüngel: Boundedness-by-entropy method. Nonlinearity 28 (2015)]. The second model describes diffusion through obstacles. The underlying cross-diffusion system can be derived from a two species mixture of Brownian hard spheres. We will discuss open questions concerning this model.

• Thematic program: Mean-field transportation models in the life and social sciences (2015/2016)
• Event: Working Group on "Multiscale transport of particles" (2015)