Velocity averaging in classical kinetic models [1,2,3] like Vlasov or Boltzmann equation is a smoothing mechanism for „macroscopic“ observables (in position space) as averages in the velocity variable of the phase space distribution function. The Wignertransform [4,5,6,7] converts the Schrödinger (or von Neumann) equation into sort of Vlasov equation for the Wignerfunction, with the standard transport term and a nonlocal (i.e. pseudodifferential) force term. It is a long standing question if one can apply velocity averaging to quantum kinetic Wigner equations, in order to obtain a gain of regularity on quantities such as the density function. In this talk we introduce kinetic equations and the classical averaging lemma and then show that this indeed works also in the quantum physics case, for special mixed states, but typically fails for pure states (similar to the situation for the global in time semiclassical limit of nonlinear Schrödinger equations [5,6]). We use a new (?) derivation of Madelung's fluid dynamic formulation of Schrödinger equations [8]. Joint work with Jakob Möller.

• Event: 27th Pauli Colloquium: François Golse (2025)

Abstract: The spontaneous emergence of tissue patterns is often attributed to biochemical reaction-diffusion systems. Strictly biochemical mechanism modelled through the LALI-type Gierer-Meinhardt-Turing system are a hallmark of Mathematical Biology and have been identified as drivers of pattern formation in a series of experimental systems. In Hydra spheroids, however, a purely biochemical model for self-organization remains elusive. In this talk, we present a general overview of mathematical models for pattern formation including LALI models, and then introduce a mechanochemical realisation of this concept based on a positive feedback loop between morphogen concentration and and tissue stretching. We briefly discuss the resulting minimal mathematical model for mechanochemical pattern formation in a closed elastic shell represents regenerating Hydra epithelial spheroids. Linear stability analysis of the 1D-version of the model and 3D simulations illustrate how mechanical forces drive axis formation and predict the organizer's location under various perturbations.

A case study of how the availability of tools controls the progress of science.

• Thematic program: Models in Plasmas, Earth and Space Science (2024/2025)