Uniqueness theorems and analysis of classical density functional theory in nonequilibrium random geometries


Description Phase 2

Describing the time evolution of various types of classical models for colloidal fluids by methods of statistical mechanics has become an important research area in physics over the last decades. This requires appropriate theories which are predictive, accurate and also numerically realizable. A promising option is classical dynamical density functional theory (DDFT), which, however, faces the challenge of being based on a sometimes uncontrolled approximation. Thus a basic understanding of DDFT from a rigorous mathematical point of view, which also explores the relation to recently proposed improvements of the theory, is of high importance. The aim of this research project is to continue our work in the first funding period of the SPP 2265 and elucidate the mathematical background of DDFT and related approaches. We shall pursue further our successful collaborations with other members of the SPP 2265. Our project in the second funding period of the SPP 2265 evolves around three pillars. (I) Development, improvement and application of approximate DDFT models for fluids under random external influences, including activity, bacterial growth, odd diffusive systems and random porous media. (II) Proofs of rigorous uniqueness theorems out of equilibrium. The corresponding work packages shall not only account for additional random one-body fields, such as a position-dependent active velocity, but also a time-dependent density-potential mapping on the two-body level. (III) Mathematical assessment of ingredients or results of DDFT. This point involves a rigorous analysis of a class of free energy functionals, which enters in many DDFT approaches and also the equilibrium limit, as well as, addressing the rising concept of hyperuniformity from a DDFT perspective.

Description Phase 1

Classical density functional theory (DFT) is a sophisticated and versatile tool to tackle fundamental physical questions regarding the behavior of various types of equilibrium fluids through a variational principle with respect to the probability density to find a particle at a certain position. The framework of DFT has a deep mathematical foundation provided by a rigorous theorem relating the density of the fluid and the external potential acting on the particles: the knowledge of the density uniquely determines the external potential. The aim of this research project is to establish the mathematical background of a dynamical DFT (DDFT), which we understand in the most general ma- thematical sense as a differential equation for the time-dependent density. The central question is whether the different random external influences on the nonequilibrium behavior of a fluid can be uniquely characterized alone in terms of the density which solves the DDFT equation, in generalization of the uniqueness theorem in equilibrium. This purely mathematical task is independent of explicit approximations required for practical purposes. It will be complemented by developing and employing novel DDFT methods to describe the complex interplay of different intrinsic and extrinsic forces acting on the particles. A particular focus will lie on nonequilibrium fluid behavior in porous (random) media.

Preprints/Publications

Anouar El Moumane, René Wittmann, Hartmut Löwen, Michael te Vrugt: Dynamical field theories for biaxial liquid crystals (03/2025)

René Wittmann, Iman Abdoli, Abhinav Sharma, Joseph M. Brader: Confined active particles with spatially dependent Lorentz force: an odd twist to the "best Fokker-Planck approximation" (10/2024) published

Tobias Plasczyk, Paul A. Monderkamp, Hartmut Löwen, René Wittmann: A Hitchhiker's Guide To Active Motion (10/2024) published

Lorenzo Caprini, Davide Breoni, Anton Ldov, Christian Scholz, Hartmut Löwen: Dynamical clustering and wetting phenomena in inertial active matter (06/2024) published

Lorenzo Caprini, Benno Liebchen, Hartmut Löwen: Self-reverting vortices in chiral active matter (04/2024) published

Marcus Witt, G. H. Philipp Nguyen, Josefine R. von Puttkamer-Luerssen, Can H. Yilderim, Johannes A. B. Wagner, Ebrahim Malek, Sabrina Juretzka, Jorge L. Meyrelles Jr., Maximilan Hofmann, Hartmut Löwen, Thomas Palberg: Accessing the Free Expansion of a Crystalline Colloidal Drop by Optical Experiments (04/2024) published

Alexander P. Antonov, Lorenzo Caprini, Anton Ldov, Christian Scholz, Hartmut Löwen: Inertial active matter with Coulomb friction (04/2024) published

Umberto Marini Bettolo Marconi, Hartmut Löwen, Lorenzo Caprini: Active fluctuations in the harmonic chain: phonons, entropons and velocity correlations (01/2024) published

Duyu Chen, Michael A. Klatt, Glenn H. Fredrickson: Emergence of Disordered Hyperuniformity in Melts of Linear Diblock Copolymers (12/2023)

René Wittmann: Layer topology of smectic grain boundaries (12/2023) published

Anouar El Moumane, Michael te Vrugt, Hartmut Löwen, René Wittmann: Biaxial nematic order in fundamental measure theory (11/2023) published

Lorenzo Caprini, Anton Ldov, Rahul Kumar Gupta, Hendrik Ellenberg, René Wittmann, Hartmut Löwen, Christian Scholz: Emergent memory from tapping collisions in active granular matter (10/2023) published

Yuanjian Zheng, Michael A. Klatt, Hartmut Löwen: Universal Hyperuniformity in Active Field Theories (10/2023)

Jens U. Neurohr, Friederike Nolle, Thomas Faidt, Samuel Grandthyll, Anton Wittig, Michael A. Klatt, Karin Jacobs, Frank Müller: Impact of geometry on chemical analysis exemplified for photoelectron spectroscopy of black silicon (08/2023)

Lorenzo Caprini, Umberto Marini Bettolo Marconi, Hartmut Löwen: Entropy production and collective excitations of crystals out of equilibrium: the concept of entropons (07/2023) published

Lorenzo Caprini, Hartmut Löwen, Umberto Marini Bettolo Marconi: Inhomogenous entropy production in active crystals with point imperfections (06/2023) published

Christian Spengler, Erik Maikranz, Bernhard Glatz, Michael Andreas Klatt, Hannah Heintz, Markus Bischoff, Ludger Santen, Andreas Fery, Karin Jacobs: The adhesion capability of Staphylococcus aureus cells is heterogeneously distributed over the cell envelope (05/2023) published

Michael Andreas Klatt, Christian Bair, Hartmut Löwen, René Wittmann: Foundation of classical dynamical density functional theory: uniqueness of time-dependent density-potential mappings (04/2023)

Lorenzo Caprini, Hartmut Löwen: Flocking without alignment interactions in attractive active Brownian particles (03/2023) published

René Wittmann, Paul A. Monderkamp, Hartmut Löwen: Statistics of carrier-cargo complexes (03/2023) published

René Wittmann, Paul A. Monderkamp, Jingmin Xia, Louis B. G. Cortes, Iago Grobas, Patrick E. Farrell, Dirk G. A. L. Aarts, Hartmut Löwen: Smectic structures in button-like confinements: experiment and theory (03/2023) published

Lorenzo Caprini, Hartmut Löwen, R. Matthias Geilhufe: Ultrafast entropy production in pump-probe experiments (02/2023) published

Lorenzo Caprini, Hartmut Löwen, Umberto Marini Bettolo Marconi: Chiral active matter in external potentials (01/2023) published

Alexander R. Sprenger, Lorenzo Caprini, Hartmut Löwen, René Wittmann: Dynamics of active particles with translational and rotational inertia (01/2023) published

Iman Abdoli, Hartmut Löwen, Jens-Uwe Sommer, Abhinav Sharma: Tailoring the escape rate of a Brownian particle by combining a vortex flow with a magnetic field (01/2023) published

René Wittmann, G. H. Philipp Nguyen, Hartmut Löwen, Anupam Sengupta, Fabian J. Schwarzendahl: Collective mechano-response dynamically tunes cell-size distributions in growing bacterial colonies (12/2022) published

Michael A. Klatt, Paul J. Steinhardt, Salvatore Torquato: Wave propagation and band tails of two-dimensional disordered systems in the thermodynamic limit (12/2022) published

Michael A. Klatt, Günter Last, Norbert Henze: A genuine test for hyperuniformity (10/2022)

René Wittmann, Sabine Jansen, Hartmut Löwen: Generalized geometric criteria for the absence of effective many-body interactions in the Asakura–Oosawa model (09/2022) published

Paul A. Monderkamp, Fabian Jan Schwarzendahl, Michael A. Klatt, Hartmut Löwen: Active particles using reinforcement learning to navigate in complex motility landscapes (09/2022) published

Segun Goh, Hartmut Löwen, Andreas M. Menzel: Group theoretical approach to elasticity under constraints and predeformations (08/2022) published

Lorenzo Caprini, Rahul Kumar Gupta, Hartmut Löwen: Role of rotational inertia for collective phenomena in active matter (06/2022) published

Erik Kalz, Hidde Derk Vuijk, Iman Abdoli, Jens-Uwe Sommer, Hartmut Löwen, Abhinav Sharma: Collisions Enhance Self-Diffusion in Odd-Diffusive Systems (06/2022) published

Lorenzo Caprini, Umberto Marini Bettolo Marconi, Andrea Puglisi, Hartmut Löwen: Entropons as collective excitations in active solids (06/2022) published

Davide Breoni, Fabian Jan Schwarzendahl, Ralf Blossey, Hartmut Löwen: A one-dimensional three-state run-and-tumble model with a ‘cell cycle’ (06/2022) published

Michael A. Klatt, Max Hörmann, Klaus Mecke: Characterization of anisotropic Gaussian random fields by Minkowski tensors (04/2022) published

Michael A. Klatt, Steffen Winter: Almost sure convergence and second moments of geometric functionals of fractal percolation (04/2022) published

Lorenzo Caprini, Umberto Marini Bettolo Marconi, René Wittmann, Hartmut Löwen: Active particles driven by competing spatially dependent self-propulsion and external force (03/2022) published

Lorenzo Caprini, Alexander R. Sprenger, Hartmut Löwen, René Wittmann: The parental active model: A unifying stochastic description of self-propulsion (01/2022) published

A. R. Sprenger, C. Bair, H. Löwen: Active Brownian motion with memory delay induced by a viscoelastic medium (01/2022) published

Lorenzo Caprini, Umberto Marini Bettolo Marconi, René Wittmann, Hartmut Löwen: Dynamics of active particles with space-dependent swim velocity (11/2021) published

Davide Breoni, Ralf Blossey, Hartmut Löwen: Brownian particles driven by spatially periodic noise (11/2021) published

Fabian Jan Schwarzendahl, Hartmut Löwen: Anomalous Cooling and Overcooling of Active Colloids (11/2021) published

Iman Abdoli, René Wittmann, Joseph Michael Brader, Jens-Uwe Sommer, Hartmut Löwen, Abhinav Sharma: Brownian magneto-gyrator as a tunable microengine (10/2021) published

G. H. Philipp Nguyen, René Wittmann, Hartmut Löwen: Active Ornstein-Uhlenbeck model for self-propelled particles with inertia (08/2021) published

M. te Vrugt, H. Löwen, R. Wittkowski: Classical dynamical density functional theory: from fundamentals to applications (12/2020) published

Michael A. Klatt, Günter Last: On strongly rigid hyperfluctuating random measures (08/2020) published

René Wittmann, Hartmut Löwen, Joseph M. Brader: Order-preserving dynamics in one dimension – single-file diffusion and caging from the perspective of dynamical density functional theory (08/2020) published

Members

  • member's portrait

    Prof. Dr. Hartmut Löwen

    Heinrich-Heine-Universität Düsseldorf
    Principal Investigator
  • member's portrait

    Dr. René Wittmann

    Max Rubner-Institut
    Principal Investigator
  • member's portrait

    Dr. Iman Abdoli

    Heinrich-Heine-Universität Düsseldorf
    Associated Scientist
  • member's portrait

    B. Sc. Christian Bair

    Heinrich-Heine-Universität Düsseldorf
    Associated Scientist
  • member's portrait

    Prof. Dr. Lorenzo Caprini

    University of Rome "La Sapienza"
    Associated Scientist
  • member's portrait

    B. Sc. Anouar El Moumane

    Heinrich-Heine-Universität Düsseldorf
    Associated Scientist
  • member's portrait

    Dr. Michael Andreas Klatt

    German Aerospace Center (DLR)
    Associated Scientist
  • member's portrait

    M. Sc. Philipp Nguyen

    Heinrich-Heine-Universität Düsseldorf
    Associated Scientist

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